diff --git a/book/0_overview/schedule.md b/book/0_overview/schedule.md index 7847f84..510e24e 100644 --- a/book/0_overview/schedule.md +++ b/book/0_overview/schedule.md @@ -4,57 +4,3 @@ Each unit runs over one week and consists of three lessons, numbered after the u [My Timetable](https://mytimetable.tudelft.nl/) carries the authoritative schedule, with rooms and any changes. -## Quarter 1 - -| Unit | Topic | Lesson | Date | Time | -| :--- | :--- | :--- | :--- | :--- | -| 1.1 | Gradient and divergence | 1.1.1 | Tue 1 Sep 2026 | 08:45-10:30 | -| | | 1.1.2 | Thu 3 Sep 2026 | 15:45-17:30 | -| | | 1.1.3 | Fri 4 Sep 2026 | 13:45-15:30 | -| 1.2 | Curl | 1.2.1 | Mon 7 Sep 2026 | 10:45-12:30 | -| | | 1.2.2 | Tue 8 Sep 2026 | 13:45-15:30 | -| | | 1.2.3 | Fri 11 Sep 2026 | 13:45-16:30 | -| 1.3 | Potential fields: history and experiments | 1.3.1 | Mon 14 Sep 2026 | 10:45-12:30 | -| | | 1.3.2 | Tue 15 Sep 2026 | 13:45-15:30 | -| | | 1.3.3 | Fri 18 Sep 2026 | 13:45-15:30 | -| 1.4 | Potential fields: gravity, magnetic field of the Earth | 1.4.1 | Mon 21 Sep 2026 | 10:45-12:30 | -| | | 1.4.2 | Tue 22 Sep 2026 | 13:45-15:30 | -| | | 1.4.3 | Fri 25 Sep 2026 | 13:45-16:30 | -| 1.5 | Electric field. Diffusion fields: hot wire | 1.5.1 | Mon 28 Sep 2026 | 10:45-12:30 | -| | | 1.5.2 | Tue 29 Sep 2026 | 13:45-15:30 | -| | | 1.5.3 | Fri 2 Oct 2026 | 13:45-15:30 | -| 1.6 | Diffusion fields: boundary conditions, heat in 2D and 3D | 1.6.1 | Mon 5 Oct 2026 | 10:45-12:30 | -| | | 1.6.2 | Tue 6 Oct 2026 | 15:45-17:30 | -| | | 1.6.3 | Fri 9 Oct 2026 | 13:45-16:30 | -| 1.7 | Mechanical waves: strings, acoustic waves, 2D and 3D | 1.7.1 | Mon 12 Oct 2026 | 10:45-12:30 | -| | | 1.7.2 | Tue 13 Oct 2026 | 13:45-15:30 | -| | | 1.7.3 | Fri 16 Oct 2026 | 13:45-15:30 | -| 1.8 | Mechanical waves: power flux | 1.8.1 | Mon 19 Oct 2026 | 10:45-12:30 | -| | | 1.8.2 | Tue 20 Oct 2026 | 13:45-15:30 | -| | | 1.8.3 | Fri 23 Oct 2026 | 13:45-16:30 | -| 1.9 | Unsupervised study | | | | -| 1.10 | Midterm week: exam and discussion of solutions | 1.10.1 | | | -| | | 1.10.2 | | | -| | | 1.10.3 | Fri 6 Nov 2026 | 13:30-16:30 | - -## Quarter 2 - -| Unit | Topic | Lesson | Date | Time | -| :--- | :--- | :--- | :--- | :--- | -| 2.1 | Electromagnetism: Maxwell's equations, plane waves, telegraph equation | 2.1.1 | Mon 9 Nov 2026 | 13:45-15:30 | -| | | 2.1.2 | Tue 10 Nov 2026 | 13:45-15:30 | -| | | 2.1.3 | Fri 13 Nov 2026 | 10:45-12:30 | -| 2.2 | 3D waves, Poynting vector, polarisation, lossy and lossless media | 2.2.1 | Mon 16 Nov 2026 | 13:45-15:30 | -| | | 2.2.2 | Tue 17 Nov 2026 | 13:45-15:30 | -| | | 2.2.3 | Fri 20 Nov 2026 | 09:45-12:30 | -| 2.3 | Reflection, transmission, refraction. Multi-layered media | 2.3.1 | Mon 23 Nov 2026 | 13:45-15:30 | -| | | 2.3.2 | Tue 24 Nov 2026 | 13:45-15:30 | -| | | 2.3.3 | Fri 27 Nov 2026 | 10:45-12:30 | -| 2.4 | Trapped and surface waves. Phase and group velocity | 2.4.1 | Mon 30 Nov 2026 | 13:45-15:30 | -| | | 2.4.2 | Tue 1 Dec 2026 | 08:45-10:30 | -| | | 2.4.3 | Fri 4 Dec 2026 | 09:45-12:30 | -| 2.8 | Unsupervised study | | | | -| 2.9 | Unsupervised study | | | | -| 2.10 | Resit exam | | Wed 27 Jan 2027 | 13:30-16:30 | - -The exam is on Wed 16 Dec 2026, 13:30-16:30. The longer practicals, running to 16:30, end with a formative assessment. diff --git a/book/3_diffusion_fields/em_diffusion.md b/book/3_diffusion_fields/em_diffusion.md index 73133a8..fc49b08 100644 --- a/book/3_diffusion_fields/em_diffusion.md +++ b/book/3_diffusion_fields/em_diffusion.md @@ -10,19 +10,24 @@ $$ \nabla\times\boldsymbol E + \mu\,\partial_t\boldsymbol H = \boldsymbol 0, $$ (eq:erel) -where the right-hand sides contain the sources that generate the electromagnetic field. In {eq}`eq:hrel`, the source is the electric current dipole $\boldsymbol J^e$ (A/m$^2$), and in {eq}`eq:erel` the source is zero, because no magnetic monopoles have been found. Some electric sources can be approximated by a magnetic current dipole source. A closed current-carrying loop antenna is such an electric current source that is often described as a magnetic dipole, which can be understood from integrating both sides of {eq}`eq:erel` over the area of the loop, performing a scalar product of each term with the unit normal to the loop, and using Stokes' theorem on the curl of the electric field. In the left-hand sides, $\boldsymbol E$ and $\boldsymbol H$ are the electric and magnetic field strengths in V/m and A/m, respectively, and the medium parameters $\varepsilon,\sigma,\mu$ are the electric permittivity and conductivity, and the magnetic permeability, with units of s/($\Omega$m), 1/($\Omega$m), and s$\Omega$/m, respectively. These equations describe in principle the electric, magnetic and electromagnetic fields for all possible time variations, including the static fields discussed before. Here we are interested in the diffusive field regime. We see that {eq}`eq:hrel`, Ampère's law modified by Maxwell, contains both the electric field and the time derivative of the electric field. If we take the divergence of this equation, we obtain +where the right-hand sides contain the sources that generate the electromagnetic field. In {eq}`eq:hrel`, the source is the electric current dipole $\boldsymbol J^e$ (A/m$^2$), and in {eq}`eq:erel` the source is zero, because no magnetic monopoles have been found. Some electric sources can be approximated by a magnetic current dipole source. A closed current-carrying loop antenna is such an electric current source that is often described as a magnetic dipole, which can be understood from integrating both sides of {eq}`eq:erel` over the area of the loop, performing a scalar product of each term with the unit normal to the loop, and using Stokes' theorem on the curl of the electric field. In the left-hand sides, $\boldsymbol E$ and $\boldsymbol H$ are the electric and magnetic field strengths in V/m and A/m, respectively, and the medium parameters $\varepsilon,\sigma,\mu$ are the electric permittivity and conductivity, and the magnetic permeability, with units of s/($\Omega$m), 1/($\Omega$m), and s$\Omega$/m, respectively. These equations describe in principle the electric, magnetic and electromagnetic fields for all possible time variations, including the static fields discussed before. Here we are interested in the diffusive field regime. + +:::{admonition} A new bridge to wave fields to distinguish them from diffusive fields +:class: important + +We see that {eq}`eq:hrel` is Ampère's law modified by Maxwell contains both the electric field and the time derivative of the electric field. We take the divergence of this equation and obtain $$ (\sigma + \varepsilon\,\partial_t)\,\nabla\cdot\boldsymbol E = -\nabla\cdot\boldsymbol J^e . $$ -We use the fact that $\boldsymbol D=\varepsilon\boldsymbol E$ and that $\nabla\cdot\boldsymbol D=\rho_f$, with $\rho_f$ being the volume density of free charge (C m$^{-3}$). Substituting this in the equation gives +If we use the fact the $\boldsymbol D=\varepsilon\boldsymbol E$ and that $\nabla\cdot\boldsymbol D=\rho_f$ with $\rho_f$ being the volume density of free charge (C m$^{-3}$). Substituting this in the equation gives $$ -\partial_t\rho_f = -\nabla\cdot(\sigma\boldsymbol E + \boldsymbol J^e), +\partial_t\rho_f = -\nabla\cdot(\sigma\boldsymbol E + \boldsymbol J^e). $$ -which is known as the equation of continuity of electric current, or the charge conservation law. We replace the source by assuming there is an accumulation of free charge $\rho_0$ at $t=0$, which removes the source and replaces it with the initial condition $\rho_f(0)=\rho_0$. If we write the electric field as $\boldsymbol E=\boldsymbol D/\varepsilon$, the equation becomes +You have seen this equation without the external source term in the section on 'Steady Currents, Charge Conservation, and a Bridge to Waves'. This is known as the equation of continuity of electric current, or the charge conservation law. Here we continue the bridge to waves and by assuming there is an accumulation of free charge of $\rho_0$ at $t=0$. We remove the source and replace it with the initial condition that $\rho_f(0)=\rho_0$. We write the electric field as $\boldsymbol E=\boldsymbol D/\varepsilon$, and find $$ \partial_t\rho_f = -\frac{\sigma}{\varepsilon}\rho_f, @@ -34,20 +39,36 @@ $$ \rho_f = \rho_0\exp\left(-\frac{\sigma}{\varepsilon}t\right) = \rho_0\exp(-t/\tau_r), $$ -where $\tau_r=\varepsilon/\sigma$ is known as the charge relaxation time. It is a measure of the time it takes the medium to return to its equilibrium state after it has been disturbed by an electromagnetic wave. First, we learn from this equation the following theorem: *within a region of non-vanishing conductivity there can be no permanent distribution of free charge*. This is the significant equation, because when alternating source currents have oscillation periods much larger than the relaxation time, the effect of $\varepsilon$ can be neglected and we can effectively assume $\varepsilon=0$ in the first Maxwell equation. For most Earth materials, $\varepsilon\lesssim 10^{-9}$ F/m and $\sigma\gtrsim 10^{-4}$ S/m, so for most earth materials under investigation the relaxation time is less than 10 $\mu$s. Hence, for source time function oscillations with periods longer than 0.3 ms, we can safely take $\varepsilon=0$. For now, we assume this is the case. Electromagnetic waves are treated later in the course. We write Maxwell's equations in the diffusive approximation out in components as +with $\tau_r=\varepsilon/\sigma$ is known as the charge relaxation time and it is a measure of the time it takes the medium to return to its equilibrium state after it has been disturbed by an electromagnetic wave. First, we learn from this equation the following theorem: + +**_Within a region of non-vanishing conductivity there can be no permanent distribution of free charge_**. + +This is the significant equation, because when alternating source currents have oscillation periods much larger than the relaxation time, the effect of $\varepsilon$ can be neglected and we can effectively assume $\varepsilon=0$ in the first Maxwell equation. For all Earth materials, $\varepsilon < 10^{-9}$ F/m and $\sigma\gtrsim 10^{-4}$ S/m for most earth materials under investigation the relaxation time is less than 10 $\mu$s. Hence, for source time function oscillations with periods less than 0.3 $\mu$s, the electromagnetic field will be a wave field, whereas for source time function oscillations with periods longer than 0.3 ms, we can safely take $\varepsilon=0$. Electromagnetic waves are treated later in the course. +::: + +We write Maxwell's equations in the diffusive approximation out in components as, $$ \begin{aligned} -\partial_y H_z + \partial_z H_y + \sigma E_x &= -J_x^e, \\ -\partial_z H_x + \partial_x H_z + \sigma E_y &= -J_y^e, \\ --\partial_x H_y + \partial_y H_x + \sigma E_z &= -J_z^e, \\ +-\partial_x H_y + \partial_y H_x + \sigma E_z &= -J_z^e, +\end{aligned} +$$ + +$$ +\begin{aligned} \partial_y E_z - \partial_z E_y + \mu\,\partial_t H_x &= 0, \\ \partial_z E_x - \partial_x E_z + \mu\,\partial_t H_y &= 0, \\ \partial_x E_y - \partial_y E_x + \mu\,\partial_t H_z &= 0 . \end{aligned} $$ -First we link these equations to the 1D heat equation by assuming a one-dimensional problem here as well. If the source is an infinite current sheet with a constant current in the $(x,y)$-plane, it means that all fields are independent of the $x$- and $y$-coordinates and all derivatives $\partial_x$ and $\partial_y$ become zero. Then, when the current has an $x$-component only, we have $J_y^e=J_z^e=0$. We substitute these findings in the equations and find +First we link these equations to the 1D heat equation by assuming a one-dimensional problem here as well. + +## Maxwell's equations in one dimension with a source + +If the source is an infinite current sheet with a constant current in the $(x,y)$-plane and the medium parameters $\sigma$ and $\mu$ are constant, it means that all fields are independent of the $x$- and $y$-coordinates and the derivatives $\partial_x$ and $\partial_y$ become zero. Then, when the current has an $x$-component only, we have $J_y^e=J_z^e=0$. We substitute these findings in the equations and find $$ \begin{aligned} @@ -58,140 +79,220 @@ $$ \partial_z E_x + \mu\,\partial_t H_y &= 0, \\ \mu\,\partial_t H_z &= 0, \end{aligned} -$$ (eq:plzHy) +$$ -which eliminates the vertical electric and magnetic field components and results in two equal but independent sets of two coupled differential equations, one for $E_x,H_y$ and one for $E_y,H_x$, but the latter has no source to generate the fields. The source is taken as $J_x^e(z,t)=\delta(z)W(t)$, which represents the plane electric current sheet in the $(x,y)$-plane at $z=0$ with a current directed along the $x$-axis and with a time function $W(t)$. This means it is a point source in the $z$-direction and of infinite extent in the $x$- and $y$-directions. In the course Signals and Time Series, the delta-function was introduced in time, but we can use it in all coordinates. Here, this leads to equations for $E_x$ and $H_y$, as +which eliminates the vertical electric and magnetic field components and results in two equal but independent sets of two coupled differential equations, one for $E_x,H_y$ and one for $E_y,H_x$, but the latter has no source to generate the fields and without a generating source, they will be zero. This leads to equations for $E_x$ and $H_y$, as $$ -\begin{aligned} -\partial_z H_y(z,t) + \sigma E_x(z,t) &= -J_x^e(z,t), \\ -\partial_z E_x(z,t) + \mu\,\partial_t H_y(z,t) &= 0 . -\end{aligned} +\partial_z H_y(z,t) + \sigma E_x(z,t) = -J_x^e(z,t), +$$ (eq:plzHy) + +$$ +\partial_z E_x(z,t) + \mu\,\partial_t H_y(z,t) = 0 . $$ -The diffusive field that satisfies these equations is called the Transverse ElectroMagnetic (TEM) diffusive field. The diffusion equation for the electric field is obtained by eliminating the magnetic field from the two equations above. We do that as follows, +The diffusive field that satisfies this equation is called the Transverse ElectroMagnetic (TEM) diffusive field, because both the electric and magnetic fields are transverse (perpendicular) to the propagation direction. The diffusion equation for the electric field is obtained by eliminating the magnetic field from the two equations above. We do that as follows, we multiply both sides of {eq}`eq:plzHy` with $\mu$ and apply a time derivative, which leads to, $$ -\begin{aligned} -\partial_z\left(\mu\,\partial_t H_y(z,t)\right) + \sigma\mu\,\partial_t E_x(z,t) &= -\mu\,\partial_t J_x^e(z,t), \\ -\mu\,\partial_t H_y(z,t) &= -\partial_z E_x(z,t), -\end{aligned} +\partial_z\left(\mu\,\partial_t H_y(z,t)\right) + \sigma\mu\,\partial_t E_x(z,t) = -\mu\,\partial_t J_x^e(z,t), +$$ (eq:plzpltHy) + $$ +\mu\,\partial_t H_y(z,t) = -\partial_z E_x(z,t). +$$ (eq:pltHy) -and substituting the right-hand side of the second equation to replace the term with the magnetic field in the first equation. The electric field is found to satisfy the following diffusion equation: +We then substitute the right-hand side of the {eq}`eq:pltHy` to replace the term with the magnetic field in {eq}`eq:plzpltHy`, we find that the electric field satisfies the following diffusion equation: $$ \partial_z\partial_z E_x - \mu\sigma\,\partial_t E_x = \mu\,\partial_t J_x^e(z,t). $$ (eq:dee) -For $J_x^e(z,t)=\delta(z)W(t)$, the solution is known and given by +{eq}`eq:dee` has the same structure as the heat equation. We now have the electric field instead of temperature and the heat conductivity $\kappa$ has been replaced by $\sigma\mu$, which is the electromagnetic diffusion coefficient and it has units of s/m$^2$, and there is a source term in the right-hand side. + +To construct the solution for this equation, we assume the source is active in an arbitrary space-time interval, given by $J_x^e(z,t)\ne0$ for $z_b0$, the integral can be evaluated explicitly and the solution is +In the course Signals and Time Series, the delta-function was introduced in time, but we can use it in all coordinates. Because {eq}`eq:dee` relates the electric field linearly to the source, we use the principle of superposition to express the electric field as $$ -E_x(z,t) = -I\sqrt{\frac{\mu}{4\pi\sigma t}}\exp\left(-\frac{\sigma\mu z^2}{4t}\right), \quad\text{for } t>0 . -$$ (eq:ExtH) +E_x(z,t) = -\mu\int_{z'=z_b}^{z_e}\int_{t'=t_b}^{t_e} G(z-z',t-t')\,\partial_{t'}J_x^e(z',t')\,\mathrm{d}z'\,\mathrm{d}t', +$$ (eq:EGJ) -You recognise the similarity with the solution to the heat equation with a Gaussian distribution for the initial temperature on an infinitely long wire. Note that here we make explicit that the expression for the electric field exists only for positive times. This is because $t=0$ is the moment the source is switched on, and physics tells us that the Earth will respond to a source action only at or after the moment the source becomes active, and not before. This is the notion of causality. +where $G$ is known as the Green's function and the minus-sign was chosen for convenience. Substituting these two expressions in {eq}`eq:dee`, removing the integrals over $z'$ and $t'$ and removing the same terms in the left-hand and right-hand sides, results in -## Solution using Fourier transformation +$$ +(\partial_z\partial_z - \mu\sigma\,\partial_t)\,G = -\delta(z-z')\,\delta(t-t'). +$$ (eq:deeg) -In the course on Signals and Time Series, you have learned Fourier series and the Fourier transformation. In the heat equation for a finite-length wire, we use Fourier series to find solutions. We then showed that for an infinitely long wire that has a Gaussian temperature distribution at zero time, a solution in terms of Fourier series is possible as well. We also showed that if we let the distance between two frequency values go to zero, the Fourier transformation is found. But we can solve partial differential equations with a non-zero source term more formally with the aid of Fourier transformations. We do this by transforming {eq}`eq:dee` to the frequency domain using +We now recognise $G$ as the impulse response. Hence, the Green's function is the impulse response of the Earth if we would generate signals with an impulse in space and time. Once $G$ has been found, the electric field can be obtained from the space-time convolution of the Green's function and the time derivative of the source as expressed in {eq}`eq:EGJ`. We find the solution for the Green's function by transforming {eq}`eq:deeg` to the frequency domain using $$ -\hat{\boldsymbol E}(z,\omega) = \int_{t=0}^{\infty}\exp(-\mathrm{i}\omega t)\,\boldsymbol E(z,t)\,\mathrm{d}t, +\hat G(z-z',\omega) = \int_{t=0}^{\infty}\exp(-\mathrm{i}\omega t)\,G(z-z',t)\,\mathrm{d}t, $$ -to obtain +The diacritical hat on the quantities denote the Fourier transform of the quantity. We find, $$ -\partial_z\partial_z\hat E_x - \mathrm{i}\omega\sigma\mu\,\hat E_x = \mathrm{i}\omega\mu\,\hat W(\omega)\,\delta(z). -$$ (eq:dees) +(\partial_z\partial_z - \mathrm{i}\omega\sigma\mu)\,\hat G = -\delta(z-z')\exp(-\mathrm{i}\omega t'). +$$ (eq:deegs) -The diacritical hat on the quantities denotes the Fourier transform of the quantity. The time derivative is gone and replaced by an algebraic multiplication with $\mathrm{i}\omega$, and the equation has been reduced to an ordinary second-order differential equation. As you have learned in the course Signals and Time Series, the exponential function is an eigenfunction, so we can propose a solution in the form +The time-derivative transforms to an algebraic multiplication with $\mathrm{i}\omega$ and the delta-function in time transforms to a multiplication with $\exp(-\mathrm{i}\omega t')$ that keeps the information on the time-shift $t'$. {eq}`eq:deegs` is an ordinary second order differential equation. You have learned in the course Signals and Time Series, and seen in the part on potential fields, that the exponential function is an eigenfunction. If there is a source at $z=z'$ and the medium parameters are constants, the field will propagate in the direction where it started from the source. This means that the field propagates in the negative $z$-direction when $z-z'<0$ and propagates in the positive $z$-direction when $z>z'$. We can assume the strength of these two fields are equal and we can propose a solution of the form, $$ -\hat E_x = \hat A\exp\left(-\sqrt{\mathrm{i}\omega\mu\sigma}\,|z|\right), +\hat G = \hat A\left\{\exp\left[-\sqrt{\mathrm{i}\omega\mu\sigma}\,(z-z')\right]u(z-z') + \exp\left[\sqrt{\mathrm{i}\omega\mu\sigma}\,(z-z')\right]u(z'-z)\right\}. $$ -where $\hat A$ is the constant that needs to be determined. The reason why $|z|$ is in the argument of the exponential function is that the field must diffuse away from the source and decrease in amplitude with increasing distance in both the positive and negative $z$-directions. Evaluating the first and second derivatives with respect to $z$ gives +where $\hat A$ is the constant that needs to be determined. The field must diffuse away from the source and decrease in amplitude with increasing distances in both the positive and negative $z$-directions, as depicted in {numref}`fig-greens1d`. This is in contrast with finite domains where we have to impose end point conditions and signals travelling in both directions can occur everywhere on the line segment. Here the fields travel only in one direction and we use physics to tell us that the field should go to zero when we move infinitely far away from the source. Evaluating the first and second derivatives with respect to $z$ gives + +```{figure} figures/Greens1D.png +:name: fig-greens1d +:width: 70% + +The generated of signals by a point source on the line at $z=z'$ with a signal that propagates in the positive and in the negative $z$-directions away from the location $z=z'$. +``` $$ \begin{aligned} -\partial_z\hat E_x &= -\sqrt{\mathrm{i}\omega\sigma\mu}\,\hat A\exp\left(-\sqrt{\mathrm{i}\omega\sigma\mu}\,|z|\right)\frac{z}{|z|}, \\ -\partial_z\partial_z\hat E_x &= \mathrm{i}\omega\mu\sigma\,\hat A\exp\left(-\sqrt{\mathrm{i}\omega\sigma\mu}\,|z|\right) - 2\sqrt{\mathrm{i}\omega\sigma\mu}\,\hat A\,\delta(z)\exp\left(-\sqrt{\mathrm{i}\omega\sigma\mu}\,|z|\right), \\ -&= \mathrm{i}\omega\sigma\mu\,\hat E_x - 2\sqrt{\mathrm{i}\omega\sigma\mu}\,\hat A\,\delta(z), +\partial_z\hat G &= -\sqrt{\mathrm{i}\omega\sigma\mu}\,\hat A\left\{\exp\left[-\sqrt{\mathrm{i}\omega\mu\sigma}\,(z-z')\right]u(z-z') - \exp\left[\sqrt{\mathrm{i}\omega\mu\sigma}\,(z-z')\right]u(z'-z)\right\}, \\ +\partial_z\partial_z\hat G &= \mathrm{i}\omega\mu\sigma\hat A\exp\left(-\sqrt{\mathrm{i}\omega\sigma\mu}\,|z-z'|\right) - 2\sqrt{\mathrm{i}\omega\sigma\mu}\,\hat A\,\delta(z-z')\exp\left(-\sqrt{\mathrm{i}\omega\sigma\mu}\,|z-z'|\right) \\ +&= \mathrm{i}\omega\sigma\mu\hat G - 2\sqrt{\mathrm{i}\omega\sigma\mu}\,\hat A\,\delta(z-z'), \end{aligned} $$ -where we have used the fact that +where we used the fact that $$ -\mathrm{sign}(z) = \frac{z}{|z|} = \left\{\begin{array}{rl} -1, & z<0, \\ 1, & z>0, \end{array}\right. +\partial_z u[\pm(z-z')] = \pm\delta(z-z') $$ -is known as the sign-function. It has a sudden step of size 2 at $z=0$, for which reason its derivative is equal to $\partial_z\,\mathrm{sign}(z)=2\delta(z)$. Because the impulse function acts only at $z=0$, we can substitute that already in the exponential function that multiplies the delta-function. If we substitute these results in {eq}`eq:dees` we find that +and the terms with those derivatives cancel in the expression of the first derivative, whereas in the second derivative they add up. We have used the fact that we can combine the two solutions as $$ -\hat A = -\frac{\sqrt{\mathrm{i}\omega\mu}}{2\sqrt{\sigma}}\hat W . +\exp\left(-\sqrt{\mathrm{i}\omega\sigma\mu}\,|z-z'|\right) = \exp\left[-\sqrt{\mathrm{i}\omega\mu\sigma}\,(z-z')\right]u(z-z') + \exp\left[\sqrt{\mathrm{i}\omega\mu\sigma}\,(z-z')\right]u(z'-z). $$ -The solution for the electric field is then found as +If we substitute these results in {eq}`eq:deegs` we find that $$ -\hat E_x(z,\omega) = -\frac{\sqrt{\mu}}{2\sqrt{\sigma}}\hat W\sqrt{\mathrm{i}\omega}\exp\left(-\sqrt{\mathrm{i}\omega\sigma\mu}\,|z|\right). -$$ (eq:eesd) +\hat A = \frac{\exp(-\mathrm{i}\omega t')}{2\sqrt{\mathrm{i}\omega\sigma\mu}}, +$$ -This is the space-frequency domain solution for the one-dimensional electric field in the Earth in response to an electric current sheet that is an impulse at $z=0$ but exists for all $x$- and $y$-values, and has a time function $W(t)$. The transformation back to time results in the expression given in {eq}`eq:extW`, and in {eq}`eq:ExtH` in case the source is a step-function switch-on, in which case its frequency transform is given by $\hat W=I/(\mathrm{i}\omega)$. Now we can explain the convolution in time, because you know from the course Signals and Time Series that a product in the frequency domain leads to a convolution in the time domain. +and the Green's function is then -## The concept of the principle of superposition +$$ +\hat G = \frac{\exp\left(-\sqrt{\mathrm{i}\omega\sigma\mu}\,|z-z'|\right)}{2\sqrt{\mathrm{i}\omega\sigma\mu}}\exp(-\mathrm{i}\omega t'). +$$ -Before we go to the three-dimensional problem, we take a look at the right-hand side of {eq}`eq:dee` for the given source time function that is switched on at $t=0$ and kept constant after that, and note that $\partial_t u(t)=\delta(t)$, where $u(t)$ is the unit-step function and $\delta(t)$ is the delta-function. In that case the equation for the electric field becomes +The space-time Green's function is obtained by the inverse Fourier transformation, which is known but beyond the scope of the course. It is given by $$ -\partial_z\partial_z E_x - \mu\sigma\,\partial_t E_x = \mu I\,\delta(t)\,\delta(z). -$$ (eq:deeg) +G(z-z',t-t') = \sqrt{\frac{1}{4\pi\sigma\mu(t-t')}}\exp\left(-\frac{\sigma\mu(z-z')^2}{4(t-t')}\right), \quad\text{for } t>t'. +$$ + +The source is taken as $J_x^e(z,t)=I\delta(z)u(t)$, which represents the plane electric current sheet in the $(x,y)$-plane at $z=0$ with a current $I$ directed along the $x$-axis and with a step switch-on time function $u(t)$. This means it is an impulse in the $z$-direction and of infinite extent in the $x$- and $y$-directions. The switch-on current source means $I$ is a constant for all times after $t=0$, $u(t)=0$ for $t<0$ and $u(t)=1$ for $t>0$. For this source, the integrals in {eq}`eq:EGJ` can be evaluated explicitly and the electric field is given by + +$$ +E_x(z,t) = -I\sqrt{\frac{\mu}{4\pi\sigma t}}\exp\left(-\frac{\sigma\mu z^2}{4t}\right), \quad\text{for } t>0 . +$$ (eq:ExtH) + +You recognise the similarity with the solution to the heat equation with a Gaussian distribution for the initial temperature on an infinitely long wire. Note that here we make explicit that the expression for the electric field exists only for positive times. This is because $t=0$ is the moment the source is switched on and physics tells us that the Earth will respond to a source action only at or after the moment the source becomes active and not before. This is the notion of causality. + +## Maxwell's equations in two dimensions with a source + +If the source is an infinitely long wire and with a constant current in the $z$-direction, and the medium parameters are constants, all fields are independent of the $z$-coordinate and the derivative $\partial_z$ becomes zero, and $J_x^e=J_y^e=0$. With these choices, Maxwell's equations become + +$$ +\begin{aligned} +-\partial_y H_z + \sigma E_x &= 0, \\ +\partial_x H_z + \sigma E_y &= 0, \\ +\partial_x H_y - \partial_y H_x + \sigma E_z &= -J_z^e, \\ +\partial_y E_z + \mu\,\partial_t H_x &= 0, \\ +-\partial_x E_z + \mu\,\partial_t H_y &= 0, \\ +\partial_x E_y - \partial_y E_x + \mu\,\partial_t H_z &= 0 . +\end{aligned} +$$ + +The fields that connect to the source are grouped as + +$$ +\partial_x H_y - \partial_y H_x + \sigma E_z = -J_z^e, +$$ (eq:TEEy) + +$$ +\mu\,\partial_t H_x = -\partial_y E_z, +$$ (eq:TEHx) + +$$ +\mu\,\partial_t H_y = \partial_x E_z. +$$ (eq:TEHz) + +Because the fields do not depend on the $z$-direction, the plane of interest is the $(x,y)$-plane. We see that the electric field is perpendicular to the plane where the fields will change while the magnetic field has a direction in that plane. For this reason this is called the Transverse Electric or TE mode. We use {eq}`eq:TEHx` and {eq}`eq:TEHz` to substitute the magnetic fields in {eq}`eq:TEEy` to find the differential equation that $E_z$ satisfies, + +$$ +(\partial_x^2 + \partial_y^2 - \sigma\mu\,\partial_t)E_z = \mu\,\partial_t J_z^e. +$$ -The function that collapses a differential equation to an impulse in space and time is called the Green's function. Physically, the Green's function describes the (scalar) response to an impulse in space and time. This is a general principle for linear partial differential equations and you have discussed this in the course Signals and Time Series. The equation for the space-time impulse response is then +We recognise that the equation is very similar to the equation in one dimension. We can also see that the two magnetic field components satisfy $$ -(\partial_z\partial_z - \mu\sigma\,\partial_t)\,G(z-z',t-t') = -\delta(t-t')\,\delta(z-z'), -$$ (eq:deg) +\begin{aligned} +(\partial_x^2 + \partial_y^2 - \sigma\mu\,\partial_t)H_x &= -\partial_y J_z^e, \\ +(\partial_x^2 + \partial_y^2 - \sigma\mu\,\partial_t)H_y &= \partial_x J_z^e. +\end{aligned} +$$ -where the minus sign in the right-hand side is a historic choice. The constant displacements $t'$ and $z'$ are chosen because the real source may have a finite length in the $z$-direction and have a certain time duration. If we now multiply both sides of this equation with a function $-\mu\,\partial_{t'}J_x^e(z',t')$, assume that $J_x^e(z',t')\ne 0$ only for $z_bt', +G(\boldsymbol r-\boldsymbol r',t-t') = \frac{(\sigma\mu)^{1/2}}{[4\pi(t-t')]^{3/2}}\exp\left(-\frac{\sigma\mu|\boldsymbol r-\boldsymbol r'|^2}{4(t-t')}\right), \quad\text{for } t>t', $$ (eq:G3D) and it satisfies @@ -209,7 +310,7 @@ $$ (\nabla^2 - \sigma\mu\,\partial_t)\,G = -\delta(\boldsymbol r-\boldsymbol r')\,\delta(t-t') . $$ -Again we recognise the similarity with the solution for the heat equation for an infinitely long wire with an initial temperature that was a Gaussian distribution. It also resembles the 1D diffusive field solution to Maxwell's equations. The major difference is that here the late-time behaviour is proportional to $t^{-3/2}$. The electric and magnetic fields have much more complicated behaviour, but that is far beyond the scope of the present course. +Again we recognise the similarity with solution for the heat equation for an infinitely long wire with an initial temperature that was a Gaussian distribution. It also resembles the 1D and 2D diffusive field solution to Maxwell's equations. The major difference is that here the late-time behaviour is proportional to $t^{-3/2}$. The electric and magnetic field have much more complicated behaviour but that is beyond the scope of the present course. In case of an electric current loop source the electric field can be written in the same way as for the infinitely long wire, but now we must integrate along the wire segments of the loop. The reason is that the electric field generated by a loop source is divergence-free. ### Brief analysis of the outcome @@ -226,3 +327,89 @@ G(\boldsymbol r-\boldsymbol r',t-t'=t_p) = \frac{3\sqrt{6}\exp(-3/2)}{4\pi^{3/2} $$ which shows that if you double the distance to the source, the peak amplitude is 8 times smaller. + +## The diffusive electric field generated by an electric loop source + +In the field, a loop is often used to generate a transient electromagnetic field and the Earth response is recorded at the ground surface with an electric field antenna or with a loop receiver. Loops are also carried in the air by a helicopter or a drone, in which case either the same loop, or another smaller loop is used to measure the time derivative of the magnetic induction in the Earth response. Here we look at the electric field that is generated in a conductive medium by a rectangular loop as indicated in {numref}`fig-loop`, with sides $L_x$ and $L_y$ and a unit amplitude step switch-on source time function. The current runs clockwise from the vertices labeled 1 to 4. The coordinates of these vertices are given by + +$$ +\begin{aligned} +\boldsymbol r_1 &= (-L_x/2,-L_y/2,0), & \boldsymbol r_2 &= (L_x/2,-L_y/2,0),\\ +\boldsymbol r_3 &= (L_x/2,L_y/2,0), & \boldsymbol r_4 &= (-L_x/2,L_y/2,0). +\end{aligned} +$$ + +The point of observation is $\boldsymbol r$. For a loop source, the electric field is divergence-free and we can write the electric field as a convolution of the Green's function and the source vector, given by + +$$ +\boldsymbol E(\boldsymbol r,t) = -\mu\oint_{\boldsymbol r'\in L} G(\boldsymbol r-\boldsymbol r',t)\,\boldsymbol\tau(\boldsymbol r')\,\mathrm{d}l, +$$ + +which is a line integral along the four line segments of the loop. The unit vector $\boldsymbol\tau$ is equal to $\hat{\boldsymbol x}$ when we integrate from $\boldsymbol r_1$ to $\boldsymbol r_2$ and it is $-\hat{\boldsymbol x}$ when we integrate from $\boldsymbol r_3$ to $\boldsymbol r_4$. Similarly, the unit vector $\boldsymbol\tau$ is equal to $\hat{\boldsymbol y}$ when we integrate from $\boldsymbol r_2$ to $\boldsymbol r_3$ and it is $-\hat{\boldsymbol y}$ when we integrate from $\boldsymbol r_4$ to $\boldsymbol r_1$. We can therefore write the two horizontal components of the electric field as + +$$ +\begin{aligned} +E_x(\boldsymbol r,t) &= -\frac{\mu}{4\pi}\sqrt{\frac{\sigma\mu}{t^3}}\left\{\exp\left(-\frac{\sigma\mu[(y+L_y/2)^2+z^2]}{4t}\right) - \exp\left(-\frac{\sigma\mu[(y-L_y/2)^2+z^2]}{4t}\right)\right\} \\ +&\quad\times\int_{-L_x/2}^{L_x/2}\exp\left(-\frac{\sigma\mu(x-x')^2}{4t}\right)\mathrm{d}x', +\end{aligned} +$$ + +$$ +\begin{aligned} +E_y(\boldsymbol r,t) &= -\frac{\mu}{4\pi}\sqrt{\frac{\sigma\mu}{t^3}}\left\{\exp\left(-\frac{\sigma\mu[(x-L_x/2)^2+z^2]}{4t}\right) - \exp\left(-\frac{\sigma\mu[(x+L_x/2)^2+z^2]}{4t}\right)\right\} \\ +&\quad\times\int_{-L_y/2}^{L_y/2}\exp\left(-\frac{\sigma\mu(y-y')^2}{4t}\right)\mathrm{d}y', +\end{aligned} +$$ + +```{figure} figures/Loopsrc.png +:name: fig-loop +:width: 80% + +An electric current square loop source with a unit step switch-on current and its vertices 1, 2, 3, and 4. +``` + +where the exponentials with argument $y\pm L_y/2$ represent the integrals from point $\boldsymbol r_2$ to $\boldsymbol r_3$ and from $\boldsymbol r_4$ to $\boldsymbol r_1$, respectively. Because the integrals are similar but just run over a different coordinate axis, we will carry out the integral in the $x$-direction and then we write down the solution for the integral in $y$-direction by substitution. In the integral, we substitute $p=\sqrt{\sigma\mu(x-x')^2/(4t)}$ and integrate over the new variable $p$. The Jacobian of the parameter transformation is $\mathrm{d}p=\sqrt{\sigma\mu/(4t)}\,\mathrm{d}x'$. + +$$ +\int_{-L_x/2}^{L_x/2}\exp\left(-\frac{\sigma\mu(x-x')^2}{4t}\right)\mathrm{d}x' = \sqrt{\frac{4t}{\sigma\mu}}\int_{\sqrt{\frac{\sigma\mu}{4t}}x_m}^{\sqrt{\frac{\sigma\mu}{4t}}x_p}\exp(-p^2)\,\mathrm{d}p, +$$ + +where the new coordinates are introduced as $x_m=x-L_x/2$ and $x_p=x+L_x/2$. This is also a good moment to introduce the diffusion distance as + +$$ +D = \sqrt{\frac{4t}{\sigma\mu}}. +$$ + +Independent from whether $x\pm L_x/2>0$ or $x\pm L_x/2<0$, we can always write the integral as a difference of two integrals that both start at $p=0$, + +$$ +\int_{Dx_m}^{Dx_p}\exp(-p^2)\,\mathrm{d}p = \int_{0}^{Dx_p}\exp(-p^2)\,\mathrm{d}p - \int_{0}^{Dx_m}\exp(-p^2)\,\mathrm{d}p = \frac{\sqrt{\pi}}{2}\left[\mathrm{erf}(Dx_p) - \mathrm{erf}(Dx_m)\right], +$$ + +where $\mathrm{erf}$ denotes the error function and the scale factor is chosen such that $\mathrm{erf}(\infty)=1$. Substituting these results in the expressions for the electric field, we finally obtain + +$$ +E_x(\boldsymbol r,t) = -\frac{\mu}{8\sqrt{\pi}\,t}\left\{\exp\left(-\frac{y_p^2+z^2}{D^2}\right) - \exp\left(-\frac{y_m^2+z^2}{D^2}\right)\right\}\left[\mathrm{erf}\left(\frac{x_p}{D}\right) - \mathrm{erf}\left(\frac{x_m}{D}\right)\right], +$$ (eq:Exloop) + +$$ +E_y(\boldsymbol r,t) = -\frac{\mu}{8\sqrt{\pi}\,t}\left\{\exp\left(-\frac{x_m^2+z^2}{D^2}\right) - \exp\left(-\frac{x_p^2+z^2}{D^2}\right)\right\}\left[\mathrm{erf}\left(\frac{y_p}{D}\right) - \mathrm{erf}\left(\frac{y_m}{D}\right)\right]. +$$ (eq:Eyloop) + +{eq}`eq:Exloop` and {eq}`eq:Eyloop` are the expressions for the electric field anywhere in space and for all times after the source has been switched on. They show that $E_x=0$ in the $(x,z)$-plane at $y=0$ because then $y_p^2=y_m^2$ and $E_y=0$ in the $(y,z)$-plane at $x=0$ because then $x_p^2=x_m^2$. That makes sense because the contributions from the two line-segments in the $x$-direction have equal distance to all the points in the $(x,z)$ plane but have opposite currents. The equations also show that $E_y$ is the strongest in the $(x,z)$-plane at $y=0$ because then $y_p=-y_m$ and the error functions add up, and similarly $E_x$ is strongest in the $(y,z)$-plane at $x=0$ because then $x_p=-x_m$. The electric field is computed for a square loop with 100 m long segments placed in a conductive medium with $\sigma=0.3$ S/m. The result can be seen in {numref}`fig-et12xz` where $E_y$ is shown for two time instances in the $(x,z)$-plane at $y=0$, which is the cross-section right through the middle of the loop, such that the wire segments between points 4 and 1 and points 2 and 3 are perpendicular to the plane of the graph. The figure shows $E_y$ as a function of $x$ and $z$ at $t=0.03$ ms (left plot) and $t=5$ ms (right plot). It is shown only for positive depth values, but the field is symmetric in $z$ and the plot can be mirrored to add the picture of negative $z$. It can be seen that at 0.03 ms, the field is very strong and concentrated around the wires of the loop, whereas at 5 ms, the field is three orders of magnitude weaker and it almost occupies all the plane where the plot is made. Hence, we see that the maximum value of the electric field moves away from the loop wire segments and the width of the Gaussian in space widens with increasing time. + +```{figure} figures/Eyt_xz.png +:name: fig-et12xz +:width: 100% + +The electric field $E_y$ generated by a unit step switch-on current in the $(x,z)$-plane at $y=0$ m, for times $0.03$ ms (left plot) and $5$ ms (right plot). +``` + +At very early times and not too far away from the loop in the $z$-direction, the electric field shows the shape of the loop but the vector points in the opposite direction of the current vector in the loop. This can be seen in {numref}`fig-et12xy` where the magnitude of the electric field vector is shown in colour in the $(x,y)$-plane at $z=10$ m, at $t=0.03$ ms (left plot) and $t=5$ ms (right plot) and the white arrows show the direction of the electric field. At $t=0.03$ ms, the left plot shows that far away from the line segments, the electric field is fully aligned with but points in opposite direction as the current direction in the line segments, but the amplitude is very small. Curvature of the field lines going around the loop can be seen closer to the loop and is again almost fully aligned with the line segments just above them, but also here the field points in the opposite direction as the source current. At the later time of 5 ms, the right plot shows that the shape of the source loop is lost and the electric field is almost circular inside and around the loop. Notice also that the field strength has decreased by six orders of magnitude, which seems to suggest that the decay is faster in the $(x,y)$-plane for fixed depth than in the $(x,z)$-plane for fixed $y$. This will be reduced when larger depth values are chosen, because then the influence of the horizontal distances decreases. + +```{figure} figures/Eyt_xy.png +:name: fig-et12xy +:width: 100% + +The electric field $E_y$ generated by a unit step switch-on current in the $(x,y)$-plane at $z=10$ m, for times $0.03$ ms (left plot) and $5$ ms (right plot). +``` diff --git a/book/3_diffusion_fields/figures/Eyt_xy.png b/book/3_diffusion_fields/figures/Eyt_xy.png new file mode 100644 index 0000000..614a07d Binary files /dev/null and b/book/3_diffusion_fields/figures/Eyt_xy.png differ diff --git a/book/3_diffusion_fields/figures/Eyt_xz.png b/book/3_diffusion_fields/figures/Eyt_xz.png new file mode 100644 index 0000000..99af180 Binary files /dev/null and b/book/3_diffusion_fields/figures/Eyt_xz.png differ diff --git a/book/3_diffusion_fields/figures/Ezt.png b/book/3_diffusion_fields/figures/Ezt.png new file mode 100644 index 0000000..85f21d9 Binary files /dev/null and b/book/3_diffusion_fields/figures/Ezt.png differ diff --git a/book/3_diffusion_fields/figures/Greens1D.png b/book/3_diffusion_fields/figures/Greens1D.png new file mode 100644 index 0000000..e54ab81 Binary files /dev/null and b/book/3_diffusion_fields/figures/Greens1D.png differ diff --git a/book/3_diffusion_fields/figures/Hxt.png b/book/3_diffusion_fields/figures/Hxt.png new file mode 100644 index 0000000..a135420 Binary files /dev/null and b/book/3_diffusion_fields/figures/Hxt.png differ diff --git a/book/3_diffusion_fields/figures/Loopsrc.png b/book/3_diffusion_fields/figures/Loopsrc.png new file mode 100644 index 0000000..4124511 Binary files /dev/null and b/book/3_diffusion_fields/figures/Loopsrc.png differ diff --git a/book/3_diffusion_fields/heat_equation_1d.md b/book/3_diffusion_fields/heat_equation_1d.md index 4f32414..2f81481 100644 --- a/book/3_diffusion_fields/heat_equation_1d.md +++ b/book/3_diffusion_fields/heat_equation_1d.md @@ -501,7 +501,7 @@ $$ (eq:HGCos) The exact equation is easier to analyse in terms of the actual field behaviour, which is not easy to find from the series solution. -As a last topic, we discuss the inhomogeneous electromagnetic diffusive field equations. This means we are going to put a source somewhere in space and solve Maxwell's diffusive field equations in one and three dimensions. +As a last topic, we discuss the inhomogeneous electromagnetic diffusive field equations. This means we are going to put a source somewhere in space and solve Maxwell's diffusive field equations in one, two, and three dimensions. ## Exercises diff --git a/book/4_mechanical_waves/acoustic_waves.md b/book/4_mechanical_waves/acoustic_waves.md new file mode 100644 index 0000000..48a52a0 --- /dev/null +++ b/book/4_mechanical_waves/acoustic_waves.md @@ -0,0 +1,105 @@ +# Wave fields in a fluid: acoustic (P) waves + +With acoustic waves, we assume a fluid in which no shear forces exist, and therefore only P-waves exist. Since waves are physical phenomena, they should have a relation to basic physical laws. The two laws applicable are the conservation of mass and Newton's second law. These two have been used in [the appendix on 1-D acoustic wave motion](./appendix_acoustic_1d.md) to derive the two equations governing the wave motion due to a P-wave. There are some simplifying assumptions in the derivation, one of them being that we consider a 1-dimensional wave. When we denote $p$ as the pressure and $v_x$ as the particle velocity, the conservation of mass leads to: + +$$ +- \frac{1}{K} \frac{\partial p}{\partial t} = \frac{\partial v_x}{\partial x} +$$ (eq:eqdeform) + +in which $K$ is called the bulk modulus. The other relation follows from an application of Newton's law: + +$$ +-\frac{\partial p}{\partial x} = \rho \frac{\partial v_x}{\partial t} +$$ (eq:eqmotion) + +where $\rho$ denotes the mass density. This equation is called the equation of motion. + +We are now going to combine these two equations. Therefore, we let the operator $\partial / \partial x$ work on the equation of motion: + +$$ +- \frac{\partial}{\partial x} \left( \frac{\partial p}{\partial x} \right) + = + \frac{\partial}{\partial x} + \left( \rho \frac{\partial v_x}{\partial t} \right). +$$ (eq:diffeqmotion) + +Now, assuming $\rho$ is constant, it can be taken in front of the $\partial / \partial x$ operator. And differentiations with respect to space $x$ and time $t$ are commutative, so: + +$$ + \frac{\partial}{\partial x} \left( \frac{\partial v_x}{\partial t} \right) + = + \frac{\partial}{\partial t} \left( \frac{\partial v_x}{\partial x} \right). +$$ (eq:commut) + +Now for $\partial v_x / \partial x$, the conservation of mass can be substituted in {eq}`eq:diffeqmotion` to give: + +$$ + -\frac{\partial^2 p}{\partial x^2} = \rho + \frac{\partial}{\partial t} + \left( - \frac{1}{K} \frac{\partial p}{\partial t} \right). +$$ + +Rewriting gives us the 1-Dimensional wave equation: + +$$ +\frac{\partial^2 p}{\partial x^2} + - \frac{1}{c^2} \frac{\partial^2p}{\partial t^2} = 0 +$$ + +in which $c$ can be seen as the velocity of the wave, for which we have: $c = \sqrt{K/ \rho}$. + +This is the basic wave equation for one-dimensional acoustic waves. Equally well, we can derive from the same two equations {eq}`eq:eqdeform` and {eq}`eq:eqmotion` (see also *EXERCISES*) that the particle velocity also satisfies the wave equation: + +$$ +\frac{\partial^2 v_x}{\partial x^2} + - \frac{1}{c^2} \frac{\partial^2 v_x}{\partial t^2} = 0 +$$ (eq:waveqacv) + +where $c$ is the same as in the wave equation of the pressure. + +The solution to the wave equation for the pressure is: + +$$ +p(x,t) = s(t \pm x/c) +$$ (eq:pxt) + +where $s(t)$ is some function. Note the dependency on space and time via $(t \pm x/c)$, which denotes that it is a travelling wave, and behaves as a direct wave as discussed in [](./snells_law.md). The sign in the argument depends on the direction the wave is travelling. + +Often, seismic responses are analyzed in terms of frequencies, i.e., Fourier spectra. The definition used here is: + +$$ +G(\omega) = \int_{-\infty}^{+\infty} g(t) \exp(- i \omega t) \, dt +$$ (eq:fourier1) + +where the radial frequency $\omega$ is used, so $\omega = 2 \pi f$ where $f$ is the (ordinary) frequency. Using this convention, it is easy to show that differentiation with respect to time is equivalent to multiplication with $i\omega$ in the Fourier domain. When we transform the solution of the wave equation ({eq}`eq:pxt`) to the Fourier domain, we obtain: + +$$ +P(x,\omega) = S(\omega) \exp (\pm i\omega x/c). +$$ (eq:Pxw) + +Note here that the time delay $x/c$ (in the time domain) becomes a *linear* phase in the Fourier domain, so the phase $\pm \omega x/c$ is linear as a function of $\omega$. + +In the above, we gave an expression for the pressure, but one can also derive an equivalent expression for the particle velocity $v_x$. For that purpose, we can use the equation of motion as expressed in {eq}`eq:eqmotion`, but then in its Fourier-transformed version, which is: + +$$ +V_x(x,\omega) = -\frac{1}{i\omega \rho} \frac{\partial P(x,\omega)}{\partial x}. +$$ + +When we substitute the solution for the pressure from above ({eq}`eq:Pxw`), we get for the negative sign: + +$$ +\begin{aligned} +V_x(x,\omega) & = -\frac{1}{i \omega \rho} + S(\omega) \frac{- i \omega}{c} \exp(-i\omega x/c) +\\ + & = S(\omega) \frac{1}{\rho c} \exp(-i\omega x/c). +\end{aligned} +$$ + +Notice that the particle velocity is a scaled version of the pressure: + +$$ +V_x(x,\omega) = \frac{P(x,\omega)}{\rho c}. +$$ + +The scaling factor is ($\rho c$), being called the *seismic impedance*. diff --git a/book/4_mechanical_waves/appendix_acoustic_1d.md b/book/4_mechanical_waves/appendix_acoustic_1d.md new file mode 100644 index 0000000..1ad1193 --- /dev/null +++ b/book/4_mechanical_waves/appendix_acoustic_1d.md @@ -0,0 +1,76 @@ +(app-waveq1d)= +# Appendix: derivation of basic equations for 1-D acoustic wave motion + +*In this appendix, the one-dimensional wave motion for an acoustic medium is derived, starting from the deformation law of Hooke and from conservation of momentum (Newton's Second Law). This results in the so-called deformation equation and the equation of motion for acoustic waves.* + +## Derivation + +Here we will derive the basic equations for wave motion in homogeneous media, using the conservation of momentum (Newton's second law) and the deformation law, known as Hooke's Law of elasticity. In this derivation, we consider a single cube of mass when it is subdued to a seismic disturbance (see {numref}`fig-cube`). This cube has a volume $\Delta V$ with sides $\Delta X, \Delta Y$ and $\Delta Z.$ + +```{figure} figures/fig_ap_ac_cube.png +:name: fig-cube +:width: 70% + +A cube of mass, subjected to one-dimensional wave motion. Drawn line: original status. Dotted line: status after deformation. +``` + +We start with an elastic deformation of the cube, using Hooke's Law. This law states that the deformation force that works on a piece of material is linearly related to the extension of that piece. Applying this to our cube of mass, the deformation force gives a relative change in volume: + +$$ +p = -K \frac{dV}{\Delta V} +$$ (eq:hooke) + +where the pressure $p$ is introduced as being the force per unit area, $K$ is called the bulk modulus, and the minus sign expresses that the pressure is opposite to the deformation direction. Now we assume that the volume change is only in one direction (1-dimensional) as shown in {numref}`fig-cube`, so then we get: + +$$ +\begin{aligned} +\frac{dV}{\Delta V} & = + \frac{u_x(x+\Delta X)\Delta Y \Delta Z- u_x(x) \Delta Y \Delta Z} + {\Delta X \Delta Y \Delta Z} \\ +& = \frac{u_x(x+\Delta X) - u_x(x)}{\Delta X} +\end{aligned} +$$ + +where $u_x$ is the displacement in the $x$-direction. Using the situation that $\Delta X$ is rather small, we can approximate $u_x(x+\Delta X)$ by $u_x(x) + ( \partial u_x/\partial x ) \Delta X$; this then gives for Hooke's Law: + +$$ +p = -K \frac{\partial u_x}{\partial x}. +$$ + +Since we are finally interested in particle velocities rather than displacements (since geophones, seismic sensors on land, measure particle velocities), we differentiate both sides of Hooke's Law with respect to time $t$, and introduce $v_x = \partial u_x/\partial t$ to give: + +$$ +\boxed{ +\frac{1}{K} \frac{\partial p}{\partial t} = - \frac{\partial v_x}{\partial x} +} +$$ + +This is the so-called deformation equation. It is one basic relation needed for describing one-dimensional wave motion. + +The other relation is obtained via Newton's Law, applied to the volume $\Delta V$ with mass $M$ in the direction $x$, since we consider 1-dimensional motion: + +$$ +\begin{aligned} +F_x & = M \frac{\partial v_x}{\partial t} \\ + & = \rho \Delta V \frac{\partial v_x}{\partial t} +\end{aligned} +$$ (eq:newton) + +where $F$ is the total force that works on the element $\Delta V$ that induces motion. Consider the total force that is working on the cube in the $x$-direction, through the pressures working on the sides with area $\Delta Y \Delta Z$: + +$$ +\begin{aligned} +F_x & = - [ p(x+\Delta X) - p(x) ] \Delta S_x \\ + & = - \frac{p(x+\Delta X) - p(x)}{\Delta X} \Delta V +\end{aligned} +$$ + +Using $p(x+\Delta X) \simeq p(x) + ( \partial p/\partial x ) \Delta X$ and combining it with Newton's Law as expressed in {eq}`eq:newton` gives: + +$$ +\boxed{ + - \frac{\partial p}{\partial x} = \rho \frac{\partial v_x}{\partial t} +} +$$ + +since $\Delta V$ cancels. This equation is called the equation of motion. It is the other basic relation (next to the deformation equation) needed to describe one-dimensional wave motion. diff --git a/book/4_mechanical_waves/appendix_strain_p_s.md b/book/4_mechanical_waves/appendix_strain_p_s.md new file mode 100644 index 0000000..2fb8610 --- /dev/null +++ b/book/4_mechanical_waves/appendix_strain_p_s.md @@ -0,0 +1,375 @@ +(app-elastic)= +# Appendix: strain, P- and S-waves + +*In this appendix, strain for an elastic solid is explained. Together with the stress (discussed in the main text), it is used to derive the basic equations for P- and S-waves: the deformation equations and the equations of motion.* + +## Strain + +Strain is a geometrical quantity. It is a *relative* displacement under applied forces, i.e., it is a measure of deformation, normalized by its original size. As for stress, strain may be divided into normal strain and shear strain, depending on the type of force applied. For a normal strain in the deformation of a volume, the strain is simply the change of volume divided by its original volume, i.e.,: + +$$ +e_{\text{normal}} = \frac{\Delta V}{V} +$$ + +where $e$ is the strain, $\Delta V$ the change in volume and $V$ the original volume itself. This volume change was also used to derive the acoustic wave equation (see [the appendix on 1-D acoustic wave motion](./appendix_acoustic_1d.md)). + +Let us now look at {numref}`fig-ap-straindispl`, where the displacement $\underline{u}$ is given for a point $\underline{x}$. Notice that the displacement is a function of $\underline{x}$, i.e., $\underline{u} = \underline{u} (\underline{x})$. When a force/stress is applied, we get a difference in displacement, described by: + +$$ +\begin{aligned} +\underline{u} (\underline{x}+ d\underline{x}) & \simeq + \underline{u} (\underline{x}) + + \frac{\partial \underline{u}}{\partial \underline{x} } d\underline{x} \\ + & = \underline{u} (\underline{x}) + d\underline{u} +\end{aligned} +$$ + +where the first term is a translation and the second term is the term associated with deformation, in accordance with the assumption of elasticity. Notice that the term $\partial \underline{u}/ \partial \underline{x}$ is a matrix since both $\underline{u}$ and $\underline{x}$ are vectors: + +$$ + \frac{\partial \underline{u}}{\partial \underline{x} } = + \begin{pmatrix} + \frac{\partial u_x}{\partial x} & \frac{\partial u_x}{\partial y} & + \frac{\partial u_x}{\partial z} \\ + \frac{\partial u_y}{\partial x} & \frac{\partial u_y}{\partial y} & + \frac{\partial u_y}{\partial z} \\ + \frac{\partial u_z}{\partial x} & \frac{\partial u_z}{\partial y} & + \frac{\partial u_z}{\partial z} \\ + \end{pmatrix} +$$ + +The term $\partial \underline{u}/ \partial \underline{x}$ can be written in a symmetric and anti-symmetric part: + +$$ + \frac{\partial \underline{u}}{\partial \underline{x} } = + \underline{\underline{e}} - \frac{1}{2} \underline{\underline{\xi}} +$$ + +in which the elements of $\underline{\underline{e}}$ and $\underline{\underline{\xi}}$ are: + +$$ + e_{ij} = \frac{1}{2} + \left( \frac{\partial u_i}{\partial x_j} + \frac{\partial u_j}{\partial x_i} + \right) +$$ (eq:straindispl) + +$$ + \xi_{ij} = + \frac{\partial u_j}{\partial x_i} - \frac{\partial u_i}{\partial x_j} +$$ + +where $\underline{\underline{e}}$ is the so-called strain tensor, and $\underline{\underline{\xi}}$ represents rigid rotation. + +```{figure} figures/straindispl.PNG +:name: fig-ap-straindispl +:width: 60% + +Displacement field under presence of stress. +``` + +## Tensile stress and strain, and P-waves in a solid + +P-waves are associated with tensile stresses, so therefore, we consider the tensile stress and strain for an infinitesimal rectangular parallelepiped as shown in {numref}`fig-ap-ptens`. The sizes of the parallelepipeds are given by $\Delta X, \Delta Y$ and $\Delta Z$, while the changes therein are given by $\delta U_x, \delta U_y$ and $\delta U_z$. We apply a tensile stress in the $x$-direction only. As was also given in the appendix on acoustic wave motion, Hooke's law states that a force needed to extend or compress a spring is linearly related to the length ($u$) of extension/compression of that spring, so $F = k u.$ Stress is force per unit area, and strain is relative displacement, so Hooke's law applied to our parallelepiped gives: + +$$ +e_{xx} = \frac{\partial u_x}{\partial x} = \frac{\delta U_x}{\Delta X} = + \frac{1}{E} \tau_{xx} +$$ + +where $\tau_{xx}$ is the tensile stress and $e_{xx}$ the tensile strain in the $x$-direction, and $E$ is called Young's modulus. When applying a tensile stress, the size in the perpendicular directions $y$ and $z$ *decreases*, i.e., a lateral contraction occurs. This is quantified by the so-called Poisson's ratio, via: + +$$ +\begin{aligned} +e_{yy} & = \frac{\partial u_y}{\partial y} = \frac{\delta U_y}{\Delta Y} = + - \sigma e_{xx} \\ +e_{zz} & = \frac{\partial u_z}{\partial z} = \frac{\delta U_z}{\Delta Z} = + - \sigma e_{xx} +\end{aligned} +$$ + +We can follow the same procedure for a tensile stress in the $y$- and $z$-directions with tensile strains $e_{yy}$ and $e_{zz}$, giving totally: + +$$ + \begin{pmatrix} + e_{xx} \\ e_{yy} \\ e_{zz} + \end{pmatrix} + = \frac{1}{E} + \begin{pmatrix} + 1 & -\sigma & -\sigma \\ + -\sigma & 1 & -\sigma \\ + -\sigma & -\sigma & 1 + \end{pmatrix} + \begin{pmatrix} + \tau_{xx} \\ \tau_{yy} \\ \tau_{zz} + \end{pmatrix} +$$ (eq:etau) + +```{figure} figures/ptens.PNG +:name: fig-ap-ptens +:width: 60% + +Infinitesimal rectangular parallelepiped deformed by tensile stress. +``` + +So far, we have written down the basic expressions, but they are not in the desired form: we want to have a form in which the partial derivatives of the particle displacements are given (and not strains), and we want to have the stress on the left-hand side of the equation and the strains/particle-displacements on the right-hand side. To address the first issue, we have: + +$$ + \begin{pmatrix} + e_{xx} \\ e_{yy} \\ e_{zz} + \end{pmatrix} + = + \begin{pmatrix} + \frac{\partial u_x}{\partial x} \\ + \frac{\partial u_y}{\partial y} \\ + \frac{\partial u_z}{\partial z} + \end{pmatrix} +$$ + +To address the second issue, we have to invert the matrix in {eq}`eq:etau`. Since the inverted matrix has terms that are not very simple, the so-called Lamé parameters are introduced: + +$$ +\begin{aligned} + \lambda & = \frac{\sigma E}{ (1-2\sigma)(1+\sigma)} \\ + \mu & = \frac{E}{2 (1+\sigma)} +\end{aligned} +$$ + +where $\mu$ is known as the shear modulus. Using these, we then get: + +$$ + \begin{pmatrix} + \tau_{xx} \\ \tau_{yy} \\ \tau_{zz} + \end{pmatrix} + = + \begin{pmatrix} + \lambda + 2\mu & \lambda & \lambda \\ + \lambda & \lambda + 2\mu & \lambda \\ + \lambda & \lambda & \lambda + 2\mu + \end{pmatrix} + \begin{pmatrix} + \frac{\partial u_x}{\partial x} \\ + \frac{\partial u_y}{\partial y} \\ + \frac{\partial u_z}{\partial z} + \end{pmatrix} +$$ + +This is the desired expression that we will use next. + +Let us now derive the equations necessary to describe 1-D wave motion of a P-wave, travelling in the $x$-direction. To that end, consider {numref}`fig-ap-pblock`, in which again an infinitesimal block with size $\Delta X$, $\Delta Y$ and $\Delta Z$ is given. The mass $\Delta M$ of the little block is given by $\rho \Delta X \Delta Y \Delta Z$. We initiate a plane displacement field with displacements in the $x$-direction only (and therefore there is no contraction $\sigma$ for this 1-D case): + +$$ + \underline{u} (x,y,z,t) = + \begin{pmatrix} + u_x(x,t) \\ 0 \\ 0 + \end{pmatrix} +$$ + +In that case, Hooke's law gives $\tau_{xx} = (\lambda + 2\mu)\partial u_x/\partial x$. As for the acoustic case, we are finally interested in particle velocities rather than displacements (since geophones measure particle velocities), we differentiate both sides of Hooke's law with respect to time $t$, and introduce $v_x = \partial u_x/\partial t$ to give: + +$$ +\boxed{ + \frac{ \partial \tau_{xx} }{\partial t} = (\lambda + 2 \mu) + \frac{ \partial v_x }{\partial x} +} +$$ + +Note that this equation has the same shape as the equation for acoustic waves, i.e., + +$$ + - \frac{ \partial p }{\partial t} = K \frac{ \partial v_x }{\partial x} +$$ + +```{figure} figures/pblock.PNG +:name: fig-ap-pblock +:width: 60% + +Infinitesimal rectangular block under plane stress, used for deriving basic equations for P-wave motion. +``` + +This equation alone does not describe wave motion; to that end we need to apply Newton's law as well. Referring to {numref}`fig-ap-pblock`, the force is, in terms of stress: + +$$ + F_x = \Delta Y \Delta Z [ \tau_{xx}(x+\Delta X) - \tau_{xx}(x) ] +$$ + +and the force is given as mass times acceleration, i.e.: + +$$ + F_x = \Delta M \frac{\partial^2 u_x}{\partial t^2}. +$$ + +Equalizing these forces gives: + +$$ + \frac{ \tau_{xx}(x+\Delta X) - \tau_{xx}(x) }{\Delta X} + = + \frac{\Delta M}{\Delta X \Delta Y \Delta Z} + \frac{\partial^2 u_x}{\partial t^2} +$$ + +or, since $\tau_{xx}(x+\Delta X) \simeq \tau_{xx}(x) + ( \partial \tau_{xx}/\partial x ) \Delta X$: + +$$ + \frac{ \partial \tau_{xx} }{\partial x} + = + \rho \frac{\partial^2 u_x}{\partial t^2}. +$$ + +Again, we need to introduce the particle velocity for the displacement, giving: + +$$ +\boxed{ + \frac{ \partial \tau_{xx} }{\partial x} = + \rho \frac{\partial v_x}{\partial t}. +} +$$ + +This is the other desired equation, the equation of motion for P-waves. As expected, this equation has the same shape as the one for acoustic waves: + +$$ + - \frac{ \partial p }{\partial x} + = + \rho \frac{\partial v_x}{\partial t}. +$$ + +## Shear stress and strain, and S-waves in a solid + +S-waves are associated with shear stresses. We will follow the same line of derivation as for the P-waves but then with shear stresses: consider the shear stress and strain for an infinitesimal rectangle, deformed into a parallelogram (2D) as shown in {numref}`fig-ap-stens`, with sizes $\Delta X$ and $\Delta Z$. Let us first consider the left-hand figure (a), with $\tau_{xz}$. The shear modulus $\mu$ is defined as the constant that relates the stress and displacement, in this case as: + +$$ + \tau_{xz} = \mu \frac{\delta U_x}{\Delta Z} +$$ + +which is the shear equivalent of Hooke's law for elastic deformation. Since the associated strain $e_{xz}$ was defined as $(1/2) ( \partial u_z/\partial x + \partial u_x/\partial z)$ (see {eq}`eq:straindispl`), which for the parallelogram becomes: + +$$ + e_{xz} = \frac{1}{2} \frac{\delta U_x}{\Delta Z}, +$$ + +the stress-strain relationship for shear deformation becomes: + +$$ + \tau_{xz} = 2 \mu e_{xz}. +$$ + +```{figure} figures/stens.PNG +:name: fig-ap-stens +:width: 70% + +Infinitesimal rectangle deformed to parallelogram by shear stress. +``` + +This was the analysis for shear deformation in the horizontal direction, but deformation can also take place in the vertical direction, as shown in {numref}`fig-ap-stens`(b). In that case we have for the stress and strain: + +$$ +\begin{aligned} + \tau_{zx} & = \mu \frac{\delta U_z}{\Delta X} \\ + e_{zx} & = \frac{1}{2} \frac{\delta U_z}{\Delta X} \\ + \tau_{zx} & = 2 \mu e_{zx}. +\end{aligned} +$$ + +The deformations as shown in {numref}`fig-ap-stens`(a) and (b) are actually deformations with a rigid-body rotation where $e_{xz} \neq e_{zx}$. We are interested in the deformation which show no rigid-body rotation, as shown in {numref}`fig-ap-stens`(c). For no rigid-body rotation $\xi_{xz}=0$, so: + +$$ + \frac{\partial u_x}{\partial z} = \frac{\partial u_z}{\partial x}, + \qquad \text{or} \qquad + \frac{\delta U_x}{\Delta Z} = \frac{\delta U_z}{\Delta X}, +$$ + +We then have: + +$$ + e_{xz} = e_{zx}, +$$ + +and therefore: + +$$ + \tau_{xz} = \tau_{zx}. +$$ + +Similarly, for no rigid-body rotation in 3-D: + +$$ +\begin{aligned} + e_{xy} & = e_{yx} \\ + e_{yz} & = e_{zy} \\ + \tau_{xy} & = \tau_{yx} \\ + \tau_{yz} & = \tau_{zy} +\end{aligned} +$$ + +Let us now derive the equations necessary to describe 1-D wave motion of an S-wave, travelling in the $x$-direction (as for the P-wave). To that end, consider {numref}`fig-ap-sblock` in which again an infinitesimal block with size $\Delta X$, $\Delta Y$ and $\Delta Z$ and mass $\Delta M = \rho \Delta X \Delta Y \Delta Z$ is given. We initiate a plane displacement field, giving displacements in the $z$-direction only (for P-waves they were in the $x$-direction only): + +$$ + \underline{u} (x,y,z,t) = + \begin{pmatrix} + 0 \\ 0 \\ u_z(x,t) + \end{pmatrix} +$$ + +This means that there are no tensile stresses (since $u_z = u_z(x,t)$ only), and the other shear stresses $\tau_{xy}$ and $\tau_{yz}$ are zero. The shear equivalent of Hooke's law gives: + +$$ + \tau_{zx} = 2\mu e_{zx} = 2\mu \cdot \frac{1}{2} \left( + \frac{\partial u_z}{\partial x} + \frac{\partial u_x}{\partial z} \right) + = \mu \frac{\partial u_z}{\partial x} +$$ + +As for P-waves, in order to introduce the particle velocity $v_z$, we differentiate both sides with respect to time $t$ and take the particle velocity as $v_z = \partial u_z/\partial t$ to give: + +$$ +\boxed{ + \frac{\partial \tau_{zx}}{\partial t} = \mu \frac{\partial v_z}{\partial x}. +} +$$ + +This is the desired expression for the deformation. + +```{figure} figures/sblock.PNG +:name: fig-ap-sblock +:width: 60% + +Infinitesimal rectangular block under plane stress, used for deriving basic equations for S-wave motion. +``` + +As before, this equation alone does not describe wave motion since we need to apply Newton's law as well. Referring to {numref}`fig-ap-sblock`, the force is, in terms of stress: + +$$ + F_z = \Delta Y \Delta Z [ \tau_{zx}(x+\Delta X) - \tau_{zx}(x) ] +$$ + +and the force is given as mass times acceleration, i.e.: + +$$ + F_z = \Delta M \frac{\partial^2 u_z}{\partial t^2}. +$$ + +Equalizing these forces gives: + +$$ + \frac{ \tau_{zx}(x+\Delta X) - \tau_{zx}(x) }{\Delta X} + = + \frac{\Delta M}{\Delta X \Delta Y \Delta Z} + \frac{\partial^2 u_z}{\partial t^2} +$$ + +or, since $\tau_{zx}(x+\Delta X) \simeq \tau_{zx}(x) + ( \partial \tau_{zx}/\partial x ) \Delta X$: + +$$ + \frac{ \partial \tau_{zx} }{\partial x} + = + \rho \frac{\partial^2 u_z}{\partial t^2}. +$$ + +Again, we need to introduce the particle velocity via the displacement, giving: + +$$ +\boxed{ + \frac{ \partial \tau_{zx} }{\partial x} = + \rho \frac{\partial v_z}{\partial t}. +} +$$ + +This is the other desired equation, the equation of motion for S-waves. diff --git a/book/4_mechanical_waves/basic_notions.md b/book/4_mechanical_waves/basic_notions.md new file mode 100644 index 0000000..6f92f15 --- /dev/null +++ b/book/4_mechanical_waves/basic_notions.md @@ -0,0 +1,53 @@ +# Basic physical notion of waves + +The seismic method makes use of the properties of the velocity of seismic waves. This velocity is different for different rocks and it is this difference which is exploited in the seismic method. When we create a wave field at or near the surface of the earth, some energy will be reflected back (bounced back). They can be viewed as echoes. From these echoes we can determine the velocities of the rocks, as well as the depths where the echoes came from. In this chapter we will discuss the basic principles behind the behaviour of seismic wave fields in solid materials. When we use the seismic method, we usually discuss two types of seismic methods, depending on whether the distance from the sound source to the detector (the "ear") is large or small with respect to the depth of interest: the first is known as refraction seismics, the other as reflection seismics. Of course, there are overlaps and differences between those two types which will be discussed in this chapter. + +Everybody knows what waves are when we are talking about waves at sea. Seismic wave fields in materials have the same kind of behaviour as these waves, only they travel much faster than the waves we see at sea. Waves can occur in several ways. We will consider two types of them, namely the longitudinal and the shear wave. Longitudinal waves behave like waves in a large spring. When we push from one side of a spring, we will observe a wave going through the spring which characterizes itself by a thickening of the wires running through the spring in time (see also {numref}`fig-pands`). A property of this type of wave is that the motion of a piece of the wire is in the same direction as the wave moves. These waves are also called Push-waves, abbreviated to P-waves, or compressional waves. Another type of wave is the shear wave. A shear wave can be compared with a chord. When we push a chord upward from one side, a wave will run along the chord to the other side. The movement of the chord itself is only up- and downward: characteristic of this wave is that a piece of the chord is moving perpendicular to the direction of that of the wave (see also {numref}`fig-pands`). These types of waves are referred to as S-waves, also called shear waves. Characteristic of this wave is that a piece of the chord is pulling its "neighbour" upward, and this can only occur when the material can support shear stress. In fluids, one can imagine that a "neighbour" cannot be pulled upward simply because it is a fluid. Therefore, in fluids only P-waves exist, while in a solid both P- and S-waves exist. + +```{figure} figures/blokjes.png +:name: fig-pands +:width: 80% + +Particle motion of P and S waves, where the particle motion for SV is in the vertical direction only and for SH in the horizontal direction only. +``` + +In the real world, we deal with three spatial dimensions, so a wave will spread in three directions. In a homogeneous medium (so the properties of the material are everywhere constant and the same), the wave will spread out like a sphere. The outer shell of this sphere is called the wave front. Another way of describing this wave front is in terms of the normal to the wavefront: the ray. We are used to rays in optics and we can use the same notion in the seismic method. When we were explaining the behaviour of P- and S-waves, we already used the term "neighbour". This was an important feature; otherwise the wave would not move forward. A fundamental notion included in this, is Huygens' principle. When a wave front arrives at a certain point, that point will behave also as a source for the wave, and so will all its neighbours. The new wavefront is then the envelope of all the waves that were generated by these points. This is illustrated in {numref}`fig-huygens`. Again, the ray can then be defined as the normal to that envelope which is also given in the figure. + +```{figure} figures/huygens.png +:name: fig-huygens +:width: 80% + +Using Huygens' principle to locate new wavefronts. +``` + +So far, we only discussed the way in which the wave moves forward, but there is also another property of the wave we have not discussed yet, namely the amplitude: how does the amplitude behave as the wave moves forward? We have already mentioned spherical spreading when the material is everywhere the same. The total energy will be spread over the area over the sphere. This type of energy loss is called spherical divergence. It simply means that if we put our "ear" at a larger distance, the sound will be less loud. There is also another type of energy loss, that is due to losses within the material, which mainly consists of internal friction losses. This means that the amplitude of a wave will be extra damped because of this property. S-waves usually show higher friction losses than P-waves. Finally, we give a table of common rocks and their seismic wave velocities in {numref}`tab-wavevel`. + +```{list-table} Seismic wave velocities for some common media, soils and rocks. +:name: tab-wavevel +:header-rows: 1 + +* - Material + - Wave velocity (m/s) + - Material + - Wave velocity (m/s) +* - Air + - 330 + - Sandstone + - 2000-4500 +* - Water + - 1500 + - Shales + - 3900-5500 +* - Soil + - 20-300 + - Limestone + - 3400-7000 +* - Sands + - 600-1850 + - Granite + - 4800-6000 +* - Clays + - 1100-2500 + - Ultra-mafic rocks + - 7500-8500 +``` diff --git a/book/4_mechanical_waves/elastic_waves.md b/book/4_mechanical_waves/elastic_waves.md new file mode 100644 index 0000000..d6a60de --- /dev/null +++ b/book/4_mechanical_waves/elastic_waves.md @@ -0,0 +1,108 @@ +# Wave fields in a solid: elastic (P and S) waves + +## Stress + +Here we define *stress*. Stress can be defined as the force per unit area exerted on a body across an imaginary surface (see also {numref}`fig-stressforce`). Since the dimension of stress is of a pressure, i.e., force per unit area, the quantities of stress are also given in the dimension of pressure, i.e., Pa. Therefore, the stress can be seen as the equivalent of pressure in an (elastic) solid. One can discern normal stress as the stress perpendicular to a surface, and shear stress as the one parallel to a surface. The normal stress is also called compressional or tensile stress. Since a stress can be in any direction to a surface, the stress can be decomposed into components perpendicular and parallel to the surface. This decomposition matrix is also called the stress tensor. In formula form, this can all be written as: + +$$ + \frac{\underline{F}}{\Delta S} = + \underline{\underline{\tau}} \, \underline{n} +$$ + +where $F$ is the force, $\Delta S$ is the surface, $\underline{\underline{\tau}}$ is the stress tensor and $\underline{n}$ is the vector normal to the surface $\Delta S$. When writing out the matrix of the stress, i.e., the stress tensor, in terms of the coordinates $x$, $y$ and $z$, we get: + +$$ + \underline{\underline{\tau}} = + \begin{pmatrix} + \tau_{xx} & \tau_{xy} & \tau_{xz} \\ + \tau_{yx} & \tau_{yy} & \tau_{yz} \\ + \tau_{zx} & \tau_{zy} & \tau_{zz} + \end{pmatrix} +$$ + +Some of these factors are visualized in {numref}`fig-tautensor`. + +```{figure} figures/stressforce.PNG +:name: fig-stressforce +:width: 60% + +Definition of stress for a force working on a surface $\Delta S$. +``` + +In order to see the equivalence with the acoustic case, consider the case where no shear stresses exist. Then the diagonal elements of the matrix are the only non-zero ones, and the stress matrix can be written as: + +$$ + \underline{\underline{\tau}}_{\text{no-shear}} = + \begin{pmatrix} + -p & 0 & 0 \\ + 0 & -p & 0 \\ + 0 & 0 & -p + \end{pmatrix} +$$ + +where $p$ is the pressure. Notice that there is a minus sign since commonly the positive sign for pressure is for a compressional force, while for stress it is for a tensile force. + +```{figure} figures/tautensor.PNG +:name: fig-tautensor +:width: 60% + +Definition of stress components for stresses working on an infinitesimal parallelepiped. +``` + +## P-waves in an elastic solid + +For 1-D wave motion in the $x$-direction, Hooke's law gives for the plane tensile stress $\tau_{xx}$ (see [the appendix on strain, P- and S-waves](./appendix_strain_p_s.md)): + +$$ + \frac{\partial \tau_{xx}}{\partial t} = (\lambda + 2\mu) + \frac{\partial v_x}{\partial x} +$$ + +where $\lambda$ and $\mu$ are the so-called Lamé parameters. $\lambda$ is the first Lamé parameter, and $\mu$ is the shear modulus; in 3-D, $\lambda$ is related to the bulk modulus $K$ through $K = \lambda + (2/3)\mu$. This equation is called the deformation equation for a P-wave in an elastic solid. Note that this equation has the same form as the deformation equation for acoustic waves (see {eq}`eq:eqdeform`). + +Then applying Newton's law gives an equation nearly equivalent to the one for the acoustic case, but now having tensile stress instead of pressure (see also [the appendix on strain, P- and S-waves](./appendix_strain_p_s.md)): + +$$ +\frac{\partial \tau_{xx}}{\partial x} = \rho \frac{\partial v_x}{\partial t} +$$ + +This is the equation of motion for a P-wave. Combining the deformation equation and the equation of motion gives, as for the acoustic case, the wave equation for the stress, assuming the density being constant: + +$$ +\frac{\partial^2 \tau_{xx}}{\partial x^2} + - \frac{1}{c^2_P} \frac{\partial^2 \tau_{xx}}{\partial t^2} = 0 +$$ + +where $c^2_P = (\lambda + 2\mu)/\rho$ is the squared P-wave velocity. This wave equation describes the wave propagation for P-waves in an elastic solid. + +## S-waves in an elastic solid + +Let us now consider shear stresses which, as we shall see, will result in a wave equation for S-waves. To that end, we consider 1-D displacement in the $z$-direction for a wave propagating in the $x$-direction for the 1-D shear motion. The equivalent of Hooke's law for shear stress results in (see [the appendix on strain, P- and S-waves](./appendix_strain_p_s.md)): + +$$ + \frac{\partial \tau_{zx}}{\partial t} = \mu \frac{\partial v_z}{\partial x} +$$ (eq:deformS) + +where $\tau_{zx}$ is the shear stress. This is the deformation equation for an S-wave. + +Then applying Newton's law for a shear stress gives (see also [the appendix on strain, P- and S-waves](./appendix_strain_p_s.md)): + +$$ + \frac{\partial \tau_{zx}}{\partial x} = \rho \frac{\partial v_z}{\partial t} +$$ (eq:eqmotionS) + +which is the equation of motion for S-waves. Combining the deformation equation and the equation of motion gives the wave equation for the shear stress, assuming the density being constant: + +$$ +\frac{\partial^2 \tau_{zx}}{\partial x^2} + - \frac{1}{c^2_S} \frac{\partial^2 \tau_{zx}}{\partial t^2} = 0 +$$ (eq:waveqStau) + +where $c^2_S = \mu / \rho$ is the squared S-wave velocity. This wave equation describes the wave propagation for S-waves in an elastic solid. + +With respect to the (1-D) reflection coefficients for P- and S-waves at the boundary of two elastic solids: the derivations go similar as for the acoustic case, as worked out in the previous section: + +- For P-waves, we require the continuity of the tensile stress $\tau_{xx}$ and the particle velocity $v_x$ at the interface, and +- For S-waves we require the continuity of the shear stress $\tau_{zx}$ and the particle velocity $v_z$ at the interface. + +The resulting expressions for the reflection and transmission coefficient are the same as for the acoustic case, only now the velocities are for the elastic solid. (See *EXERCISES*) diff --git a/book/4_mechanical_waves/field_recordings.md b/book/4_mechanical_waves/field_recordings.md new file mode 100644 index 0000000..3175135 --- /dev/null +++ b/book/4_mechanical_waves/field_recordings.md @@ -0,0 +1,60 @@ +# Seismic wavefield recordings from the field: physical interpretation using kinematic concepts + +In this section, we will look at the physical interpretation of seismic wavefield data as recorded in the field, two recordings on land (one with rocky near-surface and one with loose-soil near-surface) and one recording at sea. + +## Seismic wavefield recording on land: rocky near-surface + +Let us first consider a seismic wave field recorded on land. In {numref}`fig-raw-shot-land` this wavefield record is shown. The first event we are considering is the "first arrival". It shows itself by giving a pulse after a quiet period. This arrival can be a *direct* arrival or a *refraction*, both of which we discussed earlier. When looking closely at the record, it can be seen that there is a slight change of dip, occurring at around an offset of 800 m. Therefore, the arrival until this distance is a direct arrival, while beyond that distance, it is the refraction. By drawing a straight line through the direct arrival, the velocity with which it propagates along the surface is: 2400 meters in 670 ms, so approximately 3600 m/s. This is a relatively high velocity, so there is probably some very hard rock at the surface. The velocity of the refracted arrival can also be determined by a straight line through that arrival; it amounts to 2400 meters in around 500 ms (note that the arrival does not go through the origin, so it is the difference between the time at $x=0$ m and at $x=2400$ m), so a velocity of 4800 m/s. Below the record in {numref}`fig-raw-shot-land`, a simple model is shown that explains this first arrival; the synthetic record belonging to this model is given on the top right of the figure. + +The next events we consider are the strong events that cross 2400 meter at some 1.3 seconds. This event is interpreted as ground-roll or surface waves (they propagate along the surface). Calculating the velocity, we come to 1850 m/s. Again, the simple model below the record explains this ground roll; the synthetic record on the top right of the figure also shows this arrival. + +```{figure} figures/Figure4_1.PNG +:name: fig-raw-shot-land +:width: 100% + +Field seismic wavefield recording from land survey (top left), synthetic seismogram (top right) using model of near surface (middle) and model at larger depths (bottom). +``` + +Also in this field record, a "high-frequency" event can be observed, which goes through the 4-second mark at about 1300 meters distance. Calculating the velocity from this, we come to some 325 m/s. It may be clear that this is a wave that goes through the air. This event is also synthesized in the top right figure, using the model as given below it. + +Last but not least, are "high-frequency" events which are slightly curved, e.g. the ones at 0.9 and 1.6 seconds. These events are interpreted as reflections from layer boundaries in the deep subsurface. Those are usually the events we are interested in when we want to obtain an image of the subsurface. Using a simple model as given at the bottom of the figure (which explains the deeper part of the earth), the synthetic record for these events is also shown in the top right of the figure. + +In the above, we have interpreted five types of events, which can be captured in one combined model and are shown in one combined synthetic seismogram. These synthetics explain the most important events in the raw seismic record. Still, when looking at the resulting synthetic seismogram, we see that we are very over-simplifying the situation, since the synthetic and field record are only resembling in the arrival times of the most important events. When looking at the general characteristics, they are very different indeed. + +## Seismic wavefield recording on land: loose-soil near-surface + +The field record we discussed so far was recorded on some hard rocks where the velocities are relatively high. However, when shooting data on land with loose top soil, in this case sand on a beach, the characteristics are much different. In {numref}`fig-raw-shot-wassenaar`, a field record of this situation is given. Again, we can determine the main events in this record. Let us first consider the "first arrival", i.e. the arrival that is coming in first after a quiet period. As usual, this is interpreted as a refraction, as shown in the figure below the record. The velocity can be determined: we come to some 1600 m/s. This velocity is very close to the velocity of water, so this refraction may be due to the water table. In the figure on the right, the synthetic shows this arrival. + +The next event is the most prominent one, namely the event that goes through the 1-second mark at some 180 m, so its velocity is around 180 m/s. This arrival is interpreted as "ground-roll"/surface waves, which travel along the surface. The model which explains this arrival is given again below the record, and its synthetic is shown on the top right. + +The most important events for this wavefield recording are the "high-frequency" events that are the slightly curved arrivals, which can all be interpreted as reflections from deep layers. The number of reflections are too many; only a few are synthesized in the record on the top right, using the model as given at the bottom of the figure. + +Again, when comparing the synthetic to the field seismogram, it is obvious that we have very over-simplified the earth; the positive side is that we have probably been able to understand most of the events in the field recording. + +```{figure} figures/Figure4_2.PNG +:name: fig-raw-shot-wassenaar +:width: 100% + +Field seismic wavefield recording from land survey with loose top soil (top left), synthetic seismogram (top right) using model of near surface (middle) and model at larger depths (bottom). +``` + +## Seismic wavefield recording at sea + +{numref}`fig-raw-shot-marine` shows a seismic wavefield recording, made at sea. This record is much "cleaner" than the land record, as we measure in a water layer, which is a very homogeneous layer. + +Let us analyze some separate events again. The first event in the marine record is the faint one, going nearly through the origin. It crosses the 500 meter at some 340 ms; this means a velocity of some 1470 m/s. It may be clear that this is the direct arrival from the source to the receivers through the water, as explained in the model below the record. This direct arrival is thus a body wave, since it travels with the velocity of water. + +The next event is the first arrival at farther offsets; this arrival is interpreted as a refractive event. When analyzing the distance travelled over time, i.e. the apparent velocity, a velocity of roughly 2000 m/s is obtained. Using the equation for a refraction ({eq}`eq:trefract`), a depth of 300 meter is obtained. This is quantified in the model below the figure, and its associated synthetic seismogram in the figure on the top right. + +The third event we analyze is the first strong event that looks hyperbolic: starting at some 0.4 seconds and bending down to some 2.2 seconds at 3200 m offset. Clearly, because of its hyperbolic behaviour, it is interpreted as a reflection. When looking at later times, we have some more strong hyperbolic events, such as at 0.8 seconds (bending downward toward some 2.3 seconds), and at 1.2 seconds (bending downward toward 2.4 seconds), and even more. These events are interpreted as so-called multiply reflected waves, i.e., waves that bounce up and down in the water layer. In fact almost all events we see below 0.8 seconds are due to multiply reflected waves, or short-hand: multiples. The times at which the multiply reflected waves arrive seem to be periodic; this is indeed the case. + +Combining the interpretation of the refraction and the reflection, it must be noted that the refracted arrival does not converge to the first reflection but to a later reflection. This means that the refraction occurs at a deeper layer; the shallower layers are probably loosely consolidated, so that the velocity of sound has not changed much compared to the one from water. + +A simple model explaining all these events is shown in the figure below the record, with a water layer of 300 meter. The refracted layer is estimated at a depth of 550 meter, where we assumed that the velocity was a constant of 1500 m/s above. The resulting synthetic seismogram is shown in the top right of the figure. It may be clear that the multiply reflected waves come from the same reflective boundary in the subsurface, namely, the sea bottom (and the sea surface, of course), and are therefore superfluous. They are considered as noise; the only one being "signal" is the one at around 0.4 seconds. + +```{figure} figures/Figure4_3.PNG +:name: fig-raw-shot-marine +:width: 100% + +Seismic wavefield recording from marine survey (top left), synthetic seismogram (top right) using model of water layer and sea-water bottom, where only the path of one multiple reflection is drawn (bottom). +``` diff --git a/book/4_mechanical_waves/figures/Figure4_1.PNG b/book/4_mechanical_waves/figures/Figure4_1.PNG new file mode 100644 index 0000000..dcc2ab1 Binary files /dev/null and b/book/4_mechanical_waves/figures/Figure4_1.PNG differ diff --git a/book/4_mechanical_waves/figures/Figure4_2.PNG b/book/4_mechanical_waves/figures/Figure4_2.PNG new file mode 100644 index 0000000..c1bf4c7 Binary files /dev/null and b/book/4_mechanical_waves/figures/Figure4_2.PNG differ diff --git a/book/4_mechanical_waves/figures/Figure4_3.PNG b/book/4_mechanical_waves/figures/Figure4_3.PNG new file mode 100644 index 0000000..1c4426f Binary files /dev/null and b/book/4_mechanical_waves/figures/Figure4_3.PNG differ diff --git a/book/4_mechanical_waves/figures/blokjes.png b/book/4_mechanical_waves/figures/blokjes.png new file mode 100644 index 0000000..49013f1 Binary files /dev/null and b/book/4_mechanical_waves/figures/blokjes.png differ diff --git a/book/4_mechanical_waves/figures/fig_ap_ac_cube.png b/book/4_mechanical_waves/figures/fig_ap_ac_cube.png new file mode 100644 index 0000000..efa8be4 Binary files /dev/null and b/book/4_mechanical_waves/figures/fig_ap_ac_cube.png differ diff --git a/book/4_mechanical_waves/figures/huygens.png b/book/4_mechanical_waves/figures/huygens.png new file mode 100644 index 0000000..c1fdd40 Binary files /dev/null and b/book/4_mechanical_waves/figures/huygens.png differ diff --git a/book/4_mechanical_waves/figures/pblock.PNG b/book/4_mechanical_waves/figures/pblock.PNG new file mode 100644 index 0000000..8159fac Binary files /dev/null and b/book/4_mechanical_waves/figures/pblock.PNG differ diff --git a/book/4_mechanical_waves/figures/ptens.PNG b/book/4_mechanical_waves/figures/ptens.PNG new file mode 100644 index 0000000..b66dd25 Binary files /dev/null and b/book/4_mechanical_waves/figures/ptens.PNG differ diff --git a/book/4_mechanical_waves/figures/rayelement.png b/book/4_mechanical_waves/figures/rayelement.png new file mode 100644 index 0000000..74e68fa Binary files /dev/null and b/book/4_mechanical_waves/figures/rayelement.png differ diff --git a/book/4_mechanical_waves/figures/rays.png b/book/4_mechanical_waves/figures/rays.png new file mode 100644 index 0000000..ef418c3 Binary files /dev/null and b/book/4_mechanical_waves/figures/rays.png differ diff --git a/book/4_mechanical_waves/figures/raytimes.png b/book/4_mechanical_waves/figures/raytimes.png new file mode 100644 index 0000000..2861c0e Binary files /dev/null and b/book/4_mechanical_waves/figures/raytimes.png differ diff --git a/book/4_mechanical_waves/figures/refltrans.png b/book/4_mechanical_waves/figures/refltrans.png new file mode 100644 index 0000000..7b00c37 Binary files /dev/null and b/book/4_mechanical_waves/figures/refltrans.png differ diff --git a/book/4_mechanical_waves/figures/sblock.PNG b/book/4_mechanical_waves/figures/sblock.PNG new file mode 100644 index 0000000..f064996 Binary files /dev/null and b/book/4_mechanical_waves/figures/sblock.PNG differ diff --git a/book/4_mechanical_waves/figures/snell.png b/book/4_mechanical_waves/figures/snell.png new file mode 100644 index 0000000..21b0b54 Binary files /dev/null and b/book/4_mechanical_waves/figures/snell.png differ diff --git a/book/4_mechanical_waves/figures/stens.PNG b/book/4_mechanical_waves/figures/stens.PNG new file mode 100644 index 0000000..ad3beb0 Binary files /dev/null and b/book/4_mechanical_waves/figures/stens.PNG differ diff --git a/book/4_mechanical_waves/figures/straindispl.PNG b/book/4_mechanical_waves/figures/straindispl.PNG new file mode 100644 index 0000000..a8fdbc2 Binary files /dev/null and b/book/4_mechanical_waves/figures/straindispl.PNG differ diff --git a/book/4_mechanical_waves/figures/stressforce.PNG b/book/4_mechanical_waves/figures/stressforce.PNG new file mode 100644 index 0000000..f0abae6 Binary files /dev/null and b/book/4_mechanical_waves/figures/stressforce.PNG differ diff --git a/book/4_mechanical_waves/figures/tautensor.PNG b/book/4_mechanical_waves/figures/tautensor.PNG new file mode 100644 index 0000000..323598f Binary files /dev/null and b/book/4_mechanical_waves/figures/tautensor.PNG differ diff --git a/book/4_mechanical_waves/fundamental_laws.md b/book/4_mechanical_waves/fundamental_laws.md new file mode 100644 index 0000000..2a71385 --- /dev/null +++ b/book/4_mechanical_waves/fundamental_laws.md @@ -0,0 +1,3 @@ +# Seismic wave fields from fundamental physical laws + +Above, we discussed the kinematic properties of wave motion, using rays and traveltimes, without worrying ourselves about the more fundamental laws that were behind it. Here we start from the physical fundamentals of mechanics, i.e., Newton's second law and deformation laws. Using these laws, we can derive the basic equations for wave motion, which then result in the so-called wave equation. The solution of the wave equation shows the features as discussed previously: it turns out that rays are obtained from the high-frequency approximation to that wave equation. The attenuation of the wave due to spreading is captured by the wave equation. And reflection and transmission amplitudes are also captured by applying the resulting equations describing the wave motion. So, here we will go into this, giving the fundamental backbone to the seismic wavefield recordings. diff --git a/book/4_mechanical_waves/intro.md b/book/4_mechanical_waves/intro.md index e69de29..f5293f5 100644 --- a/book/4_mechanical_waves/intro.md +++ b/book/4_mechanical_waves/intro.md @@ -0,0 +1,5 @@ +# Seismic wavefields + +In this chapter, we discuss seismic wave fields. We start from a kinematic point of view, including basic notions of wave fronts, rays, travel times, Huygens' principle and Snell's law at boundaries. Commonly, seismic wavefield measurements are taking place at the surface and then typically direct waves, reflected waves and critically-refracted/head waves are observed, and therefore considered here. Then, some typical wavefield recordings as observed on land and at sea are interpreted, in terms of these arrivals. From these events velocities and depths can be estimated. And from those, a first model for the subsurface is obtained (where the model is here seen as the interpretation of the wavefield data). + +Next, the seismic wave fields are discussed from a dynamic point of view. There we use the fundamental laws of Newton and Hooke to derive the basic equations describing seismic wave fields. From these the wave equation can be derived. Next, reflection and transmission coefficients at boundaries are derived. And finally elastic media with longitudinal (P) and shear (S) waves are considered and which equations describe their full dynamic behaviour. diff --git a/book/4_mechanical_waves/reflection_transmission.md b/book/4_mechanical_waves/reflection_transmission.md new file mode 100644 index 0000000..b64a95f --- /dev/null +++ b/book/4_mechanical_waves/reflection_transmission.md @@ -0,0 +1,77 @@ +(subsec-wavetheory)= +# Reflection and transmission at boundaries + +In this section, we will consider what happens at a boundary between two layers with different mechanical properties. Then we can show which characteristics are responsible for, e.g., a reflection. To that end, we will focus on the basic physical and theoretical principles of reflection and transmission in this section. The basic equations describing wave motion in one direction will be used to derive an expression for the reflection and transmission coefficient at a boundary between two layers with different wave speeds and densities. In the previous section, the solution for the wave equation in the frequency domain was given, i.e.: + +$$ +P(x,\omega) = S(\omega) \exp (\pm i\omega x/c). +$$ + +```{figure} figures/refltrans.png +:name: fig-refltrans +:width: 60% + +Reflection and transmission at boundary between two media with different wave velocities $(c)$ and densities $(\rho)$. +``` + +Let us consider {numref}`fig-refltrans`. We defined a plane boundary between two regions with different wave speeds and mass densities. The boundary will reflect some of the energy back, and some will be transmitted. Above the boundary ($x<0$), we will have a so-called incident wave with some amplitude $S(\omega)$. In addition, we have a reflected wave above the boundary, which is travelling in the opposite direction as the incident field. This reflected field has a scaled version of the amplitude of the incident field. This scaling factor is called $R$, which is called the reflection coefficient. Below the boundary ($x>0$), we have a wave travelling in the same direction as the incident field, but has a scaled amplitude due to the transmission through the boundary. We call this amplitude $T$, which is the transmission coefficient. So, above the boundary ($x<0$), we have: + +$$ +P(x,\omega) = S(\omega) \exp(-i\omega x/c_1) + R S(\omega) \exp(i\omega x/c_1) +$$ + +Below the boundary ($x>0$), we have: + +$$ +P(x,\omega) = T S(\omega) \exp(-i\omega x/c_2) +$$ + +We have defined the reflection and transmission coefficient, but we still need to quantify them. This is achieved by posing the boundary conditions, which are that both the pressure and the particle velocity must be continuous, i.e.: + +$$ +\lim_{x \uparrow 0} P(x,\omega) = \lim_{x \downarrow 0} P(x,\omega) +$$ (eq:limitp) + +$$ +\lim_{x \uparrow 0} V_x(x,\omega) = \lim_{x \downarrow 0} V_x(x,\omega) +$$ (eq:limitv) + +The first boundary condition can be retrieved directly from the solutions for the pressure. However, for the second boundary condition, we need to use the equation of motion in its Fourier-transformed version, which is: + +$$ +V_x(x,\omega) = - \frac{1}{i\omega \rho} \frac{\partial P(x,\omega)}{\partial x} +$$ + +Working these out for the regions above and below the boundary, we obtain, respectively: + +$$ +\begin{aligned} +V_x(x,\omega) & = S(\omega) \frac{1}{\rho_1 c_1} \exp(-i\omega x/c_1) + - R S(\omega) \frac{1}{\rho_1 c_1} \exp(i\omega x/c_1) + & & \text{for } x < 0 \\ +V_x(x,\omega) & = T S(\omega) \frac{1}{\rho_2 c_2} \exp(-i\omega x/c_2) + & & \text{for } x > 0 +\end{aligned} +$$ + +Now we simply substitute the equations in the boundary conditions {eq}`eq:limitp` and {eq}`eq:limitv`, which give: + +$$ +\begin{aligned} +1 + R & = T +\\ +\frac{1}{\rho_1 c_1} - \frac{1}{\rho_1 c_1} R & = +\frac{1}{\rho_2 c_2} T. +\end{aligned} +$$ + +Working this out, we obtain for the reflection and transmission coefficients $R$ and $T$: + +$$ +\begin{aligned} +R & = \frac{\rho_2 c_2 - \rho_1 c_1}{\rho_2 c_2 + \rho_1 c_1} \\ +T & = \frac{2\rho_2 c_2}{\rho_2 c_2 + \rho_1 c_1} +\end{aligned} +$$ + +These are the desired expressions. First, notice that we have expressions in terms of seismic impedances, which are the product of the wave speed with the mass density, i.e., $\rho c$. Second, note that the reflection coefficient is determined by the *contrast in the seismic impedances of the different regions*. diff --git a/book/4_mechanical_waves/snells_law.md b/book/4_mechanical_waves/snells_law.md new file mode 100644 index 0000000..bb0e864 --- /dev/null +++ b/book/4_mechanical_waves/snells_law.md @@ -0,0 +1,154 @@ +(subsec-basickinematics)= +# The interface: Snell's law, refraction and reflection + +So far, we discussed waves in a material which everywhere had the same constant wave velocity. When we include a boundary between two different materials, some energy is bounced back, or reflected, and some energy is going through to the other medium. It is nice to perform Huygens' principle graphically on such a configuration to see how the wave field moves forward (propagates), especially into the second medium. From this picture, we could also derive the ray concept. In this discussion, we will only consider the notion of rays. A basic notion in the ray concept, is Snell's law. Snell's law is a fundamental relation in the seismic wave fields. It tells us the relation between angle of incidence of a wave and velocity in two adjacent layers (see {numref}`fig-snellslaw`). + +```{figure} figures/snell.png +:name: fig-snellslaw +:width: 70% + +Transmission through a boundary (for deriving Snell's law). +``` + +$AA'A''$ is part of a plane wave incident at angle $\theta_1$ to a plane interface between medium 1 of velocity $c_1,$ and medium 2 of velocity $c_2.$ Velocities $c_1$ and $c_2$ are constant. In a time $t$ the wave front $AA'$ moves to the position $BB'$ and the movement is normal to the wave front. Then, the time $t$ it takes to travel from $A'$ to $B'$ in medium 1 is equal to the time from $A$ to $B$ in medium 2, so + +$$ +t = \frac{A'B'}{c_1} = \frac{AB}{c_2} +$$ + +Considering the two triangles, this may be written as: + +$$ +t = \frac{AB' \sin \theta_1}{c_1} = \frac{AB' \sin \theta_2}{c_2} +$$ + +Hence, + +$$ +\frac{\sin \theta_1}{c_1} = \frac{\sin \theta_2}{c_2} +$$ + +which is Snell's law for transmission through a boundary between two layers with different velocities. So far, we have taken general velocities $c_1$ and $c_2$. However, in a solid, two velocities exist, namely P- and S-wave velocities. Generally, when a P-wave is incident on a boundary, it can transmit as a P-wave into the second medium, but also as a S-wave. So in the case of the latter, Snell's law reads: + +$$ +\frac{\sin \theta_P}{c_P} = \frac{\sin \theta_S}{c_S} +$$ + +where $c_P$ is the P-wave velocity, and $c_S$ the S-wave velocity. The same holds for reflection: a P-wave incident on the boundary generates a reflected P-wave and a reflected S-wave. Finally, the same holds for an incident S-wave: it generates a reflected P-wave, a reflected S-wave, a transmitted P-wave and a transmitted S-wave. + +A special case of Snell's law is of interest in refraction prospecting. If the ray is critically refracted along the interface (that is, if $\theta_2 = 90^\circ$), we have + +$$ +\frac{\sin \theta_c}{c_1} = \frac{1}{c_2} +$$ + +where $\theta_c$ is known as the critical angle. + +So far, we have looked at the basic notions of refraction and reflection at an interface. When we measure in the field, and there would be one boundary below it, we could observe several arrivals: a direct ray, a reflected ray, and a refracted ray. We will derive the arrival time of each ray as depicted in {numref}`fig-rays`. + +```{figure} figures/rays.png +:name: fig-rays +:width: 80% + +The paths of the direct, reflected and refracted rays. +``` + +The arrival time of the direct ray is very simple: it is the horizontal distance divided by the velocity of the wave, i.e.,: + +$$ +t = \frac{x}{c_1}, +$$ + +so in a time-distance graph this is a linear event. + +When we look at the reflected ray, we find that the angle of incidence is the same as the angle of reflection. This also follows from Snell's law: when the velocities are the same, the angles must also be the same. When we use Pythagoras' theorem, we obtain the following for the traveltime: + +$$ +t = \frac{ \left( 4z^2+x^2 \right)^{1/2} }{c_1} +$$ + +Squaring this equation: + +$$ +t^2 = \left( \frac{2z}{c_1} \right)^2 + \left( \frac{x}{c_1} \right)^2 , +$$ + +or + +$$ +t^2 - \left( \frac{x}{c_1} \right)^2 = \left( \frac{2z}{c_1} \right)^2, +$$ + +so for a time-distance $(t-x)$ graph this is the equation of a hyperbola. + +When we look at the refracted ray, the derivation is a bit more complicated. Take each ray element; therefore, take the paths $AB,$ $BC$ and $CD$ separately. Then, for the first element, as shown in {numref}`fig-rayelement`, we obtain the traveltime: + +$$ +\Delta t_1 = \frac{\Delta s_1 + \Delta s_2}{c_1} = +\frac{\Delta x_1 \sin \theta_c}{c_1} + \frac{z \cos \theta_c}{c_1} +$$ + +where $\theta_c$ is the critical angle. + +```{figure} figures/rayelement.png +:name: fig-rayelement +:width: 60% + +An element of the ray with critical incidence. +``` + +We can also do this for the paths $BC$ and $CD$, and we obtain the total time as: + +$$ +\begin{aligned} +t & = \Delta t_1 + \Delta t_2 + \Delta t_3 \\ + & = \frac{\Delta x_1 \sin \theta_c}{c_1} + \frac{z \cos \theta_c}{c_1} + + \frac{\Delta x_2}{c_2} + + \frac{\Delta x_3 \sin \theta_c}{c_1} + \frac{z \cos \theta_c}{c_1} +\end{aligned} +$$ + +where $\Delta x_2 = BC$ and $\Delta x_3$ is the horizontal distance between $C$ and $D$. When we now use Snell's law, i.e., $\sin \theta_c /c_1=1/c_2,$ in terms with $\Delta x_1$ and $\Delta x_3,$ then we can add all the terms with $1/c_2,$ using $x = \Delta x_1 + \Delta x_2 + \Delta x_3$ to obtain: + +$$ +t = \frac{x}{c_2} + \frac{2z \cos \theta_c}{c_1} +$$ (eq:trefract) + +We recognize this equation as the equation of a straight line when $t$ is considered as a function of the distance $x,$ the line along which we do our wavefield measurements. + +We have now derived the equations for the three rays, and we can plot their travel times as a function of distance $x$. This is done in {numref}`fig-raytimes`. This picture is important. When we measure seismic data in the field, the characteristics in this plot can most of the time be observed. + +```{figure} figures/raytimes.png +:name: fig-raytimes +:width: 60% + +Time-distance $(t,x)$ curve for the direct, reflected and refracted rays. +``` + +Based on this figure, we can specify better when we can expect reflections or refractions. In refraction seismics we are interested in the critically refracted waves, and then only in the travel times. This means that we can only observe the traveltimes well if it is not masked by reflections and/or direct waves, which means that we must measure at a relatively large distance with respect to the depth of interest. This is different for reflection seismics. There, the reflections will always be masked by refractions and/or direct waves, but there are ways to enhance the reflections. What we are interested in is the arrival at relatively small offsets, thus distances of the sound source to the detector, which are small with respect to the depth we are interested in. + +Before discussing any more differences between the refraction method and the reflection method, we would like to discuss the amplitude effects at the boundary. Let us first introduce the acoustic impedance, which is the product of the density $\rho$ with the wave velocity $c,$ i.e., $\rho c.$ When a ray encounters a boundary, some wavefield energy will be reflected back, and some will be transmitted to the next layer. The amount of energy reflected back is characterized by the reflection coefficient $R$: + +$$ +R = \frac{\rho_2 c_2 - \rho_1 c_1}{\rho_2 c_2 + \rho_1 c_1} +$$ + +That this is the case, will be derived from basic physical principles in a later section on wave theory ([](./reflection_transmission.md)). Obviously, the larger the impedance contrast between two layers, the higher the amplitude of the reflected wave. Notice that it is the impedance contrast that determines whether energy is reflected back or not; it may happen that the velocities and densities are different between two layers, but that the impedances are (nearly) the same. In that case, we will not see a reflection. We can now state another difference between refraction and reflection seismics. With refraction seismics we are only interested in traveltimes of the waves, so this means that we are interested in contrasts in velocities. This is different in reflection seismics. Then we are interested in the amplitude of the waves, and we will only measure an amplitude if there is a contrast in acoustic impedance in the subsurface. + +Finally, we tabulate the most important differences between reflection and refraction seismic in {numref}`tab-diffref`. + +```{list-table} Important differences between refraction and reflection seismics. +:name: tab-diffref +:header-rows: 1 + +* - **REFRACTION SEISMICS** + - **REFLECTION SEISMICS** +* - Based on contrasts in: seismic wave speed $(c)$ + - Based on contrasts in: seismic wave impedances $(\rho c)$ +* - Material property determined: wave speed only + - Material properties determined: wave speed and wave impedance +* - Only traveltimes used + - Traveltimes and amplitudes used +* - Source-receiver distances large compared to investigation depth + - Source-receiver distances small compared to investigation depth +``` diff --git a/book/_toc.yml b/book/_toc.yml index 8b62075..4c35deb 100644 --- a/book/_toc.yml +++ b/book/_toc.yml @@ -106,6 +106,19 @@ parts: - file: 3_diffusion_fields/heat_equation_1d.md - file: 3_diffusion_fields/em_diffusion.md - file: 3_diffusion_fields/labs/week05_diffusion.md + - caption: Mechanical waves + chapters: + - file: 4_mechanical_waves/intro.md + sections: + - file: 4_mechanical_waves/basic_notions.md + - file: 4_mechanical_waves/snells_law.md + - file: 4_mechanical_waves/field_recordings.md + - file: 4_mechanical_waves/fundamental_laws.md + - file: 4_mechanical_waves/acoustic_waves.md + - file: 4_mechanical_waves/reflection_transmission.md + - file: 4_mechanical_waves/elastic_waves.md + - file: 4_mechanical_waves/appendix_acoustic_1d.md + - file: 4_mechanical_waves/appendix_strain_p_s.md - caption: Miscellaneous chapters: - file: references.md diff --git a/book/intro.md b/book/intro.md index 400ed64..2131c2f 100644 --- a/book/intro.md +++ b/book/intro.md @@ -7,12 +7,14 @@ title: ECTB2140 Fields and Waves 2026 :::{admonition} This book is under development :class: warning -The book is being written during the 2026-2027 run of ECTB2140, and the material is incomplete beyond the first chapter. +The book is being written during the 2026-2027 run of ECTB2140, and the material is incomplete. -- **Chapter 1, Gradient, divergence and curl**, is complete: the lecture notes and both computer labs. +- **Chapter 1, Gradient, divergence and curl**, is complete. - **Chapter 2, Potential fields**, is partly written. +- **Chapter 3, Diffusion fields**, is partly written. +- **Chapter 4, Mechanical waves**, is partly written. - Later chapters are not yet available here. -- The **Exercises** section in the sidebar is left over from the book template. It explains how TeachBooks works and is not course material. + Brightspace remains the authoritative source for the schedule, assessment and announcements. Pages here change between weeks, so download a notebook again rather than relying on a copy saved earlier. :::