Add the mechanical waves chapter and update diffusive fields to the 6 October notes

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.

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
 

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*.

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.