Monopoles and free space Green’s Functions

Author: Nick Ovenden

In a quiescent medium of constant sound speed $c$, any spherically symmetric time-harmonic source centred at a point ${\mathbf{x}}_{\mathbf{s}}$ of angular frequency $\omega$ produces an externally outgoing acoustic field of the form

$$p(\mathbf{x})=\text{Re}\left(\mathcal{S}\frac{e^{i\omega |\mathbf{x}-{\mathbf{x}}_{\mathbf{s}}|/c}}{|\mathbf{x}-{\mathbf{x}}_{\mathbf{s}}|}\right),$$

where $\mathcal{S}$ is a complex number representing the strength and phase of the source. As one might expect, this expression is the solution everywhere to the following PDE

$$({\mathrm{\nabla }}^2+\frac{{\omega }^2}{c^2})p=-4\pi \mathcal{S}\delta (\mathbf{x}-{\mathbf{x}}_{\mathbf{s}})$$

in an external unbounded domain with no incoming waves from infinity and where $\delta (\mathbf{x})$ is the delta-dirac function. Setting $\mathcal{S}=-\frac{1}{4\pi }$ leads to the so-called free-field Green’s function

$$G_f(\mathbf{x},{\mathbf{x}}_{\mathbf{s}})=-\frac{1}{4\pi }\frac{e^{i\omega |\mathbf{x}-{\mathbf{x}}_{\mathbf{s}}|/c}}{|\mathbf{x}-{\mathbf{x}}_{\mathbf{s}}|},$$

that solves

$$({\mathrm{\nabla }}^2+\frac{{\omega }^2}{c^2})G_f=\delta (\mathbf{x}-{\mathbf{x}}_{\mathbf{s}})$$

in an externally unbounded domain.