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!

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

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