Wave Equation Solver

This project develops a fast direct solver for the wave equation with sound-soft boundary conditions (Dirichlet) by working in the frequency domain. The key ingredients are a broadband Helmholtz solver and am inverse Fourier transform use complex deformations and the fast sinc transform.

The interactive form below allows you to run code developed for this solver. Choose a given scatterer and incident wavefield and run the simulation!

Note that the interactive solver is actively being developed and very out of date. If you have any questions or problems, feel free to contact me.

Interactive Solver

§1  Scatterer Geometry

Scatterer type C Curve Kite Star Circles Whisper Gallery Crescent Gallery Teardrop Keyhole Magnetron Spikes Perforated Wall

No additional parameters required for the Kite.

$a$ — indent depth  (0 < a < 1)

Strictly between 0 and 1. Values above 0.85 are significantly harder to compute.

$w$ — number of petals

Integer, range [2, 10]

Center $c$ — one complex number

Format: a+bi  —  e.g. 1+1i, -2.5+3i, 0+0i, -1-2i

Radius $R$ — positive number

Must be strictly greater than 0.

$a$ — corner sharpness  (larger = sharper corners)

Range: [0.5, 15]

$b$ — channel arc length  ($0$ = no channel, $\pi$ = widest)

Range: $[0, \pi]$.  $\pi \approx 3.1416$

$c$ — fatness  (larger = fatter)

$d$ — angle of cavity opening  (radians)

$0$ = cavity opens left. $\pi \approx 3.1416$ = opens right. Any angle in $[0, 2\pi)$.

$a$

$b$

Fatness

$a$

$b$

$d$ — distance

$m$ — sharpness  (integer 1 to 4)

Must be a whole number: 1, 2, 3, or 4.

$r$ — small radius

$R$ — large radius  (must be larger than $r$)

$e$ — half tube height  (must be less than $r$)

$\theta$ — angle of hole (radians)

$\pi \approx 3.1416$

Parameters to be documented — fill in when ready.

$r$

$R$

$e$

$r_{\text{out}}$

$\Delta\theta$

$n$ — count

$\theta$

$\pi/3 \approx 1.0472$

$a$ — number of spikes  (positive integer, max 10)

Integer, range [1, 10]

$h$ — height

Must be a positive number.

$w$ — width

Must be a positive number.

$\theta$ — angle (radians)

$N$ — number of holes  (integer, max 30)

Integer, range [1, 30]

§2  Incident Wave Field

$$u_{\text{inc}} = \frac{1}{\sqrt{2\pi}\,\sigma} \, e^{i\omega_0\!\left(\mathbf{x}\cdot\mathbf{r}/c \;-\; t\right)} e^{-\dfrac{\left|\mathbf{x}\cdot\mathbf{r}/c \;-\; t\right|^2}{2\sigma^2}}$$

$\omega_0$ — base oscillatory frequency

Must be > 0.

$\mathbf{r}$ — propagation direction  (normalised automatically)

$r_x$ (horizontal)

$r_y$ (vertical)

$c$ — wave speed

Must be > 0.

$\sigma$ — wave packet width

Must be > 0. Larger $\sigma$ = narrower frequency band.

§3  Output Field Type

Total Scattered Incident

§4  Plot Value Type

Real part Absolute value Imaginary part

§5  Simulation End Time

$T$ — final time

Range: [0.1, 300]. Simulation runs from $t=0$ to $t=T$.

Preview Geometry & Wave ◈

Scatterer

Incident wavefield

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Run Solver →

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Starting Setup Universal Skeleton Density Function Helmholtz Evaluation Fast Sinc Transform GIF Done

Computation Times

Broadband Skeletonization —

Density solve —

Frequency domain evaluation —

Inverse Fourier transform —

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Simulation Result

⏸ Pause

Result auto-hides after 10 minutes. Run again to refresh.