Classical Mechanics – Why is the Amplitude of Monopole Solution in Helmholtz Equation Complex?

acousticsclassical-mechanicscomputational physicssimulationswaves

1. Background

Given an surface of vibrating object $\partial \Omega$, I am trying to simulate the outer acoustic field. I use the equivalent source method[1], which basically tries to approximate the solution (sound pressure) field by a linear combination of basis functions.

Helmholtz

We know the propagation of a wave is governed by the Helmholtz equation (homogenous):
$$
(\nabla^2 + k^2) p = 0
$$
where $ p = p(x) \in \mathbb C$ is a complex sound pressure field.

This equation has a family of solutions. One of them is the Monopole solution:
$$
p(\mathbf x; \mathbf c) = A \frac{e^{- i k r}}{4 \pi r }
$$
where $\mathbf c \in \mathbb R^3 $ is the center of monopole, $r = |\mathbf x – \mathbf c|$, $A $ is a strange complex amplitutde.

After searching for many books and papers, I still cannot figure out the following questions:

2. question

I cannot figure out why $A$ is a complex number (this paper [1] claims the "amplitude" is complex), would anyone please tell me the reason?

  • I understand $p(x)$ should be complex, because the phase shift $e^{i\phi}$ is involved in $p(x)$. But this phase shift has been considered by Monopole in its $e^{- i kr}$ item now.

  • Why A is still complex?

reference

[1] Kondapalli, P.S., Shippy, D.J. and Fairweather, G., 1992. Analysis of acoustic scattering in fluids and solids by the method of fundamental solutions. The Journal of the Acoustical Society of America, 91(4), pp.1844-1854.

Best Answer

If there is a single monopole it does not matter whether $A$ is real or not but if you have two or more sources then their relative phases, and thus the phase of $A$, do matter. The same holds if the source is distributed and not point-like. Acoustic phase is important if you care about spatial distribution of the wave field, ie., radiation pattern, but to a first (?) order human hearing is not sensitive to phase. In other words humans cannot hear the difference between $x_1(t)=K_0cos(\omega_0 t+\phi_1)$ and $x_2(t)=K_0cos(\omega_0 t+\phi_2$. Apparently, humans can "hear" $\omega_0 $ (frequency) and $|K_0|$ (amplitude) but not $\phi$ (phase). In other words, the human ear is a frequency sensitive non-coherent intensity detector, just like a spectrum analyzer. This is true whether the oscillation be damped or undamped.

A phase sensitive detector is really two frequency coherent detectors in "quadrature", that is phase is measured relative to two reference signals one is, say, $r_c(t)=Rcos(\omega_0 t+\psi)$ the other is $r_s(t)=Rcos(\omega_0 t+\psi - \pi/2) = Rsin(\omega_0 t+\psi)$ where $\psi$ and $R$ are the arbitrary phase and amplitude of the local reference, resp. While measuring the time evolution of $x(t)=K_0 cos(\omega_0 t+\phi)$ relative to the oscillations of $r_c(t)$ and $r_s(t)$ we can calculate $\phi - \psi$ and then all the other relative phases between the other sources, as well, at the same frequency. Human and I think most if not all animal hearing is very sensitive to time delay between two sources, this is how we discern the direction from which the sound is emitted. It is amazing how sensitive the directional hearing of grasshoppers or crickets is despite the fact that spatial separation between the "ears" of a grasshopper is at most a few millimeters. One wonders if the cricket's hearing may be phase sensitive then, but I am just guessing here and do not really know.

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