acousticPropagation

Abstract

This project is centered on the intricate dynamics of acoustic propagation in fluidic environments. We are reproducing and building on some of the results presented in this seminal paper.

Theory

The evolution of an acoustic pulse with wave speed $c=1$ inside a fluid with linear velocity $M$ on a 2D plane with cylindrical coordinates $(r,\theta)$ is derived from the potential function $\phi(r,\theta,t)$. This is governed by the wave equation:

$$ \frac{\partial^2 \phi}{\partial t^2 }=c^2\cdot \Delta_{r,\theta,M} \phi, \ t>0, \ (r,\theta)\in (0,\infty)\times(0,2\pi) $$

where $\Delta_{r,\theta}$ is the Laplace operator in cylindrical coordinates in the moving frame of the fluid. The initial conditions at $t=0$, for a circular pulse, are assumed to be radially symmetric, i.e., independent of $\theta$:

$$ \phi(r,0)=0, \ \frac{\partial \phi}{\partial t}(r,0)=\phi'_0(r) $$

The pressure distribution $p(r,\theta,t)$ can then be deduced as the time derivative of the potential function, i.e., $p=\phi'$. The relation between cylindrical coordinates $(r,\theta)$ and Cartesian ones $(x,y)$ is:

$$ x=r\cos \theta, \ y=r\sin\theta $$

Using the cylindrical Hankel transform (akin to the 2D Fourier transform) of the initial distribution $\phi'_0$, the equation can be solved. The pressure solution at any time $t>0$ for spatial coordinates $(r,\theta)$ is:

$$ p(r,\theta,t) = \int_{0}^{\infty} \mathcal{H}{\phi'_0}(\omega) \cdot J_0\left( \omega \sqrt{r^2+(M\cdot t)^2-2r\cdot (M\cdot t) \cdot \cos(\theta)}\right)\cdot \omega \cos(\omega t) \ d\omega $$

The function is given by:

$$ \mathcal{H}{\phi\prime_{0}}(\omega)=\int_{0}^{\infty} r'J_0(\omega r')\phi\prime_{0}(r')\ dr' $$

This is the zero-order Hankel transform of the initial pulse $\phi'_0$.

Sample Results

Below are some sample animations showcasing the results: