Analytical Solution to the acoustic / scalar (Inhomogeneous) wave equation with source term

Question

The acoustic wave equation in 2D is

$$\frac{\partial^2}{\partial t^2}p(x,z,t) = c(x,z)^2\left[\frac{\partial^2}{\partial x^2}p(x,z,t) + \frac{\partial^2}{\partial z^2}p(x,z,t)\right] + s(x,z,t) \enspace .$$

Can anyone please give an analytical solution to the above, not worrying about the boundary conditions (assuming infinite boundary)?

• $p$ is a scalar field ( pressure )
• $c$ is the velocity ( a scalar value ) , same everywhere in the domain.
• $s(x,z,t)$ or source function is at only one point in the domain ( at the center of the 2D / 3D domain).

An IPython snippet of this code for finite differences is

# it is the time step number.
# dt is the time step size.
# src is the source

f0 = 100.0    # dominant frequency of source (Hz)
T = 1.0 / f0  # dominant period
ist = 100     # shifting of source time function
src[it] = exp(-1.0 / T ** 2 * ((it - ist) * dt) ** 2)   # This is a Gaussian source
# Take the first derivative to finally define the source function.
src = np.diff(src) / dt

The green's function is a bit complicated for me.

How can it be re-defined / adapted for our case of finite differences , given this : https://en.wikipedia.org/wiki/Acoustic_wave_equation#Cartesian_coordinates

Answer 1

You can make up as many solutions to this equation as you want using the Method of Manufactured Solutions (PDF). Is there a reason you're focused on this particular case only, or could you use any solution/forcing-function pair?

Answer 2

I understand that you want to solve the differential equation

$$\nabla^2 u - \frac{1}{c^2}\frac{\partial^2 u}{\partial t^2} = f(t)\delta(\mathbf{r}) \enspace ,$$

for an unbounded domain, $\mathbf{r}$ is the position vector, and

$$f(t) = -\frac{2(t-t_0)}{T^2} e^{-\frac{\left( t-t_0\right)^2}{T^2}} \enspace ,$$

although, we don't need to write this function explicitly.

We compute then the Fourier transform of the equation, i.e.,

$$\nabla^2 u + k^2 u = \hat{f}(\omega)\delta(\mathbf{r}) \enspace ,$$

with $k^2 = \omega^2/c^2$, and $\hat{f}(\omega)$ the Fourier transform of $f(t)$ that reads

$$\hat{f}(\omega) = \frac{i\,\omega\,T\,{e}^{-\frac{{\omega}^{2}\,{T}^{2}}{4}-i\,\omega\,t0}}{\sqrt{2}} \enspace .$$

And, the solution to your problem looks like

$$u(t) = \frac{i}{4 \sqrt{2\pi}}\int_{-\infty}^{\infty} H_0^{(1)}(k|\mathbf{r}|) \hat{f} (\omega) e^{i\omega t} d\omega \enspace .$$

You should read about the solution for the Helmholtz equation, Fourier transform and Convolutions to understand the procedure.