From e2ade02fde8a060e99b2406cb6d38c9877449924 Mon Sep 17 00:00:00 2001 From: ARS183 <1028762817@qq.com> Date: Wed, 7 Oct 2026 12:47:47 +0200 Subject: [PATCH 01/11] Add field-map and isosurface helpers to the lab module --- book/3_diffusion_fields/labs/fwtools.py | 156 +++++++++++++++++++++++- 1 file changed, 151 insertions(+), 5 deletions(-) diff --git a/book/3_diffusion_fields/labs/fwtools.py b/book/3_diffusion_fields/labs/fwtools.py index e354ecf..47b73ed 100644 --- a/book/3_diffusion_fields/labs/fwtools.py +++ b/book/3_diffusion_fields/labs/fwtools.py @@ -6,8 +6,10 @@ rates, Gaussian kernels -- rather than on plotting boilerplate. The self-check functions have the same names and behaviour as in the Lab 1 -and Lab 2 module. Everything here is one-dimensional in space: fields are -arrays of shape (nt, nx), one row per time. +and Lab 2 module. The one-dimensional helpers take fields of shape (nt, nx), +one row per time. ``show_map`` and ``show_isosurface`` take two- and +three-dimensional fields, indexed as ``np.meshgrid(..., indexing='ij')`` +builds them, so ``F[ix, iy]`` sits at ``(x[ix], y[iy])``. Requires: numpy, matplotlib, plotly. """ @@ -21,6 +23,7 @@ __all__ = [ "show_profiles", "show_spacetime", "show_spacetime_slider", "show_records", + "show_map", "show_isosurface", "check", "check_close", "check_abs", "check_scalar", ] @@ -35,7 +38,8 @@ def _fmt(v, fmt): def show_profiles(x, frames, values, *, value_name="t", value_fmt="{:.3g}", unit="", reference=None, reference_name="initial condition", xlabel="x [m]", ylabel="", title="", ylim=None, height=430, - ms_per_frame=90, colors=("#00A6D6", "#EC6842", "#7B61FF")): + ms_per_frame=90, colors=("#00A6D6", "#EC6842", "#7B61FF"), + log_x=False, log_y=False): """Profiles over time, with a play button and a draggable progress bar. ``frames`` is either one array of shape (len(values), nx), or a dict @@ -74,8 +78,11 @@ def show_profiles(x, frames, values, *, value_name="t", value_fmt="{:.3g}", fig.update_layout( title=title, height=height, margin=dict(l=60, r=20, t=50, b=30), - xaxis_title=xlabel, yaxis_title=ylabel, - yaxis_range=[lo - pad, hi + pad], legend=dict(x=0.99, xanchor="right", y=0.99), + xaxis=dict(title=xlabel, type="log" if log_x else "linear"), + yaxis=dict(title=ylabel, type="log" if log_y else "linear", + range=([np.log10(max(lo, 1e-300)), np.log10(max(hi, 1e-299))] + if log_y else [lo - pad, hi + pad])), + legend=dict(x=0.99, xanchor="right", y=0.99), sliders=[dict(active=0, steps=steps, len=0.82, x=0.18, y=0, pad=dict(t=50, b=8), currentvalue=dict(prefix=f"{value_name} = ", suffix=unit, font=dict(size=13)))], @@ -182,6 +189,145 @@ def show_records(t, records, *, t_label="t [s]", y_label="T [K]", title="", plt.show() +def _arrow_lines(X, Y, U, V, length): + """Unit arrows at (X, Y), as one broken line trace. NaN separates them.""" + n = np.hypot(U, V) + n = np.where(n > 0, n, 1.0) + u, v = U / n * length, V / n * length + x0, y0 = X - 0.5 * u, Y - 0.5 * v + x1, y1 = X + 0.5 * u, Y + 0.5 * v + ang = np.arctan2(v, u) + b = 0.34 * length + lx, ly = x1 - b * np.cos(ang - 0.42), y1 - b * np.sin(ang - 0.42) + rx, ry = x1 - b * np.cos(ang + 0.42), y1 - b * np.sin(ang + 0.42) + nan = np.full_like(x0, np.nan) + xs = np.stack([x0, x1, nan, lx, x1, rx, nan], axis=-1).ravel() + ys = np.stack([y0, y1, nan, ly, y1, ry, nan], axis=-1).ravel() + return xs, ys + + +def show_map(x, y, frames, values, *, value_name="t", value_fmt="{:.3g}", unit="", + x_label="x [m]", y_label="y [m]", c_label="", title="", signed=None, + arrows=None, arrow_every=8, zmin=None, zmax=None, per_frame_scale=False, + log_c=False, log_floor=1e-4, y_down=False, equal_aspect=True, + height=560, ms_per_frame=140): + """A field over a plane, with a play button and a draggable slider. + + ``frames`` has shape (len(values), len(x), len(y)): one map per slider + position, indexed the way ``np.meshgrid(..., indexing='ij')`` builds it. + ``arrows`` is an optional pair ``(U, V)`` of the same shape, drawn as unit + arrows every ``arrow_every`` points to show direction only. + ``per_frame_scale=True`` rescales the colour axis on every frame, which + shows the shape of a field whose amplitude changes by decades. + ``log_c=True`` colours by the base-10 logarithm instead, keeping the shape + of the pattern readable where the amplitude spans several decades; values + below ``log_floor`` times the largest are clipped. + """ + x, y = np.asarray(x, float), np.asarray(y, float) + frames = np.asarray(frames, float) + if log_c: + if signed: + raise ValueError("log_c colours a magnitude, so the field must not change sign") + signed = False + scale = np.nanmax(np.abs(frames), axis=(1, 2), keepdims=True) if per_frame_scale \ + else np.nanmax(np.abs(frames)) + frames = np.log10(np.maximum(np.abs(frames) / scale, log_floor)) + per_frame_scale = False + zmin, zmax = np.log10(log_floor), 0.0 + if signed is None: + signed = np.nanmin(frames) < 0 < np.nanmax(frames) + if per_frame_scale: + lim = [float(np.nanmax(np.abs(f))) or 1.0 for f in frames] + rng = [(-L, L) if signed else (0.0, L) for L in lim] + else: + hi = float(np.nanmax(np.abs(frames))) if signed else float(np.nanmax(frames)) + lo = -hi if signed else float(min(0.0, np.nanmin(frames))) + rng = [(lo if zmin is None else zmin, hi if zmax is None else zmax)] * len(values) + if log_c: + rng = [(zmin, zmax)] * len(values) + cs = "RdBu_r" if signed else "Inferno" + + def heat(k): + return go.Heatmap(z=frames[k].T, x=x, y=y, colorscale=cs, + zmin=rng[k][0], zmax=rng[k][1], zsmooth="best", + colorbar=dict(title=c_label)) + + quiver = None + if arrows is not None: + U, V = (np.asarray(a, float) for a in arrows) + sx, sy = x[::arrow_every], y[::arrow_every] + Xa, Ya = np.meshgrid(sx, sy, indexing="ij") + step = float(min(sx[1] - sx[0], sy[1] - sy[0])) * 0.85 + quiver = [_arrow_lines(Xa, Ya, U[k][::arrow_every, ::arrow_every], + V[k][::arrow_every, ::arrow_every], step) + for k in range(len(values))] + + fig = go.Figure(heat(0)) + if quiver is not None: + fig.add_trace(go.Scatter(x=quiver[0][0], y=quiver[0][1], mode="lines", + line=dict(color="white", width=1.3), + hoverinfo="skip", showlegend=False)) + data_k = (lambda k: [heat(k), go.Scatter(x=quiver[k][0], y=quiver[k][1])]) if quiver \ + else (lambda k: [heat(k)]) + fig.frames = [go.Frame(name=str(k), data=data_k(k), + traces=list(range(len(fig.data)))) for k in range(len(values))] + + still = dict(mode="immediate", frame=dict(duration=0, redraw=True), + transition=dict(duration=0)) + steps = [dict(method="animate", label=_fmt(v, value_fmt), args=[[str(k)], still]) + for k, v in enumerate(values)] + fig.update_layout( + title=title, height=height, margin=dict(l=60, r=20, t=50, b=30), + xaxis=dict(title=x_label, constrain="domain"), + yaxis=dict(title=y_label, autorange="reversed" if y_down else True, + scaleanchor="x" if equal_aspect else None, scaleratio=1), + sliders=[dict(active=0, steps=steps, len=0.80, x=0.18, y=0, pad=dict(t=50, b=8), + currentvalue=dict(prefix=f"{value_name} = ", suffix=unit, + font=dict(size=13)))], + updatemenus=[dict(type="buttons", direction="left", showactive=False, + x=0.0, y=0, xanchor="left", yanchor="top", pad=dict(t=55, b=8), + buttons=[ + dict(label="\u25b6 play", method="animate", + args=[None, dict(mode="immediate", fromcurrent=False, + frame=dict(duration=ms_per_frame, redraw=True), + transition=dict(duration=0))]), + dict(label="\u275a\u275a", method="animate", + args=[[None], still])])]) + fig.show() + + +def show_isosurface(x, y, z, V, *, levels=(0.1, 0.3, 0.6), opacity=0.3, c_label="", + title="", wire=None, wire_name="source", x_label="x [m]", + y_label="y [m]", z_label="z [m]", z_down=True, height=600): + """Nested translucent surfaces of constant ``V``, free to rotate and zoom. + + ``V`` has shape (len(x), len(y), len(z)), built with ``indexing='ij'``. + ``levels`` are fractions of the largest value in ``V``. Transparency keeps + the inner surfaces visible through the outer ones. ``wire`` is an optional + (xs, ys, zs) polyline, e.g. the loop that generated the field. + """ + x, y, z = (np.asarray(a, float) for a in (x, y, z)) + V = np.asarray(V, float) + X, Y, Z = np.meshgrid(x, y, z, indexing="ij") + vmax = float(np.nanmax(V)) + lo, hi = sorted(float(f) * vmax for f in (min(levels), max(levels))) + + fig = go.Figure(go.Isosurface( + x=X.ravel(), y=Y.ravel(), z=Z.ravel(), value=V.ravel(), + isomin=lo, isomax=hi, surface_count=len(levels), opacity=opacity, + colorscale="Inferno", caps=dict(x_show=False, y_show=False, z_show=False), + colorbar=dict(title=c_label))) + if wire is not None: + fig.add_trace(go.Scatter3d(x=wire[0], y=wire[1], z=wire[2], mode="lines", + line=dict(color="#00A6D6", width=6), name=wire_name)) + fig.update_layout( + title=title, height=height, margin=dict(l=0, r=0, t=50, b=0), + scene=dict(xaxis_title=x_label, yaxis_title=y_label, zaxis_title=z_label, + zaxis=dict(autorange="reversed" if z_down else True), + aspectmode="data")) + fig.show() + + # -------------------------------------------------------------------------- # Self-checks (same behaviour as the Lab 1 and Lab 2 module) # -------------------------------------------------------------------------- From 1ca2c237e55b58a8d6eba3d9a3d774951d5da981 Mon Sep 17 00:00:00 2001 From: ARS183 <1028762817@qq.com> Date: Wed, 7 Oct 2026 12:47:47 +0200 Subject: [PATCH 02/11] Add the Week 6 lab on diffusive electromagnetic fields --- .../labs/figures/ELoopwsxz.mp4 | Bin 0 -> 488203 bytes .../labs/week05_diffusion.md | 150 +--- .../labs/week06_em_diffusion.md | 668 ++++++++++++++++++ book/_toc.yml | 1 + 4 files changed, 670 insertions(+), 149 deletions(-) create mode 100644 book/3_diffusion_fields/labs/figures/ELoopwsxz.mp4 create mode 100644 book/3_diffusion_fields/labs/week06_em_diffusion.md diff --git a/book/3_diffusion_fields/labs/figures/ELoopwsxz.mp4 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The prefactors differ by the constant $I/\sigma$, which sets the amplitude and not the shape. Rock is non-magnetic, so $\mu=\mu_0$. - -The sheet sits in a uniform conductor and the solution is even in $z$, so reading $z$ as depth means ignoring the air above: an idealisation, not a consequence of the derivation. The approximation also neglects $\varepsilon\,\partial_t\boldsymbol{E}$, which is justified when $t$ is much longer than the charge relaxation time $\tau_r = \varepsilon/\sigma$. This is the one-dimensional idealisation behind time-domain electromagnetic sounding; a real system uses a finite loop and records $\partial_t B$. - -### Task 8 — when does the field arrive, and in what ground? - -Write the time at which $\lvert E_x\rvert$ at depth $z$ peaks, from $\partial_t\lvert E_x\rvert = 0$. Differentiating $\ln\lvert E_x\rvert$ is easier and has the same roots. **Predict** first whether a deeper receiver peaks earlier or later, and by what power of $z$. - -Then the same receiver in three grounds: seawater at 4 S/m, saturated sediments at $10^{-2}$ S/m, crystalline basement at $10^{-4}$ S/m. **Predict**, before running, how they rank by arrival time at 300 m and by what factor $t_p$ changes from basement to seawater. - -The second blank is the margin the diffusive approximation has left, $t_p/\tau_r$ with $\tau_r=\varepsilon/\sigma$. Write it in closed form, in terms of $z$ and $\sigma$, and watch what $\sigma$ does to it. - -```{code-cell} ipython3 -sigma, eps_r, I_s = 0.01, 10.0, 1.0 # sediments [S/m], relative permittivity, current [A/m] -tau_r = eps_r * epsilon_0 / sigma # charge relaxation time [s] -t_em = np.logspace(-6, -1, 3001) # [s], from about 100 relaxation times -_depths = [100.0, 300.0, 1000.0] # [m] - -def E_step(z, t, s=sigma): - return -I_s * np.sqrt(mu_0 / (4*np.pi*s*t)) * np.exp(-s*mu_0*z**2 / (4*t)) - -# Task 8 -- two blanks. `s` is the conductivity [S/m]; it defaults to the sediments above. -def t_peak_em(z, s=sigma): - """Time at which |E_x| at depth z peaks [s].""" - return ___ - -def peak_over_relax(z, s): - """How many charge relaxation times the peak waits: t_p / tau_r. Dimensionless.""" - return ___ - -# --- given, 1: the field soaking downward. Drag the bar, or press play --- -_zs = np.linspace(1.0, 2000.0, 400) # depth [m] -_tp = np.logspace(-6, -1.3, 60) # [s] -_prof = np.array([np.abs(E_step(_zs, tv)) / np.abs(E_step(1.0, tv)) for tv in _tp]) -fw.show_profiles(_zs, _prof, _tp, value_name="t", value_fmt="{:.2g}", unit=" s", - xlabel="depth z [m]", ylabel="|Ex| / its value at the surface", - title="The field diffuses into the Earth", ylim=(0.0, 1.05)) - -# --- given, 2: depth against time, with your arrival-time curve over it --- -fw.show_spacetime(_zs, _tp, _prof, t_label="t [s]", x_label="depth z [m]", - c_label=r"$|E_x|$ / surface value", signed=False, - curve=(t_peak_em(_zs), _zs), curve_label=r"your $t_p(z)$", deeper_down=True, - title=r"Deeper means later: $t_p \propto z^2$") - -# --- given, 3: what a receiver records at three depths --- -fw.show_records(t_em, {f"z = {z:g} m": np.abs(E_step(z, t_em)) / np.max(np.abs(E_step(z, t_em))) - for z in _depths}, - peaks={f"z = {z:g} m": (t_peak_em(z), 1.0) for z in _depths}, - t_label="t [s]", y_label=r"$|E_x|$ / peak value", - title=r"Step response at depth, $\sigma$ = 0.01 S/m") -for z in _depths: - print(f"z = {z:6.0f} m: t_p = {t_peak_em(z):.3g} s, t_p / tau_r = {t_peak_em(z)/tau_r:.2g}," - f" |E| at the peak = {abs(E_step(z, t_peak_em(z))):.3g} V/m") - -# --- self-check (leave this alone) --- -for z in _depths: - fw.check_scalar(f"t_p at {z:g} m against the peak of the record", t_peak_em(z), - t_em[np.argmax(np.abs(E_step(z, t_em)))], rtol=0.01, unit=" s") -fw.check_close("t_p / tau_r against t_p and tau_r built separately, over five decades of sigma", - [peak_over_relax(300.0, sv) for sv in (1e-4, 1e-3, 1e-2, 1e-1, 1.0)], - [t_peak_em(300.0, sv) / (eps_r * epsilon_0 / sv) - for sv in (1e-4, 1e-3, 1e-2, 1e-1, 1.0)], rtol=1e-6, - hint="tau_r = eps_r eps_0 / sigma shrinks as t_p grows, so the ratio carries sigma twice") -fw.check("a tenfold drop in conductivity costs a hundredfold in t_p / tau_r", - np.isclose(peak_over_relax(300.0, 1e-3) / peak_over_relax(300.0, 1e-4), 100.0, rtol=1e-9), - "the margin goes as sigma^2, not as sigma") -``` - -```{code-cell} ipython3 -# --- given: one receiver at 300 m, three grounds ----------------------------- -media = {"seawater, 4 S/m": 4.0, - "saturated sediments, 1e-2 S/m": 0.01, - "crystalline basement, 1e-4 S/m": 1e-4} -z_s = 300.0 # the receiver depth [m] -t_s = np.logspace(-8, 1, 900) # [s] - -fw.show_records(t_s, {k: np.abs(E_step(z_s, t_s, sv)) / np.max(np.abs(E_step(z_s, t_s, sv))) - for k, sv in media.items()}, - peaks={k: (t_peak_em(z_s, sv), 1.0) for k, sv in media.items()}, - t_label="t [s]", y_label="|E| / peak value", - title=f"The same receiver at {z_s:.0f} m, in three grounds") - -print(f"at z = {z_s:.0f} m\n") -print(f"{'ground':<30}{'t_p [s]':>11}{'tau_r [s]':>11}{'t_p / tau_r':>13}") -for k, sv in media.items(): - print(f"{k:<30}{t_peak_em(z_s, sv):>11.3g}{eps_r*epsilon_0/sv:>11.3g}" - f"{peak_over_relax(z_s, sv):>13.3g}") - -print("\na survey looking 500 m down, able to record for 10 ms:") -for k, sv in media.items(): - _tp = t_peak_em(500.0, sv) - print(f" {k:<30} t_p = {_tp:.3g} s {'inside' if _tp < 1e-2 else 'beyond'} the window") - -# --- self-check (leave this alone) --- -_tf = np.logspace(-8, 1, 20001) -for k, sv in media.items(): - fw.check_scalar(f"t_p at {z_s:.0f} m in {k.split(',')[0]}", t_peak_em(z_s, sv), - _tf[np.argmax(np.abs(E_step(z_s, _tf, sv)))], rtol=0.01, unit=" s") -fw.check("basement to seawater is a factor 4e4 in arrival time", - np.isclose(t_peak_em(z_s, 4.0) / t_peak_em(z_s, 1e-4), 4e4, rtol=1e-9), - "t_p is linear in sigma, and the two conductivities differ by 4e4") -``` - -:::{admonition} Solution — Task 8 -:class: dropdown - -```python -def t_peak_em(z, s=sigma): - return s * mu_0 * z**2 / 2 - -def peak_over_relax(z, s): - return s**2 * mu_0 * z**2 / (2 * eps_r * epsilon_0) -``` -::: - -:::{admonition} Depth maps onto arrival time, as $t_p\propto z^2$ -:class: important dropdown - -A layer ten times deeper answers a hundred times later: 63 µs at 100 m, 6.3 ms at 1 km, for $\sigma=0.01$ S/m. On the map your curve $t_p(z)$ runs along the shoulder of the diffusing field, because $|E_x|$ has fallen to $1/\sqrt e$ of its surface value exactly where $t = \sigma\mu z^2/2$. Time-domain electromagnetic soundings rest on this: the later part of a record senses the deeper Earth. In a more conductive Earth the arrival is later, since $t_p\propto\sigma$. The ratio $t_p/\tau_r$ is 7000 at 100 m and grows as $z^2$, so neglecting $\varepsilon\,\partial_t\boldsymbol{E}$ is well justified at these depths. The blurring of Task 5 applies here too: deep structure arrives late, and therefore blurred. - -The peak amplitude is 0.24 V/m at 100 m and 0.024 V/m at 1 km. A factor of ten in depth costs a factor of ten in signal and a factor of a hundred in time, which is the $1/\lvert x\rvert$ law of Task 4 with depth in place of distance along the wire. - -The chapter's three-dimensional impulse response peaks at $t_p=\sigma\mu r^2/6$, three times earlier than the $\sigma\mu z^2/2$ you found here, and its amplitude falls as $1/r^3$. Both follow from the same differentiation: a prefactor $t^{-p}$ gives $t_p=\sigma\mu r^2/(4p)$, with $p=1/2$ for the sheet and $p=3/2$ for the point. -::: - -:::{admonition} Conductivity sets the clock, and it sets the margin twice over -:class: important dropdown - -At 300 m the peak arrives after 0.23 s in seawater, 0.57 ms in sediments and 5.7 µs in basement. That is a factor $4\times10^{4}$ from basement to seawater, exactly the ratio of the two conductivities, because $t_p=\sigma\mu z^2/2$ is linear in $\sigma$. A conductive ground holds the field up; a resistive one lets it through almost at once. - -The margin behaves differently. $\tau_r=\varepsilon/\sigma$ grows as the ground becomes resistive while $t_p$ shrinks, so - -$$ \frac{t_p}{\tau_r} = \frac{\sigma^2\mu z^2}{2\varepsilon} $$ - -carries $\sigma$ twice and falls a hundredfold for every tenfold drop in conductivity: $10^{10}$ in seawater, $6.4\times10^{4}$ in sediments, and **6.4 in basement**. The diffusive approximation is comfortable in the first two and running out in the third, and it is the *resistive* ground that breaks it, not the conductive one. Dropping $\varepsilon\,\partial_t\boldsymbol{E}$ was never a statement about depth. It is a statement about how fast charge can relax, and in a resistive ground it cannot relax quickly. - -The survey question resolves the same way from the other end. Looking 500 m down with a 10 ms record: seawater peaks at 0.63 s, far beyond the window, so the target is never reached in time. Sediments peak at 1.6 ms, comfortably inside. Basement peaks at 16 µs, inside the window but so early that a real transmitter is still switching off, and the useful signal is buried in the turn-off transient. Conductivity, not instrument quality, decides which depths a time-domain sounding can reach. -::: diff --git a/book/3_diffusion_fields/labs/week06_em_diffusion.md b/book/3_diffusion_fields/labs/week06_em_diffusion.md new file mode 100644 index 0000000..b7f8e0a --- /dev/null +++ b/book/3_diffusion_fields/labs/week06_em_diffusion.md @@ -0,0 +1,668 @@ +--- +jupytext: + text_representation: + extension: .md + format_name: myst + format_version: 0.13 +kernelspec: + display_name: Python 3 (ipykernel) + language: python + name: python3 +mystnb: + # Workbook page: the task cells contain `___` blanks by design, so it must + # not be executed at build time. Readers run it themselves with Live Code. + execution_mode: 'off' +--- + +# Lab 6: Diffusive electromagnetic fields + +:::{admonition} Computer lab +:class: note + +A practical companion to the lecture on [electromagnetic diffusive field equations with a source](../em_diffusion.md). Each task states a physical question, gives the formula it rests on, and ends with a picture you can read the answer off. + +Everything you write is plain NumPy, and every line that implements an equation names it in a comment. The module `fwtools` only draws: `fw.show_map` (a field over a plane, with a play button), `fw.show_isosurface` (a rotatable 3-D view), `fw.show_profiles`, `fw.show_spacetime` and `fw.show_records`. Every slider runs in the browser. +::: + +## Learning objectives + +- **One kernel, four geometries.** A sheet, a wire, a point and a loop all produce $t^{-p}\exp(-\sigma\mu r^2/4t)$. Only $p$ changes, and it fixes when the field arrives. +- **Arrival time measures distance.** $t_p=\sigma\mu r^2/4p$ turns a recorded time into a depth, and the conductivity of the ground sets the clock. +- **The magnetic field becomes a potential field.** $\nabla\times\boldsymbol H=-\sigma E_z$ while the field diffuses, and both sides die together. +- **A source forgets its shape.** Once the diffusion distance passes the size of the loop, the field no longer knows the loop was square. + +--- + +## Part 0 — Setup + +Run this once. It contains no physics: it fetches two packages the browser lacks and locates `fwtools`. + +```{code-cell} ipython3 +import sys, pathlib + +import numpy as np +import matplotlib.pyplot as plt +from scipy.constants import mu_0, epsilon_0 +from scipy.special import erf + +# --- Live Code housekeeping, not part of the physics ----------------------- +try: + import plotly.io as pio +except ModuleNotFoundError: + print("Fetching plotly. A few seconds, and only the first time...") + import micropip + await micropip.install("plotly") + import plotly.io as pio + +try: + import nbformat # noqa: F401 +except ModuleNotFoundError: + import micropip, types + try: + await micropip.install("nbformat") + except Exception: + _nb = types.ModuleType("nbformat") + _nb.__version__ = "5.10.4" + sys.modules["nbformat"] = _nb + +for _p in (".", "book/3_diffusion_fields/labs"): + if (pathlib.Path(_p) / "fwtools.py").exists(): + sys.path.insert(0, _p) + break +try: + import fwtools as fw +except ModuleNotFoundError: + from pyodide.http import pyfetch + _r = await pyfetch("fwtools.py") + pathlib.Path("fwtools.py").write_bytes(await _r.bytes()) + import fwtools as fw + +pio.renderers.default = "plotly_mimetype+notebook" +mu = mu_0 # rock and water are non-magnetic +# --------------------------------------------------------------------------- +print("ready") +``` + +--- + +## Part 1 — One dimension: a current sheet + +A horizontal current sheet carrying $I$ per unit width is switched on at $t=0$. The field a distance $z$ from it is {eq}`eq:ExtH`, + +$$ E_x(z,t) = -I\sqrt{\frac{\mu}{4\pi\sigma t}}\exp\left(-\frac{\sigma\mu z^2}{4t}\right),\qquad t>0 . $$ + +The exponential is the kernel of the heat equation with $\kappa$ replaced by $1/(\sigma\mu)$, so the field soaks into the ground as heat spreads along a wire. The prefactor falls as $t^{-1/2}$. + +Every field in this lab has that form, $t^{-p}\exp(-\sigma\mu r^2/4t)$, with a different $p$. It is worth finding the peak time once, for any $p$. + +### Task 1 — when does the field arrive, and in what ground? + +Setting $\partial_t\ln|E|=0$ is easier than differentiating $|E|$ itself and gives the same answer. With a prefactor $t^{-p}$, + +$$ -\frac{p}{t} + \frac{\sigma\mu r^2}{4t^2} = 0 . $$ + +Solve it for $t$ and write the result once, as `t_peak(r, p, s)`. + +**Predict** before running: does a deeper receiver peak earlier or later, and by what power of $z$? Then the same receiver in three grounds, seawater at 4 S/m, saturated sediments at $10^{-2}$ S/m and crystalline basement at $10^{-4}$ S/m. Rank them by arrival time. + +The second blank is the margin the diffusive approximation still has. Dropping $\varepsilon\,\partial_t\boldsymbol E$ is safe only while the peak arrives long after the charge relaxation time $\tau_r=\varepsilon/\sigma$. Write $t_p/\tau_r$ for the sheet in closed form, in terms of $z$ and $\sigma$, and watch what $\sigma$ does to it. + +```{code-cell} ipython3 +sigma_sheet, eps_r, I_sheet = 0.01, 10.0, 1.0 # sediments [S/m], relative permittivity, [A/m] +tau_r = eps_r * epsilon_0 / sigma_sheet # charge relaxation time [s] + +def E_sheet(z, t, s=sigma_sheet): + """eq:ExtH -- the field of a current sheet switched on at t = 0 [V/m].""" + return -I_sheet * np.sqrt(mu / (4*np.pi*s*t)) * np.exp(-s*mu*z**2 / (4*t)) + +# Task 1 -- two blanks. +def t_peak(r, p, s): + """Time at which t**(-p) * exp(-s*mu*r**2/(4t)) is largest [s]. + Solve -p/t + s*mu*r**2/(4*t**2) = 0 for t.""" + return ___ + +def peak_over_relax(z, s): + """How many charge relaxation times the sheet's peak waits, t_p / tau_r. + The sheet has p = 1/2, and tau_r = eps_r*epsilon_0/s.""" + return ___ + +# --- given: the field soaking downward. Drag the bar, or press play --------- +z_grid = np.linspace(1.0, 2000.0, 400) # depth [m] +t_grid = np.logspace(-6, -1.3, 60) # [s] +soak = np.array([np.abs(E_sheet(z_grid, tv)) / np.abs(E_sheet(1.0, tv)) for tv in t_grid]) +fw.show_profiles(z_grid, soak, t_grid, value_name="t", value_fmt="{:.2g}", unit=" s", + xlabel="depth z [m]", ylabel="|Ex| / its value at the surface", + title="The field diffuses into the Earth", ylim=(0.0, 1.05)) + +# --- given: depth against time, with your arrival-time curve over it -------- +fw.show_spacetime(z_grid, t_grid, soak, t_label="t [s]", x_label="depth z [m]", + c_label=r"$|E_x|$ / surface value", signed=False, + curve=(t_peak(z_grid, 0.5, sigma_sheet), z_grid), + curve_label=r"your $t_p(z)$", deeper_down=True, + title=r"Deeper means later: does your curve follow the ridge?") + +# --- given: what a receiver records at three depths ------------------------ +t_rec = np.logspace(-6, -1, 3001) # [s] +depths = [100.0, 300.0, 1000.0] # [m] +fw.show_records(t_rec, {f"z = {z:g} m": np.abs(E_sheet(z, t_rec)) / np.max(np.abs(E_sheet(z, t_rec))) + for z in depths}, + peaks={f"z = {z:g} m": (t_peak(z, 0.5, sigma_sheet), 1.0) for z in depths}, + t_label="t [s]", y_label=r"$|E_x|$ / peak value", + title=r"Step response at depth, $\sigma$ = 0.01 S/m") + +fw.check_scalar("t_p at 300 m against the peak of the record", + t_peak(300.0, 0.5, sigma_sheet), + t_rec[np.argmax(np.abs(E_sheet(300.0, t_rec)))], rtol=0.01, unit=" s") +``` + +```{code-cell} ipython3 +# --- given: one receiver at 300 m, three grounds --------------------------- +media = {"seawater, 4 S/m": 4.0, + "saturated sediments, 1e-2 S/m": 0.01, + "crystalline basement, 1e-4 S/m": 1e-4} +z_s, t_s = 300.0, np.logspace(-8, 1, 900) + +fw.show_records(t_s, {k: np.abs(E_sheet(z_s, t_s, sv)) / np.max(np.abs(E_sheet(z_s, t_s, sv))) + for k, sv in media.items()}, + peaks={k: (t_peak(z_s, 0.5, sv), 1.0) for k, sv in media.items()}, + t_label="t [s]", y_label="|E| / peak value", + title=f"The same receiver at {z_s:.0f} m, in three grounds") + +print(f"at z = {z_s:.0f} m\n") +print(f"{'ground':<30}{'t_p [s]':>11}{'tau_r [s]':>11}{'t_p / tau_r':>13}") +for k, sv in media.items(): + print(f"{k:<30}{t_peak(z_s, 0.5, sv):>11.3g}{eps_r*epsilon_0/sv:>11.3g}" + f"{peak_over_relax(z_s, sv):>13.3g}") + +fw.check("a tenfold drop in conductivity costs a hundredfold in margin", + np.isclose(peak_over_relax(300.0, 1e-3) / peak_over_relax(300.0, 1e-4), 100.0, rtol=1e-9), + "the margin goes as sigma squared, not as sigma") +``` + +:::{admonition} Solution — Task 1 +:class: dropdown + +```python +def t_peak(r, p, s): + return s * mu * r**2 / (4 * p) + +def peak_over_relax(z, s): + return s**2 * mu * z**2 / (2 * eps_r * epsilon_0) +``` +::: + +:::{admonition} Depth maps onto arrival time, and conductivity sets the clock +:class: important dropdown + +A layer ten times deeper answers a hundred times later: 63 µs at 100 m, 6.3 ms at 1 km, for $\sigma=0.01$ S/m. Your curve runs along the shoulder of the diffusing field, because $|E_x|$ has fallen to $1/\sqrt e$ of its surface value exactly where $t=\sigma\mu z^2/2$. Time-domain electromagnetic soundings rest on this: the later part of a record senses the deeper Earth. + +At 300 m the peak arrives after 0.23 s in seawater, 0.57 ms in sediments and 5.7 µs in basement, a factor $4\times10^4$ from basement to seawater. That is exactly the ratio of the two conductivities, because $t_p=\sigma\mu z^2/2$ is linear in $\sigma$. A conductive ground holds the field up; a resistive one lets it through at once. + +The margin behaves differently. $\tau_r=\varepsilon/\sigma$ grows as the ground becomes resistive while $t_p$ shrinks, so + +$$ \frac{t_p}{\tau_r} = \frac{\sigma^2\mu z^2}{2\varepsilon} $$ + +carries $\sigma$ twice: $10^{10}$ in seawater, $6.4\times10^4$ in sediments, and **6.4** in basement. The diffusive approximation is comfortable in the first two and running out in the third, and it is the *resistive* ground that breaks it. Dropping $\varepsilon\,\partial_t\boldsymbol E$ was never a statement about depth. It is a statement about how fast charge can relax. +::: + +--- + +## Part 2 — Two dimensions: an infinite wire + +A long wire along $z$ carries a current switched on at $t=0$. Nothing depends on $z$, so the field lives in the $(x,y)$-plane: the electric field is $E_z$, perpendicular to that plane, and the magnetic field lies in it. This is the Transverse Electric mode of the lecture, and the chapter gives + +$$ E_z(\varrho,t) = \frac{\mu I}{4\pi t}\exp\left(-\frac{\sigma\mu\varrho^2}{4t}\right),\qquad \varrho=\sqrt{x^2+y^2}. $$ + +The kernel is the same as the sheet's. The prefactor is now $t^{-1}$, so $p=1$. + +### Task 2 — the field of the wire, and when it arrives + +Write $E_z$. Then **predict**, before running: at the same distance, does the wire's field peak earlier or later than the sheet's, and by what factor? + +```{code-cell} ipython3 +sigma_wire, I_wire = 3.0, 1.0 # seawater [S/m], wire current [A] + +# Task 2 -- one blank. +def E_wire(rho, t, s=sigma_wire): + """E_z of an infinitely long wire with a step switch-on current [V/m]. + Prefactor t**(-1), times the kernel exp(-s*mu*rho**2/(4t)).""" + return ___ + +# --- given: the field spreading out from the wire, on a logarithmic colour scale --- +n_w = 91 +x_w = np.linspace(-50.0, 50.0, n_w) # [m] +X_w, Y_w = np.meshgrid(x_w, x_w, indexing="ij") # F[ix, iy] sits at (x[ix], y[iy]) +rho_w = np.hypot(X_w, Y_w) +rho_w[rho_w == 0] = x_w[1] - x_w[0] # keep the wire out of the formula +t_w = np.logspace(-4, 0, 16) # [s] + +E_frames = np.array([E_wire(rho_w, tv) for tv in t_w]) +fw.show_map(x_w, x_w, E_frames, t_w, value_name="t", value_fmt="{:.2g}", unit=" s", + x_label="x [m]", y_label="y [m]", c_label="log10(Ez / max)", + log_c=True, per_frame_scale=True, title="The electric field spreads out from the wire") + +# --- given: three receivers, with your arrival times marked ---------------- +t_r = np.logspace(-6, 1, 3001) +dists = [0.36, 15.0, 30.0] # [m] +fw.show_records(t_r, {f"{d:g} m": E_wire(d, t_r) / np.max(E_wire(d, t_r)) for d in dists}, + peaks={f"{d:g} m": (t_peak(d, 1, sigma_wire), 1.0) for d in dists}, + t_label="t [s]", y_label=r"$E_z$ / peak value", + title=r"The wire, $\sigma$ = 3 S/m: close in, the field is already fading") + +print(f"the sheet and the wire at the same 15 m, in the same ground:") +print(f" sheet, p = 1/2 : t_p = {t_peak(15.0, 0.5, sigma_wire)*1e3:.3f} ms") +print(f" wire, p = 1 : t_p = {t_peak(15.0, 1.0, sigma_wire)*1e3:.3f} ms") + +_tp15 = t_peak(15.0, 1, sigma_wire) +fw.check_scalar("t_p at 15 m against the peak of the record", _tp15, + t_r[np.argmax(E_wire(15.0, t_r))], rtol=0.01, unit=" s") +fw.check_scalar("the peak value, against mu I / (4 pi t_p) times 1/e", E_wire(15.0, _tp15), + mu*I_wire/(4*np.pi*_tp15)*np.exp(-1.0), rtol=0.01, unit=" V/m") +``` + +:::{admonition} Solution — Task 2 +:class: dropdown + +```python +def E_wire(rho, t, s=sigma_wire): + return mu * I_wire / (4*np.pi*t) * np.exp(-s*mu*rho**2 / (4*t)) +``` +::: + +### Task 3 — the magnetic field climbs to its static value + +The wire's magnetic field circles it. Long after switch-on it must settle at the static result of the curl chapter, $\boldsymbol H = I\hat{\boldsymbol\varphi}/(2\pi\varrho)$. The chapter shows that it approaches that value through the same exponential that shapes $E_z$, with no power of $t$ in front: + +$$ \frac{|\boldsymbol H|}{|\boldsymbol H|_{\text{static}}} = \exp\left(-\frac{\sigma\mu\varrho^2}{4t}\right),\qquad |\boldsymbol H|_{\text{static}} = \frac{I}{2\pi\varrho}. $$ + +Write that ratio. It runs from 0 at switch-on to 1 at late time, so the picture shows how far the field has got, not how strong it is. + +Then move the conductivity slider on the second figure. **Predict** first: in which ground does the magnetic field reach its static value soonest? + +```{code-cell} ipython3 +# Task 3 -- one blank. +def H_over_static(rho, t, s=sigma_wire): + """|H| divided by its static value. Dimensionless, 0 at switch-on and 1 at late time.""" + return ___ + +# --- given: the magnetic field filling in, with its direction --------------- +H_frames = np.array([H_over_static(rho_w, tv) for tv in t_w]) +Hx_dir = np.broadcast_to(-Y_w / rho_w, (len(t_w),) + rho_w.shape) # the unit vector phi-hat +Hy_dir = np.broadcast_to( X_w / rho_w, (len(t_w),) + rho_w.shape) +fw.show_map(x_w, x_w, H_frames, t_w, arrows=(Hx_dir, Hy_dir), arrow_every=10, + value_name="t", value_fmt="{:.2g}", unit=" s", zmin=0.0, zmax=1.0, + x_label="x [m]", y_label="y [m]", c_label="|H| / static value", + title="The magnetic field is already circular. It is only filling in") + +# --- given: how far it still has to go, in three grounds ------------------- +t_h = np.logspace(-5, 2, 400) +sigmas = np.array([0.01, 0.1, 1.0, 3.0, 10.0]) # [S/m] +gap = {f"{d:g} m": np.array([1.0 - H_over_static(d, t_h, sv) for sv in sigmas]) + for d in (15.0, 30.0)} +fw.show_profiles(t_h, gap, sigmas, value_name="σ", value_fmt="{:.3g}", unit=" S/m", + xlabel="t [s]", ylabel="1 − |H| / static value", log_x=True, log_y=True, + title="The distance still to go falls as 1/t, and a conductive ground is slower") + +fw.check("the ratio climbs from 0 at switch-on to 1 at late time", + H_over_static(15.0, 1e-5) < 1e-6 and abs(H_over_static(15.0, 1e2) - 1.0) < 1e-3, + "it is exp(-a/t), not 1 - exp(-a/t): at t -> 0 there is no field yet") +``` + +:::{admonition} Solution — Task 3 +:class: dropdown + +```python +def H_over_static(rho, t, s=sigma_wire): + return np.exp(-s*mu*rho**2 / (4*t)) +``` +::: + +### Task 4 — while it diffuses, the magnetic field has curl + +The second of the chapter's two-dimensional equations is $\partial_xH_y-\partial_yH_x+\sigma E_z=0$ away from the wire. The curl of $\boldsymbol H$ is not an abstraction here: it is the electric field, times $-\sigma$. As $E_z$ dies the curl dies with it, and the magnetic field becomes a potential field. + +Write the $z$-component of the curl. The two derivatives of each component are already computed for you. + +```{code-cell} ipython3 +t_c = 1e-3 # one instant [s] +dx = x_w[1] - x_w[0] +H_mag = I_wire / (2*np.pi*rho_w) * H_over_static(rho_w, t_c) +Hx_c, Hy_c = -Y_w/rho_w * H_mag, X_w/rho_w * H_mag # [A/m] + +dHx_dx, dHx_dy = np.gradient(Hx_c, dx, dx) # indexing='ij', so [0] is d/dx +dHy_dx, dHy_dy = np.gradient(Hy_c, dx, dx) + +# Task 4 -- one blank: the z-component of the curl of H. +curl_z = ___ + +# --- given: the curl beside the electric field it equals ------------------- +shown = rho_w > 8.0 # H goes as 1/rho, so the grid cannot follow it at the wire +fw.show_map(x_w, x_w, np.array([np.where(shown, curl_z, np.nan), + np.where(shown, -sigma_wire*E_wire(rho_w, t_c), np.nan)]), + ["curl of H", "minus sigma times E_z"], value_name="showing", value_fmt="{}", + x_label="x [m]", y_label="y [m]", c_label="A/m²", + title=f"Two ways to the same picture, at t = {t_c*1e3:g} ms") + +# The comparison is made on the outer ring, where a 1.1 m grid resolves 1/rho. +ring = (rho_w > 20.0) & (rho_w < 45.0) +fw.check_close("the curl of H against minus sigma E_z, away from the wire", + curl_z[ring], -sigma_wire*E_wire(rho_w, t_c)[ring], rtol=0.05, + hint="eq:TEEy with no source: d_x H_y - d_y H_x + sigma E_z = 0") +``` + +:::{admonition} Solution — Task 4 +:class: dropdown + +```python +curl_z = dHy_dx - dHx_dy +``` +::: + +:::{admonition} The wire answers twice as fast as the sheet, and the curl dies with the electric field +:class: important dropdown + +At 15 m in seawater the sheet peaks at 0.42 ms and the wire at 0.21 ms, a factor of two, because $t_p=\sigma\mu r^2/4p$ and $p$ went from $1/2$ to $1$. A steeper prefactor pulls the peak earlier: the field is being drained faster, so the moment at which arrival stops winning over decay comes sooner. + +The two magnetic pictures say different things. The map never changes shape, because the direction $\hat{\boldsymbol\varphi}$ is circular from the first instant and only the amount fills in. What takes time is the *amount*, and the second figure shows how far there is still to go. Expanding the exponential for $t\gg\sigma\mu\varrho^2/4$, + +$$ 1 - \exp\left(-\frac{\sigma\mu\varrho^2}{4t}\right) \approx \frac{\sigma\mu\varrho^2}{4t}, $$ + +so the gap closes as $1/t$, which is the straight line of slope $-1$ on the logarithmic axes. A conductive ground is slower, and doubling the distance costs a factor of four in waiting. + +The curl map is the electric field map, rescaled. That is {eq}`eq:TEEy` read as a picture, and it is the sharpest statement of what makes a diffusive field different from a static one: the static magnetic field of the curl chapter is curl-free outside the wire, and this one is not, for exactly as long as there is an electric field to drive it. Let $t$ run on and both die, which is why the chapter says the magnetic field becomes a potential field at infinite time. +::: + +--- + +## Part 3 — Three dimensions, and one law for all of them + +A point source in three dimensions has the impulse response {eq}`eq:G3D`, + +$$ G(R,t) = \frac{(\sigma\mu)^{1/2}}{[4\pi t]^{3/2}}\exp\left(-\frac{\sigma\mu R^2}{4t}\right). $$ + +Same kernel again. The prefactor is $t^{-3/2}$, so $p=3/2$. + +### Task 5 — the ladder + +Write $G$. Your `t_peak` already covers it: the three geometries differ only in $p$, so at the same distance their peak times are in a fixed ratio. **Predict** that ratio before you run the last figure. + +```{code-cell} ipython3 +sigma_pt = 0.01 # sediments [S/m] + +# Task 5 -- one blank. +def G_point(R, t, s=sigma_pt): + """eq:G3D -- the impulse response in three dimensions [1/m].""" + return ___ + +# --- given: a rotatable view of the field at one instant ------------------- +# Change t_show and run the cell again to watch the shell move outward. +t_show = 1.0e-3 # [s] +g_xy = np.linspace(-400.0, 400.0, 32) # [m] +g_z = np.linspace(0.0, 800.0, 32) # depth [m] +GX, GY, GZ = np.meshgrid(g_xy, g_xy, g_z, indexing="ij") +G_vol = G_point(np.sqrt(GX**2 + GY**2 + GZ**2), t_show) +fw.show_isosurface(g_xy, g_xy, g_z, G_vol, levels=(0.08, 0.25, 0.6), opacity=0.32, + c_label="G", z_label="depth z [m]", + title=f"The point source at t = {t_show*1e3:g} ms. Drag to rotate") + +# --- given: the three geometries at the same distance, same ground --------- +r0 = 300.0 # [m] +t_l = np.logspace(-6, -1, 4001) +ladder = {"sheet, p = 1/2": np.abs(E_sheet(r0, t_l, sigma_pt)), + "wire, p = 1 ": E_wire(r0, t_l, sigma_pt), + "point, p = 3/2": G_point(r0, t_l)} +fw.show_records(t_l, {k: v / np.max(v) for k, v in ladder.items()}, + peaks={k: (t_peak(r0, p, sigma_pt), 1.0) + for k, p in zip(ladder, (0.5, 1.0, 1.5))}, + t_label="t [s]", y_label="field / its own peak", + title=f"The same {r0:.0f} m in the same ground, three geometries") + +print(f"{'geometry':<12}{'p':>6}{'t_p [ms]':>12}{'ratio':>8}") +for name, p in zip(("sheet", "wire", "point"), (0.5, 1.0, 1.5)): + tp = t_peak(r0, p, sigma_pt) + print(f"{name:<12}{p:>6}{tp*1e3:>12.3f}{tp/t_peak(r0, 1.5, sigma_pt):>8.1f}") + +fw.check_scalar("the point source peaks at sigma mu R^2 / 6", + t_peak(r0, 1.5, sigma_pt), t_l[np.argmax(G_point(r0, t_l))], + rtol=0.01, unit=" s") +_R = np.linspace(0.5, 6000.0, 60000) # a radial grid [m] +fw.check_scalar("the integral of G over all space, which fixes the constant in front", + np.sum(G_point(_R, 1e-3) * 4*np.pi*_R**2) * (_R[1] - _R[0]), + 1.0 / (sigma_pt*mu), rtol=0.01, unit=" m/S") +``` + +:::{admonition} Solution — Task 5 +:class: dropdown + +```python +def G_point(R, t, s=sigma_pt): + return np.sqrt(s*mu) / (4*np.pi*t)**1.5 * np.exp(-s*mu*R**2 / (4*t)) +``` +::: + +:::{admonition} Six, three, two +:class: important dropdown + +$$ t_p = \frac{\sigma\mu r^2}{4p}:\qquad \frac{\sigma\mu r^2}{2},\quad \frac{\sigma\mu r^2}{4},\quad \frac{\sigma\mu r^2}{6} $$ + +for $p=\tfrac12,1,\tfrac32$. The three peak times are in the ratio **6 : 3 : 2**, whatever the distance and whatever the ground, because the kernel is identical and only the prefactor differs. At 300 m in sediments that is 0.57 ms, 0.28 ms and 0.19 ms. + +The reason is worth saying in words. The exponential is the arrival: it rises from nothing as the diffusing field reaches $r$, and it does not care about the geometry. The prefactor is the decay: it is how fast the energy is spreading out, over a line, a cylinder or a sphere. The peak is where the two balance, so a geometry that spreads faster peaks earlier. The sphere spreads fastest, and answers first. + +The three-dimensional amplitude falls as $1/R^3$: double the distance and the peak is eight times smaller, against four for the wire and two for the sheet. Volume is expensive. +::: + +--- + +## Part 4 — A loop on the surface + +A real transmitter is a loop, not an infinite wire. A square loop of side $L_x=L_y=100$ m lies on the surface and its current is switched on at $t=0$. Integrating the three-dimensional Green's function along the four segments gives {eq}`eq:Exloop` and {eq}`eq:Eyloop`: + +$$ E_y(\boldsymbol r,t) = -\frac{\mu}{8\pi t}\left\{\exp\left(-\frac{x_m^2+z^2}{D^2}\right) - \exp\left(-\frac{x_p^2+z^2}{D^2}\right)\right\}\left[\mathrm{erf}\left(\frac{y_p}{D}\right) - \mathrm{erf}\left(\frac{y_m}{D}\right)\right] $$ + +with $x_p=x+L_x/2$, $x_m=x-L_x/2$, and the **diffusion distance** + +$$ D = \sqrt{\frac{4t}{\sigma\mu}} , $$ + +which is how far the field has spread by time $t$. The two exponentials are the two segments that run along $y$, at $x=\pm L_x/2$. They carry opposite currents, so they enter with opposite signs, and that difference is the only part you have to write. + +### Task 6 — the loop forgets its shape + +Fill in the brace of $E_y$. Then read three things off the pictures. + +1. Run the time slider on the $(x,y)$ view. At which $D$ does the square stop looking square? Compare $D$ with $L_x$. +2. Move the depth slider on the third figure. **Predict first** what going deeper does to the picture, then check. The answer is not what most people expect. +3. Rotate the 3-D view. + +```{code-cell} ipython3 +sigma_loop, Lx, Ly = 0.3, 100.0, 100.0 # [S/m], loop sides [m] + +def D_of(t, s=sigma_loop): + """The diffusion distance [m]: how far the field has spread by time t.""" + return np.sqrt(4*t / (s*mu)) + +def Ex_loop(x, y, z, t, s=sigma_loop): + """eq:Exloop -- given, as the pattern to follow [V/m].""" + D = D_of(t, s) + xp, xm = x + Lx/2, x - Lx/2 # to the two segments that run along y + yp, ym = y + Ly/2, y - Ly/2 # to the two segments that run along x + return (-mu / (8*np.pi*t) + * (np.exp(-(yp**2+z**2)/D**2) - np.exp(-(ym**2+z**2)/D**2)) + * (erf(xp/D) - erf(xm/D))) + +# Task 6 -- one blank. eq:Eyloop is eq:Exloop with x and y exchanged, except +# that the two exponentials come in the opposite order, because the two +# segments at x = +-Lx/2 carry opposite currents. +def Ey_loop(x, y, z, t, s=sigma_loop): + """eq:Eyloop [V/m].""" + D = D_of(t, s) + xp, xm = x + Lx/2, x - Lx/2 + yp, ym = y + Ly/2, y - Ly/2 + return ___ + +def E_loop(x, y, z, t, s=sigma_loop): + """Both horizontal components, as a pair.""" + return Ex_loop(x, y, z, t, s), Ey_loop(x, y, z, t, s) + +# --- given: the vertical plane through the middle of the loop -------------- +n_L = 91 +x_L = np.linspace(-300.0, 300.0, n_L) # [m] +z_L = np.linspace(0.0, 300.0, n_L) +XZ_x, XZ_z = np.meshgrid(x_L, z_L, indexing="ij") +t_L = np.logspace(-5, -1, 16) # [s] + +xz = np.array([E_loop(XZ_x, 0.0, XZ_z, tv)[1] for tv in t_L]) +fw.show_map(x_L, z_L, xz, t_L*1e3, value_name="t", value_fmt="{:.3g}", unit=" ms", + x_label="x [m]", y_label="depth z [m]", c_label="Ey [V/m]", + per_frame_scale=True, y_down=True, equal_aspect=False, + title="Two lobes of opposite sign, over the two wires, moving apart and down") +``` + +The same field, finely sampled, from the lecturer's own script. It runs over 200 instants instead of the 16 on your slider. + +```{video} figures/ELoopwsxz.mp4 +:width: 80% +``` + +```{code-cell} ipython3 +# --- given: the plane at 10 m depth, seen from above. Arrows give direction --- +XY_x, XY_y = np.meshgrid(x_L, x_L, indexing="ij") +EE = [E_loop(XY_x, XY_y, 10.0, tv) for tv in t_L] +mag = np.array([np.hypot(ex, ey) for ex, ey in EE]) +fw.show_map(x_L, x_L, mag, t_L*1e3, + arrows=(np.array([e[0] for e in EE]), np.array([e[1] for e in EE])), + arrow_every=10, log_c=True, per_frame_scale=True, y_down=True, + value_name="t", value_fmt="{:.3g}", unit=" ms", + x_label="x [m]", y_label="y [m]", c_label="log10(|E| / max)", + title="At z = 10 m: watch the square become a circle") + +print(f"{'t [ms]':>9}{'D [m]':>9}{'D / Lx':>9}") +for tv in (1e-5, 3e-5, 1e-4, 3e-4, 1e-3, 5e-3): + print(f"{tv*1e3:>9.3g}{D_of(tv):>9.1f}{D_of(tv)/Lx:>9.2f}") +``` + +```{code-cell} ipython3 +# --- given: the same instant, at a depth you choose ------------------------- +t_fix = 3e-4 # [s] +z_try = np.array([0.0, 50.0, 100.0, 200.0, 400.0]) # depth [m] +deep = np.array([np.hypot(*E_loop(XY_x, XY_y, zv, t_fix)) for zv in z_try]) +fw.show_map(x_L, x_L, deep, z_try, value_name="z", value_fmt="{:.0f}", unit=" m", + log_c=True, per_frame_scale=False, y_down=True, + x_label="x [m]", y_label="y [m]", c_label="log10(|E| / max)", + title=f"The same instant, t = {t_fix*1e3:g} ms, at five depths") + +print(f"at t = {t_fix*1e3:g} ms the diffusion distance is D = {D_of(t_fix):.0f} m\n") +print(f"{'depth [m]':>11}{'|E| / its surface value':>26}") +for zv in z_try: + f = np.exp(-zv**2 / D_of(t_fix)**2) + print(f"{zv:>11.0f}{f:>26.4f}") +``` + +```{code-cell} ipython3 +# --- given: the loop in three dimensions. Drag to rotate -------------------- +t_3d = 3e-5 # [s] +g3 = np.linspace(-180.0, 180.0, 36) # [m] +gz3 = np.linspace(0.0, 180.0, 36) # depth [m] +L3x, L3y, L3z = np.meshgrid(g3, g3, gz3, indexing="ij") +E3x, E3y = E_loop(L3x, L3y, L3z, t_3d) +loop_wire = ([-Lx/2, Lx/2, Lx/2, -Lx/2, -Lx/2], + [-Ly/2, -Ly/2, Ly/2, Ly/2, -Ly/2], [0, 0, 0, 0, 0]) +fw.show_isosurface(g3, g3, gz3, np.hypot(E3x, E3y), levels=(0.06, 0.18, 0.45), opacity=0.3, + wire=loop_wire, wire_name="the loop", c_label="|E| [V/m]", + z_label="depth z [m]", + title=f"t = {t_3d*1e3:g} ms, so D = {D_of(t_3d):.0f} m. The loop is still square") + +fw.check("the field opposes the source current over the x = +Lx/2 wire", + E_loop(Lx/2, 0.0, 10.0, 3e-5)[1] < 0 < E_loop(-Lx/2, 0.0, 10.0, 3e-5)[1], + "the current there runs in +y, so an opposing field has E_y < 0") +``` + +:::{admonition} Solution — Task 6 +:class: dropdown + +```python +def Ey_loop(x, y, z, t, s=sigma_loop): + D = D_of(t, s) + xp, xm = x + Lx/2, x - Lx/2 + yp, ym = y + Ly/2, y - Ly/2 + return (-mu / (8*np.pi*t) + * (np.exp(-(xm**2+z**2)/D**2) - np.exp(-(xp**2+z**2)/D**2)) + * (erf(yp/D) - erf(ym/D))) +``` +::: + +:::{admonition} Time changes the shape. Depth does not +:class: important dropdown + +**The square survives while $D$ is small.** At $t=0.01$ ms, $D=10$ m against a 100 m loop and the four wires are separate bright lines. By $t=0.1$ ms, $D=33$ m, and the pattern is already round. The loop stops looking square at about $D\approx L_x/3$, which is $t\approx\sigma\mu L_x^2/36$, here 0.1 ms. After that no measurement can tell this loop from a round one of the same area. A source forgets its shape once the field has diffused further than the source is big, and this is the same blurring that merged two Gaussians in the last lab. + +**Depth only dims it.** Put $z$ into {eq}`eq:Eyloop` and the depth appears as $\exp[-(x_m^2+z^2)/D^2]$, which factors into $\exp(-z^2/D^2)$ times the surface expression, identically for both components. So + +$$ \boldsymbol E(x,y,z,t) = \exp\left(-\frac{z^2}{D^2}\right)\boldsymbol E(x,y,0,t). $$ + +Going deeper multiplies the whole picture by one number. It does not blur it, does not rotate it, does not change which way the arrows point. The five frames of the depth figure are the same image at five brightnesses, and the printed column is that one factor. + +That factor is the whole of depth sounding. At $t=0.3$ ms, $D=56$ m, so 50 m down you still have 46% of the surface field and 200 m down you have $3.5\times10^{-6}$ of it. You can only see about as deep as $D$, and $D$ grows as $\sqrt t$. To look twice as deep you must wait four times as long, which is Part 1's $t_p\propto z^2$ arriving from the other direction. + +**The arrows oppose the current.** Over the wire at $x=+L_x/2$ the current runs in $+\hat{\boldsymbol y}$ and $E_y$ is negative. The switched-on current drives a field that opposes it, which is Lenz's law seen in a conductor. +::: + +--- + +## Part 5 — Designing a sounding + +You now have everything a time-domain electromagnetic survey rests on. This part asks you to put it together. Answer the four questions first, from the formulas and not from a new calculation, then run the cell. + +1. A target lies 400 m down in ground of conductivity $10^{-2}$ S/m. Roughly when does the response from that depth arrive? Which of Part 1's three formulas did you use, and does the choice matter at the factor-of-two level? +2. Your recording window is 10 µs to 10 ms. Which of the three grounds of Part 1 puts a 400 m target inside that window? +3. You have two loops, 50 m and 400 m on a side. The target is 400 m down. Which do you choose, and what decides it: the loop size, or the time you record for? +4. The field at the target is $\exp(-z^2/D^2)$ of its surface value. Below which amplitude would you say the target is not being illuminated at all, and what does that make the deepest usable depth? + +```{code-cell} ipython3 +# --- given: the window, the ground and the depth, all on one picture ------- +z_target = 400.0 # [m] +t_win = (1e-5, 1e-2) # the recording window [s] +t_d = np.logspace(-7, 1, 2001) + +fig, ax = plt.subplots(figsize=(8.2, 3.6)) +for k, (name, sv) in enumerate(media.items()): + ax.plot(t_d, np.exp(-z_target**2 * sv * mu / (4*t_d)), color=f"C{k}", lw=2, label=name) + ax.axvline(t_peak(z_target, 0.5, sv), color=f"C{k}", ls=":", lw=1.2) +ax.axvspan(*t_win, color="0.85", zorder=0, label="recording window") +ax.axhline(1/np.e, color="k", lw=0.8, ls="--") +ax.text(1.3e-7, 1/np.e*1.1, "z = D", fontsize=8) +ax.set_xscale("log"); ax.set_xlabel("t [s]") +ax.set_ylabel(f"illumination at {z_target:.0f} m, exp(-z²/D²)") +ax.set_title("Which ground puts a 400 m target inside the window?") +ax.grid(alpha=0.3); ax.legend(fontsize=8, loc="upper left") +plt.show() + +print(f"a target at {z_target:.0f} m, window {t_win[0]*1e6:.0f} us to {t_win[1]*1e3:.0f} ms\n") +print(f"{'ground':<30}{'t_p [s]':>11}{'in window?':>12}{'illum. at 10 ms':>18}") +for name, sv in media.items(): + tp = t_peak(z_target, 0.5, sv) + lit = np.exp(-z_target**2 * sv * mu / (4*t_win[1])) + print(f"{name:<30}{tp:>11.3g}{'yes' if t_win[0] < tp < t_win[1] else 'no':>12}{lit:>18.3g}") +``` + +:::{admonition} Answers +:class: dropdown + +**1.** $t_p=\sigma\mu z^2/2 = 1.0$ ms for the sheet. The wire gives 0.50 ms and the point source 0.34 ms. The choice changes the answer by a factor of three at most, and a survey design is not accurate to a factor of three anyway, so any of them sets the scale. What matters is $t\sim\sigma\mu z^2$, not the number in the denominator. + +**2.** Sediments, at 1.0 ms, sit comfortably inside. Seawater peaks at 0.40 s, forty times beyond the end of the window: the window closes long before the response from 400 m arrives. Basement peaks at 10 µs, at the very start of the window, where a real transmitter is still switching off and the useful signal is buried in the turn-off transient. Only the middle ground is workable, and that is typical. + +**3.** The time you record for. Depth of investigation is set by $D=\sqrt{4t/\sigma\mu}$, which contains no loop size at all. The loop controls how much signal you get and how far the near-field pattern reaches, not how deep you see. A bigger loop is still worth having, because the amplitude you are fighting to detect is tiny, but you cannot buy depth with it. You buy depth with late time and a quiet receiver. + +**4.** Any threshold in the range 1% to 10% gives nearly the same rule, because the Gaussian is so steep. Taking $1/e$ gives $z_{\max}=D=\sqrt{4t/\sigma\mu}$ exactly, which is the standard definition of the diffusion depth. With the 10 ms end of the window in sediments, $D=564$ m, so a 400 m target is at $\exp(-0.50)=0.60$ of the surface field and is well illuminated. In seawater the same 10 ms gives $D=28$ m and the target is at $\exp(-200)$, which is nothing at all. +::: + +:::{admonition} What this lab was about +:class: important dropdown + +One kernel, $\exp(-\sigma\mu r^2/4t)$, appeared in every part. It came from the heat equation last week with $\kappa$ replaced by $1/(\sigma\mu)$, and the only thing that changed from a sheet to a wire to a point was the power of $t$ in front, which is a statement about geometry and not about electromagnetism. + +From that one kernel came three results a geophysicist uses directly. Arrival time measures distance, $t_p=\sigma\mu r^2/4p$. Depth of investigation is the diffusion distance, $z\approx D=\sqrt{4t/\sigma\mu}$, and it is bought with time rather than with equipment. And a source is blurred into a point once $D$ exceeds its size, which is why the shape of a transmitter stops mattering, and also why deep structure is always seen blurred. + +The magnetic field made the fourth point. For as long as the electric field is there, $\nabla\times\boldsymbol H=-\sigma E_z$ is not zero, and the magnetic field is not a potential field. It becomes one only in the limit, which is where the magnetostatics of the earlier chapters lives. +::: diff --git a/book/_toc.yml b/book/_toc.yml index 4c35deb..d873a49 100644 --- a/book/_toc.yml +++ b/book/_toc.yml @@ -106,6 +106,7 @@ parts: - file: 3_diffusion_fields/heat_equation_1d.md - file: 3_diffusion_fields/em_diffusion.md - file: 3_diffusion_fields/labs/week05_diffusion.md + - file: 3_diffusion_fields/labs/week06_em_diffusion.md - caption: Mechanical waves chapters: - file: 4_mechanical_waves/intro.md From e4452ff588080050a457de8f7a16829ed7a5d855 Mon Sep 17 00:00:00 2001 From: ARS183 <1028762817@qq.com> Date: Wed, 7 Oct 2026 14:37:19 +0200 Subject: [PATCH 03/11] Add model-setup diagrams to the Week 6 lab --- .../labs/figures/make_week06_figures.py | 241 +++ .../labs/figures/setup_ladder.svg | 1423 +++++++++++++++++ .../labs/figures/setup_loop.svg | 1342 ++++++++++++++++ .../labs/figures/setup_sheet.svg | 1242 ++++++++++++++ .../labs/figures/setup_wire.svg | 1130 +++++++++++++ 5 files changed, 5378 insertions(+) create mode 100644 book/3_diffusion_fields/labs/figures/make_week06_figures.py create mode 100644 book/3_diffusion_fields/labs/figures/setup_ladder.svg create mode 100644 book/3_diffusion_fields/labs/figures/setup_loop.svg create mode 100644 book/3_diffusion_fields/labs/figures/setup_sheet.svg create mode 100644 book/3_diffusion_fields/labs/figures/setup_wire.svg diff --git a/book/3_diffusion_fields/labs/figures/make_week06_figures.py b/book/3_diffusion_fields/labs/figures/make_week06_figures.py new file mode 100644 index 0000000..e13f41f --- /dev/null +++ b/book/3_diffusion_fields/labs/figures/make_week06_figures.py @@ -0,0 +1,241 @@ +"""Regenerate the four model-setup schematics of the Week 6 lab. + +Run: python make_week06_figures.py + +Same palette and conventions as make_figures.py: transparent SVG, grey ink so the +theme can invert it in dark mode, TU Delft cyan for the field under study and +orange for the source. Depth runs downward throughout, as it does on every figure +in the chapter. Long captions go in figure coordinates, so they never fight the +equal aspect ratio the circular panels need. +""" +import matplotlib +matplotlib.use("Agg") +import numpy as np +import matplotlib.pyplot as plt +from matplotlib.patches import Rectangle, FancyArrowPatch, Circle, Polygon, Arc + +INK, GND, SRC, FLD = "#6b7280", "#8a9199", "#EC6842", "#00A6D6" + + +def arrow(ax, p0, p1, color=SRC, lw=2.0, scale=13, z=5): + ax.add_patch(FancyArrowPatch(p0, p1, arrowstyle="-|>", mutation_scale=scale, + color=color, lw=lw, zorder=z, shrinkA=0, shrinkB=0)) + + +def bare(ax, xlim, ylim, equal=False): + ax.set_xlim(*xlim); ax.set_ylim(*ylim) + ax.set_xticks([]); ax.set_yticks([]) + for s in ax.spines.values(): + s.set_visible(False) + if equal: + ax.set_aspect("equal") + + +def caption(fig, text, y=0.035, size=8.5): + fig.text(0.5, y, text, ha="center", va="bottom", color=INK, fontsize=size) + + +# --------------------------------------------------------------------------- +# Part 1: an infinite current sheet, seen edge-on +# --------------------------------------------------------------------------- +def sheet(): + fig, ax = plt.subplots(figsize=(6.8, 3.1)) + ax.add_patch(Rectangle((-1.0, -1.0), 2.0, 1.0, fc=GND, ec="none", alpha=0.16)) + ax.text(0.97, -0.96, "conductor $\\sigma,\\ \\mu$", ha="right", va="bottom", + color=INK, fontsize=9) + + # the sheet itself + ax.plot([-1.0, 1.0], [0, 0], color=SRC, lw=3, solid_capstyle="butt", zorder=4) + for x in np.linspace(-0.88, 0.72, 7): + arrow(ax, (x, 0), (x + 0.16, 0), scale=9, lw=1.3) + ax.text(0.0, 0.07, "current sheet at $z=0$: current $I$ per unit width along $x$," + " infinite in $x$ and $y$", + ha="center", va="bottom", color=SRC, fontsize=9) + + # depth axis + arrow(ax, (-1.18, 0.0), (-1.18, -0.95), color=INK, lw=1.2, scale=11) + ax.text(-1.23, -0.48, "depth $z$", ha="right", va="center", color=INK, + fontsize=9.5, rotation=90) + + # spreading front, then the three receivers + for z in np.linspace(-0.10, -0.94, 8): # the spreading front, left margin only + ax.plot([-0.97, -0.45], [z, z], color=FLD, lw=0.9, ls=":", alpha=0.55, zorder=2) + for z, lab in ((-0.22, "100 m"), (-0.45, "300 m"), (-0.82, "1000 m")): + ax.add_patch(Polygon([(-0.06, z), (0.06, z), (0.0, z + 0.055)], closed=True, + fc=FLD, ec=FLD, zorder=6)) + ax.text(0.11, z, f"receiver at {lab}", ha="left", va="center", color=FLD, + fontsize=9) + arrow(ax, (-0.36, z), (-0.11, z), color=FLD, lw=1.5, scale=10) + ax.text(-0.40, -0.22, "$E_x$", ha="right", va="center", color=FLD, fontsize=10) + + bare(ax, (-1.45, 1.35), (-1.12, 0.26)) + caption(fig, "The field diffuses downward. The solution is even in $z$, so reading $z$" + " as depth idealises away the air above.") + fig.tight_layout(rect=(0, 0.08, 1, 1)) + fig.savefig("setup_sheet.svg", transparent=True, metadata={"Date": None}) + plt.close(fig) + + +# --------------------------------------------------------------------------- +# Part 2: an infinite wire, and the plane the fields live in +# --------------------------------------------------------------------------- +def wire(): + fig, (a0, a1) = plt.subplots(1, 2, figsize=(7.4, 3.2), + gridspec_kw=dict(width_ratios=[0.85, 1.15])) + + # left: the wire, with the plane of interest cutting it + a0.plot([0, 0], [-1.1, 1.1], color=SRC, lw=3, zorder=5) + for y in (-0.8, -0.05, 0.6): + arrow(a0, (0, y), (0, y + 0.26), scale=11, lw=1.6) + a0.text(0.1, 1.05, "wire along $z$,\ncurrent $I$", ha="left", va="top", + color=SRC, fontsize=9.5) + a0.add_patch(Polygon([(-0.95, -0.38), (0.5, -0.64), (0.95, -0.02), (-0.5, 0.24)], + closed=True, fc=FLD, ec=FLD, alpha=0.20, lw=1.2, zorder=2)) + a0.text(-1.08, -0.62, "the $(x,y)$-plane", ha="left", va="top", color=FLD, + fontsize=9.5) + a0.set_title("side view", color=INK, fontsize=9.5, pad=6) + bare(a0, (-1.15, 1.35), (-1.35, 1.25)) + + # right: that plane, seen along the wire + for r in (0.34, 0.60, 0.86): + a1.add_patch(Circle((0, 0), r, fc="none", ec=FLD, lw=1.3, ls="--", alpha=0.85)) + a1.add_patch(Arc((0, 0), 1.20, 1.20, theta1=22, theta2=68, color=FLD, lw=1.9)) + th = np.deg2rad(68) + arrow(a1, (0.60 * np.cos(th), 0.60 * np.sin(th)), + (0.60 * np.cos(th + 0.22), 0.60 * np.sin(th + 0.22)), color=FLD, lw=1.9, + scale=13) + a1.text(0.34, 0.80, "$\\boldsymbol{H}$ circles the wire", ha="left", va="bottom", + color=FLD, fontsize=9.5) + + a1.add_patch(Circle((0, 0), 0.085, fc="none", ec=SRC, lw=2, zorder=6)) + a1.add_patch(Circle((0, 0), 0.026, fc=SRC, ec=SRC, zorder=6)) + a1.text(0.0, -0.14, "$I$, $E_z$", ha="center", va="top", color=SRC, fontsize=9) + + ta = np.deg2rad(-32) + arrow(a1, (0, 0), (0.86 * np.cos(ta), 0.86 * np.sin(ta)), color=INK, lw=1.3, + scale=11) + a1.text(0.50, -0.40, "$\\varrho$", ha="left", va="top", color=INK, fontsize=11) + a1.set_title("seen along the wire", color=INK, fontsize=9.5, pad=6) + bare(a1, (-1.1, 1.25), (-1.1, 1.1), equal=True) + + caption(fig, "Nothing depends on $z$, so the fields live in the $(x,y)$-plane:" + " $I$ and $E_z$ point out of it, $\\boldsymbol{H}$ lies in it.\n" + "Receivers sit at three distances from the wire.", y=0.02) + fig.tight_layout(rect=(0, 0.15, 1, 1)) + fig.savefig("setup_wire.svg", transparent=True, metadata={"Date": None}) + plt.close(fig) + + +# --------------------------------------------------------------------------- +# Part 3: the same kernel, spreading over a plane, a cylinder and a sphere +# --------------------------------------------------------------------------- +def ladder(): + fig, axes = plt.subplots(1, 3, figsize=(7.6, 3.0)) + titles = ("sheet, 1-D", "wire, 2-D", "point, 3-D") + pees = ("$p = 1/2$", "$p = 1$", "$p = 3/2$") + tps = ("$t_p = \\sigma\\mu r^2 / 2$", "$t_p = \\sigma\\mu r^2 / 4$", + "$t_p = \\sigma\\mu r^2 / 6$") + spreads = ("spreads one way", "spreads over a circle", "spreads over a sphere") + + for k, ax in enumerate(axes): + if k == 0: + ax.plot([-0.85, 0.85], [0, 0], color=SRC, lw=3, zorder=5) + for z in (0.26, 0.5, 0.74): + for s in (1, -1): + ax.plot([-0.85, 0.85], [s * z, s * z], color=FLD, lw=1.2, ls="--", + alpha=0.85) + arrow(ax, (0.0, 0.0), (0.0, -0.72), color=FLD, lw=1.6) + ax.text(0.09, -0.38, "$r$", color=FLD, fontsize=11, va="center") + else: + fills = (0.0, 0.11)[k == 2] + for r, al in ((0.30, 0.9), (0.54, 0.65), (0.78, 0.4)): + if fills: + ax.add_patch(Circle((0, 0), r, fc=FLD, ec="none", alpha=fills)) + ax.add_patch(Circle((0, 0), r, fc="none", ec=FLD, lw=1.2, ls="--", + alpha=al)) + ax.add_patch(Circle((0, 0), 0.05, fc=SRC, ec=SRC, zorder=6)) + aa = np.deg2rad(38) + arrow(ax, (0, 0), (0.78 * np.cos(aa), 0.78 * np.sin(aa)), color=FLD, lw=1.6) + ax.text(0.20, 0.44, "$r$", color=FLD, fontsize=11, + ha="right", va="bottom") + + ax.set_title(titles[k], color=INK, fontsize=10.5, pad=8) + ax.text(0.0, -1.06, spreads[k], ha="center", va="top", color=INK, fontsize=8.5) + ax.text(0.0, -1.34, pees[k], ha="center", va="top", color=SRC, fontsize=11.5) + ax.text(0.0, -1.66, tps[k], ha="center", va="top", color=INK, fontsize=10.5) + bare(ax, (-1.0, 1.0), (-2.0, 1.0), equal=True) + + caption(fig, "One kernel, $\\exp(-\\sigma\\mu r^2/4t)$. Only the prefactor $t^{-p}$" + " changes, and it is what fixes the arrival time.", y=0.025) + fig.tight_layout(rect=(0, 0.07, 1, 1)) + fig.savefig("setup_ladder.svg", transparent=True, metadata={"Date": None}) + plt.close(fig) + + +# --------------------------------------------------------------------------- +# Part 4: the loop, and the two planes the lab cuts through it +# --------------------------------------------------------------------------- +def loop(): + # axonometric projection: x right, y toward the viewer (down-left), z down + def P(x, y, z): + return (x - 0.52 * y, -0.30 * y - 0.92 * z) + + L, zc = 1.0, 0.72 # half-side, and the depth of the (x,y) cut + fig, ax = plt.subplots(figsize=(6.8, 3.6)) + + # the vertical cut at y = 0, drawn first so the rest sits on top + ax.add_patch(Polygon([P(-1.75, 0, -0.05), P(1.75, 0, -0.05), P(1.75, 0, 1.05), + P(-1.75, 0, 1.05)], closed=True, fc=FLD, ec=FLD, lw=1.0, + alpha=0.13, zorder=1)) + # the ground surface + ax.add_patch(Polygon([P(-1.9, -1.8, 0), P(1.9, -1.8, 0), P(1.9, 1.8, 0), + P(-1.9, 1.8, 0)], closed=True, fc=GND, ec=INK, lw=0.8, + alpha=0.13, zorder=2)) + ax.text(*P(1.92, -1.8, 0), s=" surface", ha="left", va="center", color=INK, + fontsize=9) + # the horizontal cut at z = 10 m + ax.add_patch(Polygon([P(-1.7, -1.6, zc), P(1.7, -1.6, zc), P(1.7, 1.6, zc), + P(-1.7, 1.6, zc)], closed=True, fc=FLD, ec=FLD, lw=1.1, + alpha=0.20, zorder=3)) + ax.text(*P(1.72, -1.6, zc), s=" the $(x,y)$-plane at $z=10$ m", ha="left", + va="center", color=FLD, fontsize=9) + ax.text(*P(-1.78, 0, 0.62), s="the $(x,z)$-plane\nat $y=0$ ", ha="right", + va="center", color=FLD, fontsize=9) + + # the loop, 1 -> 2 -> 3 -> 4: clockwise seen from above, with z downward + verts = [(-L, -L), (L, -L), (L, L), (-L, L)] + for k in range(4): + x0, y0 = verts[k] + x1, y1 = verts[(k + 1) % 4] + ax.plot(*zip(P(x0, y0, 0), P(x1, y1, 0)), color=SRC, lw=2.6, zorder=6, + solid_capstyle="round") + arrow(ax, P(x0 + 0.40 * (x1 - x0), y0 + 0.40 * (y1 - y0), 0), + P(x0 + 0.60 * (x1 - x0), y0 + 0.60 * (y1 - y0), 0), scale=13, z=7) + for (vx, vy), k in zip(verts, "1234"): + px, py = P(vx, vy, 0) + ax.text(px + (0.10 if vx > 0 else -0.10), py + 0.06, k, ha="center", + va="bottom", color=SRC, fontsize=10.5, zorder=8) + ax.text(-2.40, 0.88, "square loop on the surface\n$L_x = L_y = 100$ m\n" + "$I$ switched on at $t=0$", ha="left", va="top", color=SRC, + fontsize=9.5, zorder=8, linespacing=1.45) + + # axes, from the centre of the loop + for (dx, dy, dz), lab, off in (((0.70, 0, 0), "$x$", (0.10, 0.02)), + ((0, 0.70, 0), "$y$", (-0.10, -0.05)), + ((0, 0, 0.95), "$z$", (0.10, 0.0))): + arrow(ax, P(0, 0, 0), P(dx, dy, dz), color=INK, lw=1.3, scale=12, z=9) + px, py = P(dx, dy, dz) + ax.text(px + off[0], py + off[1], lab, color=INK, fontsize=11, zorder=9, + ha="center", va="center") + + bare(ax, (-2.45, 3.15), (-1.45, 0.95)) + caption(fig, "Depth enters only through $\\exp(-z^2/D^2)$, the same factor for both" + " components, with $D = \\sqrt{4t/\\sigma\\mu}$.") + fig.tight_layout(rect=(0, 0.08, 1, 1)) + fig.savefig("setup_loop.svg", transparent=True, metadata={"Date": None}) + plt.close(fig) + + +if __name__ == "__main__": + sheet(); wire(); ladder(); loop() + print("wrote setup_sheet.svg, setup_wire.svg, setup_ladder.svg, setup_loop.svg") diff --git a/book/3_diffusion_fields/labs/figures/setup_ladder.svg b/book/3_diffusion_fields/labs/figures/setup_ladder.svg new file mode 100644 index 0000000..3c5fbe8 --- /dev/null +++ b/book/3_diffusion_fields/labs/figures/setup_ladder.svg @@ -0,0 +1,1423 @@ + + + + + + + + image/svg+xml + + + Matplotlib v3.11.1, https://matplotlib.org/ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + diff --git a/book/3_diffusion_fields/labs/figures/setup_loop.svg b/book/3_diffusion_fields/labs/figures/setup_loop.svg new file mode 100644 index 0000000..72046a5 --- /dev/null +++ b/book/3_diffusion_fields/labs/figures/setup_loop.svg @@ -0,0 +1,1342 @@ + + + + + + + + image/svg+xml + + + Matplotlib v3.11.1, https://matplotlib.org/ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + diff --git a/book/3_diffusion_fields/labs/figures/setup_sheet.svg b/book/3_diffusion_fields/labs/figures/setup_sheet.svg new file mode 100644 index 0000000..f776855 --- /dev/null +++ b/book/3_diffusion_fields/labs/figures/setup_sheet.svg @@ -0,0 +1,1242 @@ + + + + + + + + image/svg+xml + + + Matplotlib v3.11.1, https://matplotlib.org/ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + diff --git a/book/3_diffusion_fields/labs/figures/setup_wire.svg b/book/3_diffusion_fields/labs/figures/setup_wire.svg new file mode 100644 index 0000000..a12f08d --- /dev/null +++ b/book/3_diffusion_fields/labs/figures/setup_wire.svg @@ -0,0 +1,1130 @@ + + + + + + + + image/svg+xml + + + Matplotlib v3.11.1, https://matplotlib.org/ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + From d0df7014bbae6fe3ebb0cd30567e35286b6329e4 Mon Sep 17 00:00:00 2001 From: ARS183 <1028762817@qq.com> Date: Wed, 7 Oct 2026 14:37:19 +0200 Subject: [PATCH 04/11] Make every Week 6 output follow its parameters, and introduce each cell --- .../labs/week06_em_diffusion.md | 194 +++++++++++++----- 1 file changed, 142 insertions(+), 52 deletions(-) diff --git a/book/3_diffusion_fields/labs/week06_em_diffusion.md b/book/3_diffusion_fields/labs/week06_em_diffusion.md index b7f8e0a..da4ab8b 100644 --- a/book/3_diffusion_fields/labs/week06_em_diffusion.md +++ b/book/3_diffusion_fields/labs/week06_em_diffusion.md @@ -95,6 +95,13 @@ The exponential is the kernel of the heat equation with $\kappa$ replaced by $1/ Every field in this lab has that form, $t^{-p}\exp(-\sigma\mu r^2/4t)$, with a different $p$. It is worth finding the peak time once, for any $p$. +```{figure} figures/setup_sheet.svg +:name: fig-lab6-sheet +:width: 85% + +The model of this part. An infinite sheet of current switched on at $t=0$, in a uniform conductor, with receivers at three depths. +``` + ### Task 1 — when does the field arrive, and in what ground? Setting $\partial_t\ln|E|=0$ is easier than differentiating $|E|$ itself and gives the same answer. With a prefactor $t^{-p}$, @@ -107,6 +114,8 @@ Solve it for $t$ and write the result once, as `t_peak(r, p, s)`. The second blank is the margin the diffusive approximation still has. Dropping $\varepsilon\,\partial_t\boldsymbol E$ is safe only while the peak arrives long after the charge relaxation time $\tau_r=\varepsilon/\sigma$. Write $t_p/\tau_r$ for the sheet in closed form, in terms of $z$ and $\sigma$, and watch what $\sigma$ does to it. +The cell evaluates {eq}`eq:ExtH` over a grid of depths and times. Dividing by the value at the shallowest depth cancels the common $t^{-1/2}$ and leaves $\exp(-\sigma\mu z^2/4t)$, which says how far the field has penetrated rather than how strong it is. A receiver keeps the $t^{-1/2}$, so there the arriving exponential and the decaying prefactor compete, and that competition is what you are solving for. + ```{code-cell} ipython3 sigma_sheet, eps_r, I_sheet = 0.01, 10.0, 1.0 # sediments [S/m], relative permittivity, [A/m] tau_r = eps_r * epsilon_0 / sigma_sheet # charge relaxation time [s] @@ -115,6 +124,11 @@ def E_sheet(z, t, s=sigma_sheet): """eq:ExtH -- the field of a current sheet switched on at t = 0 [V/m].""" return -I_sheet * np.sqrt(mu / (4*np.pi*s*t)) * np.exp(-s*mu*z**2 / (4*t)) +def t_for_D(D, s): + """When the diffusion distance sqrt(4t/(s*mu)) reaches D [s]. Used only to + frame the figures, so that every axis follows the parameters you set.""" + return s * mu * D**2 / 4 + # Task 1 -- two blanks. def t_peak(r, p, s): """Time at which t**(-p) * exp(-s*mu*r**2/(4t)) is largest [s]. @@ -128,8 +142,10 @@ def peak_over_relax(z, s): # --- given: the field soaking downward. Drag the bar, or press play --------- z_grid = np.linspace(1.0, 2000.0, 400) # depth [m] -t_grid = np.logspace(-6, -1.3, 60) # [s] -soak = np.array([np.abs(E_sheet(z_grid, tv)) / np.abs(E_sheet(1.0, tv)) for tv in t_grid]) +t_grid = np.logspace(np.log10(t_for_D(z_grid[-1], sigma_sheet)) - 4, + np.log10(t_for_D(z_grid[-1], sigma_sheet)), 60) # [s] +soak = np.array([np.abs(E_sheet(z_grid, tv)) / np.abs(E_sheet(z_grid[0], tv)) + for tv in t_grid]) fw.show_profiles(z_grid, soak, t_grid, value_name="t", value_fmt="{:.2g}", unit=" s", xlabel="depth z [m]", ylabel="|Ex| / its value at the surface", title="The field diffuses into the Earth", ylim=(0.0, 1.05)) @@ -142,25 +158,31 @@ fw.show_spacetime(z_grid, t_grid, soak, t_label="t [s]", x_label="depth z [m]", title=r"Deeper means later: does your curve follow the ridge?") # --- given: what a receiver records at three depths ------------------------ -t_rec = np.logspace(-6, -1, 3001) # [s] depths = [100.0, 300.0, 1000.0] # [m] +t_rec = np.logspace(np.log10(t_for_D(min(depths), sigma_sheet)) - 2, + np.log10(t_for_D(max(depths), sigma_sheet)) + 2, 3001) # [s] fw.show_records(t_rec, {f"z = {z:g} m": np.abs(E_sheet(z, t_rec)) / np.max(np.abs(E_sheet(z, t_rec))) for z in depths}, peaks={f"z = {z:g} m": (t_peak(z, 0.5, sigma_sheet), 1.0) for z in depths}, t_label="t [s]", y_label=r"$|E_x|$ / peak value", - title=r"Step response at depth, $\sigma$ = 0.01 S/m") + title=f"Step response at depth, sigma = {sigma_sheet:g} S/m") -fw.check_scalar("t_p at 300 m against the peak of the record", - t_peak(300.0, 0.5, sigma_sheet), - t_rec[np.argmax(np.abs(E_sheet(300.0, t_rec)))], rtol=0.01, unit=" s") +_zc = depths[1] +fw.check_scalar(f"t_p at {_zc:g} m against the peak of the record", + t_peak(_zc, 0.5, sigma_sheet), + t_rec[np.argmax(np.abs(E_sheet(_zc, t_rec)))], rtol=0.01, unit=" s") ``` +The same expression evaluated at three conductivities. $t_p$ is linear in $\sigma$ while $\tau_r=\varepsilon/\sigma$ is inverse in it, so as the ground changes the arrival time and the margin behind the approximation move in opposite directions. + ```{code-cell} ipython3 # --- given: one receiver at 300 m, three grounds --------------------------- media = {"seawater, 4 S/m": 4.0, "saturated sediments, 1e-2 S/m": 0.01, "crystalline basement, 1e-4 S/m": 1e-4} -z_s, t_s = 300.0, np.logspace(-8, 1, 900) +z_s = 300.0 # the receiver depth [m] +_tp_all = [t_peak(z_s, 0.5, sv) for sv in media.values()] +t_s = np.logspace(np.log10(min(_tp_all)) - 3, np.log10(max(_tp_all)) + 2, 900) fw.show_records(t_s, {k: np.abs(E_sheet(z_s, t_s, sv)) / np.max(np.abs(E_sheet(z_s, t_s, sv))) for k, sv in media.items()}, @@ -175,7 +197,7 @@ for k, sv in media.items(): f"{peak_over_relax(z_s, sv):>13.3g}") fw.check("a tenfold drop in conductivity costs a hundredfold in margin", - np.isclose(peak_over_relax(300.0, 1e-3) / peak_over_relax(300.0, 1e-4), 100.0, rtol=1e-9), + np.isclose(peak_over_relax(z_s, 1e-3) / peak_over_relax(z_s, 1e-4), 100.0, rtol=1e-9), "the margin goes as sigma squared, not as sigma") ``` @@ -215,10 +237,19 @@ $$ E_z(\varrho,t) = \frac{\mu I}{4\pi t}\exp\left(-\frac{\sigma\mu\varrho^2}{4t} The kernel is the same as the sheet's. The prefactor is now $t^{-1}$, so $p=1$. +```{figure} figures/setup_wire.svg +:name: fig-lab6-wire +:width: 92% + +The model of this part. The wire is a point in the plane you will plot, the electric field points out of that plane, and the magnetic field circles the wire within it. +``` + ### Task 2 — the field of the wire, and when it arrives Write $E_z$. Then **predict**, before running: at the same distance, does the wire's field peak earlier or later than the sheet's, and by what factor? +The wire carries the same kernel with $t^{-1}$ in front of it. The cell evaluates it over the plane perpendicular to the wire at a sequence of instants, then at three fixed distances through time. The colour is logarithmic and rescaled each frame because the field spans ten decades across one picture and four more across the run. + ```{code-cell} ipython3 sigma_wire, I_wire = 3.0, 1.0 # seawater [S/m], wire current [A] @@ -229,12 +260,14 @@ def E_wire(rho, t, s=sigma_wire): return ___ # --- given: the field spreading out from the wire, on a logarithmic colour scale --- -n_w = 91 -x_w = np.linspace(-50.0, 50.0, n_w) # [m] +n_w, half_w = 91, 50.0 +x_w = np.linspace(-half_w, half_w, n_w) # [m] X_w, Y_w = np.meshgrid(x_w, x_w, indexing="ij") # F[ix, iy] sits at (x[ix], y[iy]) rho_w = np.hypot(X_w, Y_w) rho_w[rho_w == 0] = x_w[1] - x_w[0] # keep the wire out of the formula -t_w = np.logspace(-4, 0, 16) # [s] +# from when the field has spread a fifth of the box to when it has long left it +t_w = np.logspace(np.log10(t_for_D(half_w / 5, sigma_wire)), + np.log10(t_for_D(half_w * 20, sigma_wire)), 16) # [s] E_frames = np.array([E_wire(rho_w, tv) for tv in t_w]) fw.show_map(x_w, x_w, E_frames, t_w, value_name="t", value_fmt="{:.2g}", unit=" s", @@ -242,22 +275,25 @@ fw.show_map(x_w, x_w, E_frames, t_w, value_name="t", value_fmt="{:.2g}", unit=" log_c=True, per_frame_scale=True, title="The electric field spreads out from the wire") # --- given: three receivers, with your arrival times marked ---------------- -t_r = np.logspace(-6, 1, 3001) dists = [0.36, 15.0, 30.0] # [m] +d_mid = dists[1] +t_r = np.logspace(np.log10(t_for_D(min(dists), sigma_wire)) - 2, + np.log10(t_for_D(max(dists), sigma_wire)) + 3, 3001) # [s] fw.show_records(t_r, {f"{d:g} m": E_wire(d, t_r) / np.max(E_wire(d, t_r)) for d in dists}, peaks={f"{d:g} m": (t_peak(d, 1, sigma_wire), 1.0) for d in dists}, t_label="t [s]", y_label=r"$E_z$ / peak value", - title=r"The wire, $\sigma$ = 3 S/m: close in, the field is already fading") - -print(f"the sheet and the wire at the same 15 m, in the same ground:") -print(f" sheet, p = 1/2 : t_p = {t_peak(15.0, 0.5, sigma_wire)*1e3:.3f} ms") -print(f" wire, p = 1 : t_p = {t_peak(15.0, 1.0, sigma_wire)*1e3:.3f} ms") - -_tp15 = t_peak(15.0, 1, sigma_wire) -fw.check_scalar("t_p at 15 m against the peak of the record", _tp15, - t_r[np.argmax(E_wire(15.0, t_r))], rtol=0.01, unit=" s") -fw.check_scalar("the peak value, against mu I / (4 pi t_p) times 1/e", E_wire(15.0, _tp15), - mu*I_wire/(4*np.pi*_tp15)*np.exp(-1.0), rtol=0.01, unit=" V/m") + title=f"The wire, sigma = {sigma_wire:g} S/m:" + f" close in, the field is already fading") + +print(f"the sheet and the wire at the same {d_mid:g} m, in the same ground:") +print(f" sheet, p = 1/2 : t_p = {t_peak(d_mid, 0.5, sigma_wire)*1e3:.3f} ms") +print(f" wire, p = 1 : t_p = {t_peak(d_mid, 1.0, sigma_wire)*1e3:.3f} ms") + +_tpm = t_peak(d_mid, 1, sigma_wire) +fw.check_scalar(f"t_p at {d_mid:g} m against the peak of the record", _tpm, + t_r[np.argmax(E_wire(d_mid, t_r))], rtol=0.01, unit=" s") +fw.check_scalar("the peak value, against mu I / (4 pi t_p) times 1/e", E_wire(d_mid, _tpm), + mu*I_wire/(4*np.pi*_tpm)*np.exp(-1.0), rtol=0.01, unit=" V/m") ``` :::{admonition} Solution — Task 2 @@ -279,6 +315,8 @@ Write that ratio. It runs from 0 at switch-on to 1 at late time, so the picture Then move the conductivity slider on the second figure. **Predict** first: in which ground does the magnetic field reach its static value soonest? +Faraday's law turns this $E_z$ into a magnetic field that circles the wire. Its direction is settled from the first instant; only its size grows, from zero towards the static $I/(2\pi\varrho)$, through the very same exponential that shapes $E_z$. The second figure plots what is left of that climb, which for $t\gg\sigma\mu\varrho^2/4$ falls as $\sigma\mu\varrho^2/4t$. + ```{code-cell} ipython3 # Task 3 -- one blank. def H_over_static(rho, t, s=sigma_wire): @@ -295,16 +333,23 @@ fw.show_map(x_w, x_w, H_frames, t_w, arrows=(Hx_dir, Hy_dir), arrow_every=10, title="The magnetic field is already circular. It is only filling in") # --- given: how far it still has to go, in three grounds ------------------- -t_h = np.logspace(-5, 2, 400) sigmas = np.array([0.01, 0.1, 1.0, 3.0, 10.0]) # [S/m] +# wide enough to show the 1/t tail for every curve, and no wider: far below this +# the exponential is zero to machine precision and nothing is left to plot +_a_lo = t_for_D(min(dists[1:]), sigmas.min()) +_a_hi = t_for_D(max(dists[1:]), sigmas.max()) +t_h = np.logspace(np.log10(_a_hi / 500), np.log10(_a_lo * 1e7), 400) # [s] gap = {f"{d:g} m": np.array([1.0 - H_over_static(d, t_h, sv) for sv in sigmas]) - for d in (15.0, 30.0)} + for d in dists[1:]} fw.show_profiles(t_h, gap, sigmas, value_name="σ", value_fmt="{:.3g}", unit=" S/m", xlabel="t [s]", ylabel="1 − |H| / static value", log_x=True, log_y=True, title="The distance still to go falls as 1/t, and a conductive ground is slower") +# "early" and "late" written in terms of the problem, so the test survives a change +# of sigma_wire or of the distance: the field has spread a fifth of d_mid, then a hundred times it. +_early, _late = t_for_D(d_mid / 5, sigma_wire), t_for_D(d_mid * 100, sigma_wire) fw.check("the ratio climbs from 0 at switch-on to 1 at late time", - H_over_static(15.0, 1e-5) < 1e-6 and abs(H_over_static(15.0, 1e2) - 1.0) < 1e-3, + H_over_static(d_mid, _early) < 1e-6 and abs(H_over_static(d_mid, _late) - 1.0) < 1e-3, "it is exp(-a/t), not 1 - exp(-a/t): at t -> 0 there is no field yet") ``` @@ -323,8 +368,13 @@ The second of the chapter's two-dimensional equations is $\partial_xH_y-\partial Write the $z$-component of the curl. The two derivatives of each component are already computed for you. +{eq}`eq:TEEy` with no source says $\partial_xH_y-\partial_yH_x=-\sigma E_z$. The cell builds $\boldsymbol H$ on the grid, differentiates it numerically, and lays the result beside $-\sigma E_z$. Where they agree, the diffusing magnetic field is demonstrably not curl-free; near the wire they will not agree, because a finite grid cannot follow $1/\varrho$. + ```{code-cell} ipython3 -t_c = 1e-3 # one instant [s] +# One instant, chosen so the field has spread about as far as the ring the check uses. +# That makes the accuracy below independent of sigma_wire. +r_ring = (0.4*half_w, 0.9*half_w) # the comparison ring [m] +t_c = t_for_D(np.mean(r_ring), sigma_wire) # [s] dx = x_w[1] - x_w[0] H_mag = I_wire / (2*np.pi*rho_w) * H_over_static(rho_w, t_c) Hx_c, Hy_c = -Y_w/rho_w * H_mag, X_w/rho_w * H_mag # [A/m] @@ -336,15 +386,16 @@ dHy_dx, dHy_dy = np.gradient(Hy_c, dx, dx) curl_z = ___ # --- given: the curl beside the electric field it equals ------------------- -shown = rho_w > 8.0 # H goes as 1/rho, so the grid cannot follow it at the wire +shown = rho_w > 7*dx # H goes as 1/rho, so the grid cannot follow it at the wire fw.show_map(x_w, x_w, np.array([np.where(shown, curl_z, np.nan), np.where(shown, -sigma_wire*E_wire(rho_w, t_c), np.nan)]), ["curl of H", "minus sigma times E_z"], value_name="showing", value_fmt="{}", x_label="x [m]", y_label="y [m]", c_label="A/m²", title=f"Two ways to the same picture, at t = {t_c*1e3:g} ms") -# The comparison is made on the outer ring, where a 1.1 m grid resolves 1/rho. -ring = (rho_w > 20.0) & (rho_w < 45.0) +# The comparison is made on the outer ring, where the grid resolves 1/rho. Push t_c +# up and it will fail: the true curl is on its way to zero, and round-off takes over. +ring = (rho_w > r_ring[0]) & (rho_w < r_ring[1]) fw.check_close("the curl of H against minus sigma E_z, away from the wire", curl_z[ring], -sigma_wire*E_wire(rho_w, t_c)[ring], rtol=0.05, hint="eq:TEEy with no source: d_x H_y - d_y H_x + sigma E_z = 0") @@ -382,10 +433,19 @@ $$ G(R,t) = \frac{(\sigma\mu)^{1/2}}{[4\pi t]^{3/2}}\exp\left(-\frac{\sigma\mu R Same kernel again. The prefactor is $t^{-3/2}$, so $p=3/2$. +```{figure} figures/setup_ladder.svg +:name: fig-lab6-ladder +:width: 95% + +The three models of this lab side by side. The source spreads over a plane, a cylinder or a sphere, which is the only thing that changes between them. +``` + ### Task 5 — the ladder Write $G$. Your `t_peak` already covers it: the three geometries differ only in $p$, so at the same distance their peak times are in a fixed ratio. **Predict** that ratio before you run the last figure. +The three-dimensional impulse response is the same kernel again, now with $t^{-3/2}$. The cell evaluates it on a cube to draw surfaces of constant $G$, then puts the sheet, the wire and the point on one time axis at the same distance and in the same ground, so that the only thing still different between them is the power of $t$. + ```{code-cell} ipython3 sigma_pt = 0.01 # sediments [S/m] @@ -397,8 +457,9 @@ def G_point(R, t, s=sigma_pt): # --- given: a rotatable view of the field at one instant ------------------- # Change t_show and run the cell again to watch the shell move outward. t_show = 1.0e-3 # [s] -g_xy = np.linspace(-400.0, 400.0, 32) # [m] -g_z = np.linspace(0.0, 800.0, 32) # depth [m] +D_show = np.sqrt(4*t_show / (sigma_pt*mu)) # how far it has spread [m] +g_xy = np.linspace(-1.5*D_show, 1.5*D_show, 32) # the box follows the field +g_z = np.linspace(0.0, 3.0*D_show, 32) # depth [m] GX, GY, GZ = np.meshgrid(g_xy, g_xy, g_z, indexing="ij") G_vol = G_point(np.sqrt(GX**2 + GY**2 + GZ**2), t_show) fw.show_isosurface(g_xy, g_xy, g_z, G_vol, levels=(0.08, 0.25, 0.6), opacity=0.32, @@ -407,7 +468,8 @@ fw.show_isosurface(g_xy, g_xy, g_z, G_vol, levels=(0.08, 0.25, 0.6), opacity=0.3 # --- given: the three geometries at the same distance, same ground --------- r0 = 300.0 # [m] -t_l = np.logspace(-6, -1, 4001) +t_l = np.logspace(np.log10(t_for_D(r0, sigma_pt)) - 2, + np.log10(t_for_D(r0, sigma_pt)) + 2, 4001) # [s] ladder = {"sheet, p = 1/2": np.abs(E_sheet(r0, t_l, sigma_pt)), "wire, p = 1 ": E_wire(r0, t_l, sigma_pt), "point, p = 3/2": G_point(r0, t_l)} @@ -425,9 +487,10 @@ for name, p in zip(("sheet", "wire", "point"), (0.5, 1.0, 1.5)): fw.check_scalar("the point source peaks at sigma mu R^2 / 6", t_peak(r0, 1.5, sigma_pt), t_l[np.argmax(G_point(r0, t_l))], rtol=0.01, unit=" s") -_R = np.linspace(0.5, 6000.0, 60000) # a radial grid [m] +# integrate out to ten diffusion distances, where the Gaussian has nothing left +_R = np.linspace(1e-3, 10*D_show, 60000) # a radial grid [m] fw.check_scalar("the integral of G over all space, which fixes the constant in front", - np.sum(G_point(_R, 1e-3) * 4*np.pi*_R**2) * (_R[1] - _R[0]), + np.sum(G_point(_R, t_show) * 4*np.pi*_R**2) * (_R[1] - _R[0]), 1.0 / (sigma_pt*mu), rtol=0.01, unit=" m/S") ``` @@ -466,6 +529,13 @@ $$ D = \sqrt{\frac{4t}{\sigma\mu}} , $$ which is how far the field has spread by time $t$. The two exponentials are the two segments that run along $y$, at $x=\pm L_x/2$. They carry opposite currents, so they enter with opposite signs, and that difference is the only part you have to write. +```{figure} figures/setup_loop.svg +:name: fig-lab6-loop +:width: 88% + +The model of this part, and the two planes you will cut through it. The current runs clockwise from vertex 1 to vertex 4, seen from above with $z$ pointing down. +``` + ### Task 6 — the loop forgets its shape Fill in the brace of $E_y$. Then read three things off the pictures. @@ -474,6 +544,8 @@ Fill in the brace of $E_y$. Then read three things off the pictures. 2. Move the depth slider on the third figure. **Predict first** what going deeper does to the picture, then check. The answer is not what most people expect. 3. Rotate the 3-D view. +Integrating the point response along the four sides of the loop leaves a Gaussian across each pair of parallel wires and an error function along them. The cell evaluates $E_y$ in the vertical plane through the middle of the loop, where the two segments running along $y$ are what you see, entering with opposite signs because they carry opposite currents. + ```{code-cell} ipython3 sigma_loop, Lx, Ly = 0.3, 100.0, 100.0 # [S/m], loop sides [m] @@ -506,10 +578,12 @@ def E_loop(x, y, z, t, s=sigma_loop): # --- given: the vertical plane through the middle of the loop -------------- n_L = 91 -x_L = np.linspace(-300.0, 300.0, n_L) # [m] -z_L = np.linspace(0.0, 300.0, n_L) +x_L = np.linspace(-3*Lx, 3*Lx, n_L) # three loop widths either way [m] +z_L = np.linspace(0.0, 3*Lx, n_L) XZ_x, XZ_z = np.meshgrid(x_L, z_L, indexing="ij") -t_L = np.logspace(-5, -1, 16) # [s] +# from when the field has spread a tenth of the loop to when it has spread ten times it +t_L = np.logspace(np.log10(t_for_D(Lx/10, sigma_loop)), + np.log10(t_for_D(Lx*10, sigma_loop)), 16) # [s] xz = np.array([E_loop(XZ_x, 0.0, XZ_z, tv)[1] for tv in t_L]) fw.show_map(x_L, z_L, xz, t_L*1e3, value_name="t", value_fmt="{:.3g}", unit=" ms", @@ -524,32 +598,38 @@ The same field, finely sampled, from the lecturer's own script. It runs over 200 :width: 80% ``` +The same expression in a horizontal plane, with both components so the direction can be drawn. The diffusion distance $D=\sqrt{4t/\sigma\mu}$ is the only length the field itself has, so holding it against $L_x$ is what decides whether the pattern still remembers the shape of its source. + ```{code-cell} ipython3 # --- given: the plane at 10 m depth, seen from above. Arrows give direction --- +z_xy = 10.0 # the depth of this slice [m] XY_x, XY_y = np.meshgrid(x_L, x_L, indexing="ij") -EE = [E_loop(XY_x, XY_y, 10.0, tv) for tv in t_L] +EE = [E_loop(XY_x, XY_y, z_xy, tv) for tv in t_L] mag = np.array([np.hypot(ex, ey) for ex, ey in EE]) fw.show_map(x_L, x_L, mag, t_L*1e3, arrows=(np.array([e[0] for e in EE]), np.array([e[1] for e in EE])), arrow_every=10, log_c=True, per_frame_scale=True, y_down=True, value_name="t", value_fmt="{:.3g}", unit=" ms", x_label="x [m]", y_label="y [m]", c_label="log10(|E| / max)", - title="At z = 10 m: watch the square become a circle") + title=f"At z = {z_xy:g} m: watch the square become a circle") print(f"{'t [ms]':>9}{'D [m]':>9}{'D / Lx':>9}") -for tv in (1e-5, 3e-5, 1e-4, 3e-4, 1e-3, 5e-3): +for tv in t_L[::3]: print(f"{tv*1e3:>9.3g}{D_of(tv):>9.1f}{D_of(tv)/Lx:>9.2f}") ``` +Now the time is held fixed and the depth is stepped instead. One colour scale is shared by every frame, so the frames can be compared directly. Make your prediction before you run it, then read the printed column. + ```{code-cell} ipython3 # --- given: the same instant, at a depth you choose ------------------------- t_fix = 3e-4 # [s] -z_try = np.array([0.0, 50.0, 100.0, 200.0, 400.0]) # depth [m] +# depths spaced against the diffusion distance, so the five frames always span the drop +z_try = np.round(D_of(t_fix) * np.array([0.0, 0.9, 1.8, 3.5, 7.0])) # depth [m] deep = np.array([np.hypot(*E_loop(XY_x, XY_y, zv, t_fix)) for zv in z_try]) fw.show_map(x_L, x_L, deep, z_try, value_name="z", value_fmt="{:.0f}", unit=" m", log_c=True, per_frame_scale=False, y_down=True, x_label="x [m]", y_label="y [m]", c_label="log10(|E| / max)", - title=f"The same instant, t = {t_fix*1e3:g} ms, at five depths") + title=f"The same instant, t = {t_fix*1e3:g} ms, at {len(z_try)} depths") print(f"at t = {t_fix*1e3:g} ms the diffusion distance is D = {D_of(t_fix):.0f} m\n") print(f"{'depth [m]':>11}{'|E| / its surface value':>26}") @@ -558,11 +638,15 @@ for zv in z_try: print(f"{zv:>11.0f}{f:>26.4f}") ``` +The magnitude of the same field on a cube around the loop, drawn as nested surfaces of constant $|\boldsymbol E|$ so the interior stays visible. The box is sized from $D$, so it keeps framing the field at whatever instant you set. + ```{code-cell} ipython3 # --- given: the loop in three dimensions. Drag to rotate -------------------- t_3d = 3e-5 # [s] -g3 = np.linspace(-180.0, 180.0, 36) # [m] -gz3 = np.linspace(0.0, 180.0, 36) # depth [m] +# wide enough for the loop, and for the field once it has spread past it +half3 = max(1.8*Lx/2, 1.5*D_of(t_3d)) # [m] +g3 = np.linspace(-half3, half3, 36) +gz3 = np.linspace(0.0, half3, 36) # depth [m] L3x, L3y, L3z = np.meshgrid(g3, g3, gz3, indexing="ij") E3x, E3y = E_loop(L3x, L3y, L3z, t_3d) loop_wire = ([-Lx/2, Lx/2, Lx/2, -Lx/2, -Lx/2], @@ -570,10 +654,11 @@ loop_wire = ([-Lx/2, Lx/2, Lx/2, -Lx/2, -Lx/2], fw.show_isosurface(g3, g3, gz3, np.hypot(E3x, E3y), levels=(0.06, 0.18, 0.45), opacity=0.3, wire=loop_wire, wire_name="the loop", c_label="|E| [V/m]", z_label="depth z [m]", - title=f"t = {t_3d*1e3:g} ms, so D = {D_of(t_3d):.0f} m. The loop is still square") + title=f"t = {t_3d*1e3:g} ms: D = {D_of(t_3d):.0f} m, " + f"D / Lx = {D_of(t_3d)/Lx:.2f}") fw.check("the field opposes the source current over the x = +Lx/2 wire", - E_loop(Lx/2, 0.0, 10.0, 3e-5)[1] < 0 < E_loop(-Lx/2, 0.0, 10.0, 3e-5)[1], + E_loop(Lx/2, 0.0, z_xy, t_3d)[1] < 0 < E_loop(-Lx/2, 0.0, z_xy, t_3d)[1], "the current there runs in +y, so an opposing field has E_y < 0") ``` @@ -618,11 +703,15 @@ You now have everything a time-domain electromagnetic survey rests on. This part 3. You have two loops, 50 m and 400 m on a side. The target is 400 m down. Which do you choose, and what decides it: the loop size, or the time you record for? 4. The field at the target is $\exp(-z^2/D^2)$ of its surface value. Below which amplitude would you say the target is not being illuminated at all, and what does that make the deepest usable depth? +Everything here is $\exp(-z^2/D^2)$ at the target depth, with $D$ built from each ground's conductivity, against time. A target is worth looking for where that curve is still high inside the window you can actually record in. + ```{code-cell} ipython3 # --- given: the window, the ground and the depth, all on one picture ------- z_target = 400.0 # [m] t_win = (1e-5, 1e-2) # the recording window [s] -t_d = np.logspace(-7, 1, 2001) +_tp_t = [t_peak(z_target, 0.5, sv) for sv in media.values()] +t_d = np.logspace(min(np.log10(min(_tp_t)), np.log10(t_win[0])) - 2, + max(np.log10(max(_tp_t)), np.log10(t_win[1])) + 2, 2001) fig, ax = plt.subplots(figsize=(8.2, 3.6)) for k, (name, sv) in enumerate(media.items()): @@ -630,15 +719,16 @@ for k, (name, sv) in enumerate(media.items()): ax.axvline(t_peak(z_target, 0.5, sv), color=f"C{k}", ls=":", lw=1.2) ax.axvspan(*t_win, color="0.85", zorder=0, label="recording window") ax.axhline(1/np.e, color="k", lw=0.8, ls="--") -ax.text(1.3e-7, 1/np.e*1.1, "z = D", fontsize=8) +ax.text(0.01, 1/np.e + 0.02, "z = D", fontsize=8, transform=ax.get_yaxis_transform()) ax.set_xscale("log"); ax.set_xlabel("t [s]") ax.set_ylabel(f"illumination at {z_target:.0f} m, exp(-z²/D²)") -ax.set_title("Which ground puts a 400 m target inside the window?") +ax.set_title(f"Which ground puts a {z_target:.0f} m target inside the window?") ax.grid(alpha=0.3); ax.legend(fontsize=8, loc="upper left") plt.show() print(f"a target at {z_target:.0f} m, window {t_win[0]*1e6:.0f} us to {t_win[1]*1e3:.0f} ms\n") -print(f"{'ground':<30}{'t_p [s]':>11}{'in window?':>12}{'illum. at 10 ms':>18}") +print(f"{'ground':<30}{'t_p [s]':>11}{'in window?':>12}" + f"{f'illum. at {t_win[1]*1e3:g} ms':>18}") for name, sv in media.items(): tp = t_peak(z_target, 0.5, sv) lit = np.exp(-z_target**2 * sv * mu / (4*t_win[1])) From 3232d7aa214492f0bf645999d1e28e103ba9180f Mon Sep 17 00:00:00 2001 From: ARS183 <1028762817@qq.com> Date: Wed, 7 Oct 2026 16:25:45 +0200 Subject: [PATCH 05/11] Bring the seismic wavefields chapter up to the 7 October lecture notes --- book/4_mechanical_waves/acoustic_waves.md | 51 ++---------------- book/4_mechanical_waves/elastic_waves.md | 2 +- book/4_mechanical_waves/exercises.md | 29 ++++++++++ .../figures/seismicwave_1layer_vz.mp4 | Bin 0 -> 394063 bytes .../reflection_transmission.md | 8 ++- book/4_mechanical_waves/snells_law.md | 8 ++- .../wave_equation_solutions.md | 48 +++++++++++++++++ book/_toc.yml | 2 + 8 files changed, 99 insertions(+), 49 deletions(-) create mode 100644 book/4_mechanical_waves/exercises.md create mode 100644 book/4_mechanical_waves/figures/seismicwave_1layer_vz.mp4 create mode 100644 book/4_mechanical_waves/wave_equation_solutions.md diff --git a/book/4_mechanical_waves/acoustic_waves.md b/book/4_mechanical_waves/acoustic_waves.md index 48a52a0..16ca9fa 100644 --- a/book/4_mechanical_waves/acoustic_waves.md +++ b/book/4_mechanical_waves/acoustic_waves.md @@ -48,7 +48,7 @@ $$ in which $c$ can be seen as the velocity of the wave, for which we have: $c = \sqrt{K/ \rho}$. -This is the basic wave equation for one-dimensional acoustic waves. Equally well, we can derive from the same two equations {eq}`eq:eqdeform` and {eq}`eq:eqmotion` (see also *EXERCISES*) that the particle velocity also satisfies the wave equation: +This is the basic wave equation for one-dimensional acoustic waves. Equally well, we can derive from the same two equations {eq}`eq:eqdeform` and {eq}`eq:eqmotion` (see also the [*EXERCISES*](./exercises.md)) that the particle velocity also satisfies the wave equation: $$ \frac{\partial^2 v_x}{\partial x^2} @@ -57,49 +57,8 @@ $$ (eq:waveqacv) where $c$ is the same as in the wave equation of the pressure. -The solution to the wave equation for the pressure is: +Actually, the movie we showed before was generated using the above equations (but then discretized), so the equation of deformation {eq}`eq:eqdeform` and the equation of motion {eq}`eq:eqmotion`. It is shown again below. -$$ -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}. -$$ - -The scaling factor is ($\rho c$), being called the *seismic impedance*. +```{video} figures/seismicwave_1layer_vz.mp4 +:width: 80% +``` diff --git a/book/4_mechanical_waves/elastic_waves.md b/book/4_mechanical_waves/elastic_waves.md index d6a60de..51b03e7 100644 --- a/book/4_mechanical_waves/elastic_waves.md +++ b/book/4_mechanical_waves/elastic_waves.md @@ -105,4 +105,4 @@ With respect to the (1-D) reflection coefficients for P- and S-waves at the boun - For P-waves, we require the continuity of the tensile stress $\tau_{xx}$ and the particle velocity $v_x$ at the interface, and - For S-waves we require the continuity of the shear stress $\tau_{zx}$ and the particle velocity $v_z$ at the interface. -The resulting expressions for the reflection and transmission coefficient are the same as for the acoustic case, only now the velocities are for the elastic solid. (See *EXERCISES*) +The resulting expressions for the reflection and transmission coefficient are the same as for the acoustic case, only now the velocities are for the elastic solid. (See the [*EXERCISES*](./exercises.md)) diff --git a/book/4_mechanical_waves/exercises.md b/book/4_mechanical_waves/exercises.md new file mode 100644 index 0000000..e09167b --- /dev/null +++ b/book/4_mechanical_waves/exercises.md @@ -0,0 +1,29 @@ +# Exercises on fundamental physical laws, P- and S-waves + +1. From the deformation equation and the equation of motion for acoustic waves ({eq}`eq:eqdeform` and {eq}`eq:eqmotion`) show that the particle velocity $v_x$ also satisfies the wave equation, i.e., {eq}`eq:waveqacv`. + +2. Here we look at deriving the wave equation in a different way: + + - Show that from substituting the deformation equation {eq}`eq:eqdeform` and the equation of motion {eq}`eq:eqmotion` in the commutative property for the pressure, i.e., + + $$ + \frac{\partial}{\partial x} \left( \frac{\partial p}{\partial t} \right) + = + \frac{\partial}{\partial t} \left( \frac{\partial p}{\partial x} \right), + $$ + + the wave equation for the particle velocity is nicely obtained. + - In the same way show that from substituting the deformation equation {eq}`eq:eqdeform` and the equation of motion {eq}`eq:eqmotion` in the commutative property for the *particle velocity*, as expressed in {eq}`eq:commut`, the wave equation for the pressure is nicely obtained. + +3. From the deformation equation and the equation of motion for shear waves ({eq}`eq:deformS` and {eq}`eq:eqmotionS`) show that the particle velocity $v_z$ also satisfies the wave equation. + +4. Here we consider shear waves: + + - How can the solution of the wave equation for the shear stress ({eq}`eq:waveqStau`) be written? + - Based on this solution, *derive* the solution for the particle velocity $v_z$. + - What is the difference between the two? And how is that called? + +5. Here we consider shear waves arriving at an interface: + + - When we look at a reflection and transmission across an interface between two materials: we have an incident S-wave coming from above. How would you write the wave field (shear stress and particle velocity) above the interface? And how below? + - At the interface, we require the continuity of the shear stress and particle velocity, as given above: from this, *derive* the reflection and transmission coefficients. diff --git a/book/4_mechanical_waves/figures/seismicwave_1layer_vz.mp4 b/book/4_mechanical_waves/figures/seismicwave_1layer_vz.mp4 new file mode 100644 index 0000000000000000000000000000000000000000..18a2de9804a0e5c28bde505e9e8293c886edf9f4 GIT binary patch literal 394063 zcmX`R18^oy7byJ1)+U>cZQI7iwr$(CZQHhO+jg?C{pbDee{a=H_vxcP=$h_2(*OVf z&)Ct;*37}i3IG5G{MY}y40_H6^j5Y^^Z)<=(%9a}2mtn~ZDpwE_`_2T4)**_-JK{0W(J@jp&@nLoAWa<|Z8>OZU0hsfoXrf4Y%KMx zX>9CGX#bmq#?;Zu@`uO9*3rzy+JOVlKu=%KfSVrA-pH7n5zo*_-_pjwf}5U$j)M+Q z&sxvY&B2J9&Xtja&Xt~?5zoqq+tkPv&%sIm2V%vub#VI;{b_aV4Y}!Q=zfHL8ayjA zS0h86|1{G7Na)z>S(_Mf)3e|inA+P|>FNB4(&IVW8(CVKIs8Cuu55+|jz55bofS9T zj|qB)?l#s&+zj+o^z?YfdJc{{whk6%w*PVb9|d-{IyT0}4n~gL)C_ozruIJ)KZ=;} zENyHo^h|$7I{z2RfahRoX7FRo|3%Q@S=;|_5(6_UJ;(nHF|&3wvbWUxVfw-ATRPe6 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