Aeroacoustics/Acoustic Sources

Introduction

The existence of any source which generates a volumetric expansion/contraction, force or circulation, results in generation of acoustic waves. These sources show up on the right hand side of the wave equation. In this section, we introduce three elementary category of sources (monopoles, dipoles, and quadruples) which represent a wide spectrum of aerodynamic noise sources.

Monopole

Assume that we have a source at point $\mathbf{x}_0$ which generates some volumetric expansion or contraction. The corresponding wave equation for such case is

$\frac{1}{c_{0}^2}\frac{\partial^{2} \phi}{\partial\, t^2}-\nabla^2\phi=-q(t)\delta(\mathbf{x-x_0})$

Using the free space Green's function that was introduced before [1] , the velocity potential will be

$\phi(x,t)=\frac{1}{4\pi}\int_{-\infty}^{\infty} \frac{q(\mathbf{y},t-|\mathbf{x}-\mathbf{y}|/c_0)\delta(\mathbf{y-x_0})}{|\mathbf{x}-\mathbf{y}|} d^3\,\mathbf{y}d\tau=-\frac{q(t-|\mathbf{x-x_0}|/c_0)}{4\pi\,|\mathbf{x-x_0}|}$

The above solution looks like the following picture if the source is located at the origin. As you can see this type of source generates symmetric spherical waves. This omni-direction source is called a monopole.

Dipole

A dipole is a source which appears in the wave equation as

$\frac{1}{c_{0}^2}\frac{\partial^{2} p}{\partial\, t^2}-\nabla^2 p=\frac{\partial}{\partial x_j}\left(f_j(t)\delta(\mathbf{x-x_0})\right)$

Quadrupoles

A quadruple is a source distribution involving two space derivatives

$\frac{1}{c_{0}^2}\frac{\partial^{2} p}{\partial\, t^2}-\nabla^2 p=\frac{\partial^2 T_{ij}}{\partial x_i \partial x_j}$

An acoustic field generated by a point quadrupole is shown below