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 \(\mathbf{x_s}\) of angular frequency \(\omega\) produces an externally outgoing acoustic field of the form

\[ p(\mathbf{x})=\mbox{Re} \left( \mathcal{S} \frac{e^{i\omega |\mathbf{x} - \mathbf{x_s}|/c}}{|\mathbf{x} - \mathbf{x_s}|} \right), \]

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

\[ \left(\nabla^2 + \frac{\omega^2}{c^2} \right) p = -4\pi \mathcal{S} \delta(\mathbf{x} -\mathbf{x_s}) \]

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

\[ G_f(\mathbf{x},\mathbf{x_s})= -\frac{1}{4 \pi} \frac{e^{i\omega |\mathbf{x} - \mathbf{x_s}|/c}}{|\mathbf{x} - \mathbf{x_s}|},\]

that solves

\[ \left(\nabla^2 + \frac{\omega^2}{c^2} \right) G_f = \delta(\mathbf{x} -\mathbf{x_s}) \]

in an externally unbounded domain.