/
standard_pde_manifold.tex
542 lines (481 loc) · 18.7 KB
/
standard_pde_manifold.tex
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
523
524
525
526
527
528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
\title{The standard PDE on a manifold}
We state and solve the standard second-order linear PDE on a compact
Riemannian manifold: the potential, diffusion, diffraction and wave
equations.
\newcommand{\R}{\mathbf{R}}
\newcommand{\Z}{\mathbf{Z}}
%\section{0. Overview}
%
%A standard course on partial differential equations should start always with the
%transport equation
%
%$$
%\begin{cases}
% u_t = u_x &(x,y)\in\R\times\R \\
% u(x,0) = f(x) &x\in\R
%\end{cases}
%$$
%
%whose solution is~$u(x,t)=f(x-t)$.
%
%This is pretty much a model for any linear first-order PDE with constant
%coefficients. Then, there are two possible roads: (1) study
%nonlinear first-order PDE, the method of characteristics, hamilton-jacobi,
%hyperbolic systems of conservation laws and so on; (2) study linear
%second-order PDE. Here we take the second road.
%
%These are the typical cases of linear second order PDE with constant
%coefficients on two variables (ignoring by now any boundary conditions):
%
%\begin{tabular}{ll}
% $u_t = u_{xx}$ & diffusion, blurring or heat equation \\
% $u_t = -u_{xx}$ & sharpening, or reverse heat equation \\
% $u_t = iu_{xx}$ & diffraction or Schrödinger equation \\
% $u_{tt} = u_{xx}$ & wave equation \\
% $u_{xx} + u_{yy} = 0$ & Laplace or harmonic equation
%\end{tabular}
%
%The first three cases are particular cases of the form~$u_t=cu_{xx}$ which is
%the~\emph{parabolic} case. The Laplace equation is the~\emph{elliptic} case
%and the wave equation is the~\emph{hyperbolic} case.
%
%Typically, there is also a non-constant data term that gives some flavour to
%the problem:
%
%\begin{tabular}{ll}
% $u_t = u_{xx} + f$ & heat equation with a heat source \\
% $u_{tt} = u_{xx} + f$ & wave equation with an additional force \\
% $u_{xx} + u_{yy} = f$ & Poisson or potential equation
%\end{tabular}
%
%
%To determine a unique solution, these equations require often boundary
%conditions, for example
%
%$$
%\begin{cases}
% $u_t = c u_{xx} + f$ & \Omega \\
%
%\end{cases}
%$$
\section{The Laplace-Beltrami spectrum}
Let~$M$ be a compact Riemannian manifold (with or without boundary), and
let~$\Delta$ be its Laplace-Beltrami operator, defined
as~$\Delta=*d*d$, where~$d$ is the exterior derivative (which is independent
of the metric) and~$*$ is the Hodge duality between~$p$-forms
and~$d-p$-forms (which is defined using the metric).
The following are standard results in differential geometry (see e.g.
Warner's book chapter
6~\url{https://link.springer.com/content/pdf/10.1007\%2F978-1-4757-1799-0_6.pdf})
\begin{itemize}
\item[(1)] There is a sequence of~$\mathcal{C}^\infty(M)$
functions~$\varphi_n$ and positive
numbers~$\lambda_n\to\infty$ such that
$$\Delta\varphi_n=-\lambda_n\varphi_n$$
\item[(2)] The functions~$\varphi_n$, suitably normalized, are an
orthonormal basis of~$L^2(M)$.
\end{itemize}
These results generalize Fourier series to an arbitrary smooth manifold~$M$.
Any square-integrable function~$f:M\to\R$ is written uniquely as
$$f(x)=\sum_nf_n\varphi_n(x)$$ and the coefficients~$f_n$ are computed by
$$f_n=\int_Mf\varphi_n.$$ Some particular cases are the habitual Fourier and
sine bases (but not the cosine basis), bessel functions for the disk, and
spherical harmonics for the surface of a sphere.
\begin{tabular}{lccr}
&$M$ & $\varphi_n$ & $-\lambda_n$ \\
\hline
interval & $[0,2\pi]$ & $\sin\left(\frac{nx}{2}\right)$ & $n^2/4$ \\
circle & $S^1$ & $\sin(n\theta),\cos(n\theta)$ & $n^2$ \\
square & $[0,2\pi]^2$ &
$\sin\left(\frac{nx}{2}\right)\sin\left(\frac{m\theta}{2}\right)$ &
$\frac{n^2+m^2}{4}$ \\
torus & $(S^1)^2$ & $\sin(nx)\sin(my),\ldots$ & $n^2+m^2$ \\
disk & $|r|\le1$ & $\sin,\cos(n\theta)J_n(\rho_{m,n}r)$ &
$\rho_{m,n}$ roots of~$J_n$ \\
sphere & $S^2$ & $Y^m_l(\theta,\varphi)$ & $l^2+l$
\end{tabular}
The eigenfunctions~$\varphi_n$ are called the vibration modes of~$M$, and the
eigenvalues~$\lambda_n$ are called the (squared) fundamental frequencies of~$M$.
Several geometric properties of~$M$ can be interpreted in terms of the
Laplace-Beltrami spectrum. For example, if~$M$ has~$k$ connected components,
the first~$k$ eigenfuntions will be supported successively on each connected
component. On a connected manifold~$M$, the first vibration mode can be
taken to be positive~$\varphi_1\ge0$, thus all the other modes have
non-constant signs (because they are orthogonal to~$\varphi_1$). In
particular, the sign of~$\varphi_2$ cuts~$M$ in two parts in an optimal way,
it is the Cheeger cut of~$M$, maximizing the perimeter/area ratio of the cut.
The zeros of~$\varphi_n$ are called the nodal curves (or nodal sets) of~$M$,
or also the Chladni patterns. If~$M$ is a subdomain of the plane, these
patterns can be found by cutting an object in the shape of~$M$, pouring a
layer of sand over it, and letting it vibrate by high-volume sound waves at
different frequencies. For most frequencies, the sand will not form any
particular pattern, but when the frequency coincides with
a~$\sqrt{\lambda_n}$, the sand will accumulate over the set~$[\varphi_n=0]$,
which is the set of points of the surface that do not move when the surface
vibrates at this frequency. In the typical case, the number of connected
components of~$[\varphi_n>0]$ grows linearly with~$n$, thus the
functions~$\varphi_n$ become more oscillating (less regular) as~$n$ grows.
Generally, symmetries of~$M$ arise as multiplicities of eigenvalues.
The Laplace-Beltrami spectrum~${\lambda_1,\lambda_2,\lambda_3,\ldots}$ is
closely related, but not identical, to the geodesic length spectrum, that
measures the sequence of lengths of all closed geodesics of~$M$. The grand
old man of this theory is Yves Colin de Verdière, student of Marcel Berger.
Geometry is not in general a spectral invariant, but non-isometric manifolds
with the same spectrum are difficult to come by. The first pair of distinct
but isospectral manifolds was wound in 1964 by John Milnor, in dimension 16.
The first example in dimension 2 was found in 1992 by Gordon, Webb and
Wolperd, and it answered negatively the famous question of Marc Kac ``Can you
hear the shape of a drum?'.
In 2018, we have many ways to construct discrete and continuous families of
isospectral manifolds in dimensions two and above.
\section{The standard equations and their explicit solutions}
The classical linear second order equations (potential, heat, wave and
Schrödinger) are all defined in terms of the Laplacian operator in space.
Thus, they can be defined readily on an arbitrary Riemannian manifold~$M$.
If~$M$ is compact, the solution can be found explicitly in terms of the
Laplace-Beltrami eigenfunctions. Henceforth we will call the expression of a
function~$f:M\to\R$ as~$f=\sum_nf_n\varphi_n$ the Fourier series of~$f$, the
numbers~$f_n$ the Fourier coefficients of~$f$ and so on.
The simplest case is {\bf Poisson equation}
$$
\Delta u = f
$$
The solution is found by expressing~$u$ and~$f$ as Fourier series and
identifying the coefficients:
$$
u(x) = \sum_n\frac{-f_n}{\lambda_n}\varphi_n(x)
$$
Notice that since~$\lambda_n\to\infty$, the Fourier coefficients of~$u$ tend
to zero faster than those of~$f$, thus~$u$ is more regular than~$f$
(this is obvious from the equation, since~$\Delta u$ is less regular than~$u$).
Another simple case is the {\bf screened Poisson equation}
$$
\Delta u = \alpha u + f
$$
and the solution is found by the same technique:
$$
u(x) = \sum_n\frac{-f_n}{\alpha+\lambda_n}\varphi_n(x)
$$
This is like the regular Poisson equation, but the regularity is enhanced
by~$\alpha$.
The next case is the {\bf heat equation}, also called {\bf diffusion} or {\bf
smoothing} equation:
$$
\begin{cases}
u_t = \Delta u & (x,t)\in M\times[0,T] \\
u(x,0)=g(x) & x\in M\\
\end{cases}
$$
This equation requires an initial condition~$g$. The solution is found by
separation of variables, which leads to a trivial ODE, resulting in
$$
u(x,t)=\sum_ng_ne^{-{\lambda_n}t}\varphi_n(x)
$$
It is immediate to check that this expression is a solution of the heat
equation with initial condition~$g$. Several properties of the solution are
visible from this form, most notably that~$u(x,\infty)=u_1$ if~$\lambda_1=0$,
or~$1$ otherwise. A pure vibration mode~$\varphi_n$ decays exponentially to
zero, and the speed of the exponential decay is~$\lambda_n$.
By combining the heat and Poisson equations, we get the {\bf heat equation
with source}:
$$
\begin{cases}
u_t = \Delta u + f & (x,t)\in M\times[0,T] \\
u(x,0)=g(x) & x\in M\\
\end{cases}
$$
whose solution is
$$
u(x,t)=\sum_n\left(
\frac{f_n}{\lambda_n}+g_ne^{-{\lambda_n}t}
\right)\varphi_n(x)
$$
The solution of the {\bf reverse heat equation}
$$
\begin{cases}
u_t = -\Delta u & (x,t)\in M\times[0,T] \\
u(x,0)=g(x) & x\in M\\
\end{cases}
$$
is formally similar
$$
u(x,t)=\sum_ng_ne^{{\lambda_n}t}\varphi_n(x)
$$
but notice that it blows up, often in a finite time.
Both direct and reverse heat equations are of the form~$u_t=c\Delta u$, whose
solution is~$u(x,t)=\sum_n g_n e^{-c\lambda_n t}\varphi_n(x)$. The
constant~$c$ is the speed of transmission of heat. An intermediate behaviour
between~$c>0$ and~$c<0$ happens when~$c=i$.
The {\bf linear Schrödinger equation}, also called {\bf diffraction equation}
$$
\begin{cases}
w_t = i\Delta w & (x,t)\in M\times[0,T] \\
w(x,0)=g(x) & x\in M\\
\end{cases}
$$
describes the evolution of a complex-valued function~$w$. It can be
interpreted as a system of two coupled real equations by writing~$w=u+iv$
(here, assuming a real-valued initial condition~$g$):
$$
\begin{cases}
u_t = -\Delta v & (x,t)\in M\times[0,T] \\
v_t = \Delta u & (x,t)\in M\times[0,T] \\
u(x,0)=g(x) & x\in M\\
v(x,0)=0 & x\in M\\
\end{cases}
$$
The solution is then
$$
w(x,t)=\sum_n g_n e^{-i\lambda_n t}\varphi_n(x)
$$
or, in terms of~$u$ and~$v$:
$$
\begin{cases}
u(x,t) = \sum_n g_n\cos\left(\lambda_nt\right)\varphi_n(x) \\
v(x,t) = \sum_n-g_n\sin\left(\lambda_nt\right)\varphi_n(x) \\
\end{cases}
$$
thus, a pure vibration mode~$\varphi_n$ oscillates periodically, at a
frequency~$\lambda_n$. In terms of~$|w|$, this phenomenon is called
\emph{diffraction}.
The {\bf wave equation} is
$$
\begin{cases}
u_{tt} = \Delta u & (x,t)\in M\times[0,T] \\
u(x,0) = g(x) & x\in M \\
u_t(x,0) = h(x) & x\in M \\
\end{cases}
$$
notice that it requires an initial condition and an initial speed. By
linearity, we can deal with these separately, and then sum the
results. The solution is then
$$
u(x,t)=\sum_n\left(
g_n\cos\left(\sqrt{\lambda_n} t\right)
+
\frac{h_n}{\sqrt{\lambda_n}}\sin\left(\sqrt{\lambda_n} t\right)
\right)\varphi_n(x)
$$
Thus, a pure vibration mode~$\varphi_n$ oscillates with
frequency~$\sqrt{\lambda_n}$.
Finally the {\bf wave equation with a force} is the most complex case we treat here:
$$
\begin{cases}
u_{tt} = \Delta u +f& (x,t)\in M\times[0,T] \\
u(x,0) = g(x) & x\in M \\
u_t(x,0) = h(x) & x\in M \\
\end{cases}
$$
The solution
$$
u(x,t)=\sum_n\left(
\frac{f_n}{\lambda_n}
+
g_n\cos\left(\sqrt{\lambda_n} t\right)
+
\frac{h_n}{\sqrt{\lambda_n}}\sin\left(\sqrt{\lambda_n} t\right)
\right)\varphi_n(x)
$$
is found by the same methods as above.
\section{Discretization and implementation in Octave}
Except in emblematic cases (rectangle, torus, sphere) the eigenfunctions of
an arbitrary manifold~$M$ do not have a closed-form expression.
For practical computations, we are thus restricted to numerical methods in the
discrete case. The most convenient form for this discretization is to
representd~$M$ as a graph with weights in their edges. In this context, we
have the following objects
\begin{itemize}
\item The weighted graph~$G=(V,E)$ where~$V$ is a set of~$n$ vertices.
\item The Laplacian matrix~$L$ of this graph, which is of
size~$n\times n$
\item The space~$\R^n$ is identified with functions~$V\to\R$.
Thus~$\R^n$ is the discrete version of~$\mathcal{C}^\infty(M)$.
\end{itemize}
typically~$L$ will be a matrix of rank~$n-1$ with a constant eigenvector of
eigenvalue 0.
We can find the eigensystem of~$L$ by calling~\verb+eigs(L)+ in octave, and
transfer the solutions obtained above using the obtained eigenvectors and
eigenvalues. However, in most cases the solution is more easily obtained by
solving a linear problem.
To fix the ideas we start with a concrete example: a square domain with flat
metric. The following is a complete program that computes the chladni
figures of a square domain.
\begin{verbatim}
w = 128; # width and height of the domain
p = sparse(1:w-1, 2:w, 1, w, w); # path graph of length p
A = kron(p, speye(w)) + kron(speye(w), p); # kronecker sum
L = A+A' - diag(sum(A+A')); # graph laplacian
[f,l] = eigs(L, 64, "sm"); # eigs of smallest magnitude
\end{verbatim}
%SCRIPT mkdir -p o
%SCRIPT octave <<END
%SCRIPT w = 96;
%SCRIPT p = sparse(1:w-1, 2:w, 1, w, w);
%SCRIPT A = kron(p, speye(w)) + kron(speye(w), p);
%SCRIPT L = A+A' - diag(sum(A+A'));
%SCRIPT [f,l] = eigs(-L'*L, 64, "sm");
%SCRIPT for i=1:64
%SCRIPT n = sprintf("o/chladni_%03d.png", i);
%SCRIPT x = uint8(reshape(200*double(0<f(:,i)),w,w));
%SCRIPT imwrite(x, n);
%SCRIPT endfor
%SCRIPT END
After running this code, the ith eigenfunction is \verb+f(:,i)+ and the
eigenvalues are on~\verb+diag(l)+.
\includegraphics{o/chladni_001.png}
\includegraphics{o/chladni_002.png}
\includegraphics{o/chladni_003.png}
\includegraphics{o/chladni_004.png}
\includegraphics{o/chladni_005.png}
\includegraphics{o/chladni_006.png}
\includegraphics{o/chladni_007.png}
\includegraphics{o/chladni_008.png}
\includegraphics{o/chladni_009.png}
\includegraphics{o/chladni_010.png}
\includegraphics{o/chladni_011.png}
\includegraphics{o/chladni_012.png}
\includegraphics{o/chladni_013.png}
\includegraphics{o/chladni_014.png}
\includegraphics{o/chladni_015.png}
\includegraphics{o/chladni_016.png}
\includegraphics{o/chladni_017.png}
\includegraphics{o/chladni_018.png}
\includegraphics{o/chladni_019.png}
\includegraphics{o/chladni_020.png}
\includegraphics{o/chladni_021.png}
\includegraphics{o/chladni_022.png}
\includegraphics{o/chladni_023.png}
\includegraphics{o/chladni_024.png}
\includegraphics{o/chladni_025.png}
\includegraphics{o/chladni_026.png}
\includegraphics{o/chladni_027.png}
\includegraphics{o/chladni_028.png}
\includegraphics{o/chladni_029.png}
\includegraphics{o/chladni_030.png}
\includegraphics{o/chladni_031.png}
\includegraphics{o/chladni_032.png}
\includegraphics{o/chladni_033.png}
\includegraphics{o/chladni_034.png}
\includegraphics{o/chladni_035.png}
\includegraphics{o/chladni_036.png}
\includegraphics{o/chladni_037.png}
\includegraphics{o/chladni_038.png}
\includegraphics{o/chladni_039.png}
\includegraphics{o/chladni_040.png}
\includegraphics{o/chladni_041.png}
\includegraphics{o/chladni_042.png}
\includegraphics{o/chladni_043.png}
\includegraphics{o/chladni_044.png}
\includegraphics{o/chladni_045.png}
\includegraphics{o/chladni_046.png}
\includegraphics{o/chladni_047.png}
\includegraphics{o/chladni_048.png}
\includegraphics{o/chladni_049.png}
\includegraphics{o/chladni_050.png}
\includegraphics{o/chladni_051.png}
\includegraphics{o/chladni_052.png}
\includegraphics{o/chladni_053.png}
\includegraphics{o/chladni_054.png}
\includegraphics{o/chladni_055.png}
\includegraphics{o/chladni_056.png}
\includegraphics{o/chladni_057.png}
\includegraphics{o/chladni_058.png}
\includegraphics{o/chladni_059.png}
\includegraphics{o/chladni_060.png}
\includegraphics{o/chladni_061.png}
\includegraphics{o/chladni_062.png}
\includegraphics{o/chladni_063.png}
\includegraphics{o/chladni_064.png}
And now, with Dirichlet boundary conditions (slightly different code)
\begin{verbatim}
w = 128; # width and height of the domain
p = sparse(1:w-1, 2:w, 1, w, w) - speye(w); # path graph of length p
A = kron(p, speye(w)) + kron(speye(w), p); # kronecker sum
L = A + A'; # graph laplacian
[f,l] = eigs(L, 64, "sm"); # eigs of smallest magnitude
\end{verbatim}
%SCRIPT octave <<END
%SCRIPT w = 96;
%SCRIPT p = sparse(1:w-1, 2:w, 1, w, w) - speye(w);
%SCRIPT A = kron(p, speye(w)) + kron(speye(w), p);
%SCRIPT L = A + A';
%SCRIPT [f,l] = eigs(-L'*L, 64, "sm");
%SCRIPT for i=1:64
%SCRIPT n = sprintf("o/dchladni_%03d.png", i);
%SCRIPT x = uint8(reshape(200*double(0<f(:,i)),w,w));
%SCRIPT imwrite(x, n);
%SCRIPT endfor
%SCRIPT END
\includegraphics{o/dchladni_001.png}
\includegraphics{o/dchladni_002.png}
\includegraphics{o/dchladni_003.png}
\includegraphics{o/dchladni_004.png}
\includegraphics{o/dchladni_005.png}
\includegraphics{o/dchladni_006.png}
\includegraphics{o/dchladni_007.png}
\includegraphics{o/dchladni_008.png}
\includegraphics{o/dchladni_009.png}
\includegraphics{o/dchladni_010.png}
\includegraphics{o/dchladni_011.png}
\includegraphics{o/dchladni_012.png}
\includegraphics{o/dchladni_013.png}
\includegraphics{o/dchladni_014.png}
\includegraphics{o/dchladni_015.png}
\includegraphics{o/dchladni_016.png}
\includegraphics{o/dchladni_017.png}
\includegraphics{o/dchladni_018.png}
\includegraphics{o/dchladni_019.png}
\includegraphics{o/dchladni_020.png}
\includegraphics{o/dchladni_021.png}
\includegraphics{o/dchladni_022.png}
\includegraphics{o/dchladni_023.png}
\includegraphics{o/dchladni_024.png}
\includegraphics{o/dchladni_025.png}
\includegraphics{o/dchladni_026.png}
\includegraphics{o/dchladni_027.png}
\includegraphics{o/dchladni_028.png}
\includegraphics{o/dchladni_029.png}
\includegraphics{o/dchladni_030.png}
\includegraphics{o/dchladni_031.png}
\includegraphics{o/dchladni_032.png}
\includegraphics{o/dchladni_033.png}
\includegraphics{o/dchladni_034.png}
\includegraphics{o/dchladni_035.png}
\includegraphics{o/dchladni_036.png}
\includegraphics{o/dchladni_037.png}
\includegraphics{o/dchladni_038.png}
\includegraphics{o/dchladni_039.png}
\includegraphics{o/dchladni_040.png}
\includegraphics{o/dchladni_041.png}
\includegraphics{o/dchladni_042.png}
\includegraphics{o/dchladni_043.png}
\includegraphics{o/dchladni_044.png}
\includegraphics{o/dchladni_045.png}
\includegraphics{o/dchladni_046.png}
\includegraphics{o/dchladni_047.png}
\includegraphics{o/dchladni_048.png}
\includegraphics{o/dchladni_049.png}
\includegraphics{o/dchladni_050.png}
\includegraphics{o/dchladni_051.png}
\includegraphics{o/dchladni_052.png}
\includegraphics{o/dchladni_053.png}
\includegraphics{o/dchladni_054.png}
\includegraphics{o/dchladni_055.png}
\includegraphics{o/dchladni_056.png}
\includegraphics{o/dchladni_057.png}
\includegraphics{o/dchladni_058.png}
\includegraphics{o/dchladni_059.png}
\includegraphics{o/dchladni_060.png}
\includegraphics{o/dchladni_061.png}
\includegraphics{o/dchladni_062.png}
\includegraphics{o/dchladni_063.png}
\includegraphics{o/dchladni_064.png}
For completenes, this is the octave code that saves the figures above
\begin{verbatim}
for i=1:64
n = sprintf("o/chladni_%03d.png", i);
x = reshape(200*double(0<f(:,i)),w,w);
iio_write(n, x);
endfor
\end{verbatim}
% vim:set tw=77 filetype=tex spell spelllang=en: