A point monopole source in air has velocity potential

PHYSICS · Subjective · Fermi University Physics · Oscillations, Waves & Acoustics · Acoustics of fluids: generation, propagation and attenuation · Acoustics of fluids: generation, propagation and attenuation

Signal Speeds in Fluids, Ch Sound in a moving, absorbing fluid: sources, refraction and loss

Problem

\phi(r,t) = -\dfrac{Q_0}{4\pi r}\cos(kr-\omega t)

, where

Q_0

is the amplitude of the volume flow it injects,

k=\omega/c_0

, and

r

is distance from the source. The chapter's power formula used only the far field; here we keep every term. (a) Find the pressure perturbation

p'=-\rho_0\,\partial\phi/\partial t

and the radial velocity

u_r'=\partial\phi/\partial r

exactly, at any

r

, with no far-field approximation. (b) Split

u_r'

into a part in phase with

p'

and a part a quarter-cycle out of phase with it. Show that only the in-phase part contributes to

\langle p'u_r'\rangle

, and hence that

4\pi r^2\langle p'u_r'\rangle=\rho_0\omega^2Q_0^2/8\pi c_0

at every radius, close to the source as well as far away. (c) For

Q_0=1.3\times10^{-3}\,\mathrm{m^3\,s^{-1}}

and

f=100\,\mathrm{Hz}

in air (

\rho_0=1.204\,\mathrm{kg\,m^{-3}}

c_0=343.2\,\mathrm{m\,s^{-1}}

), evaluate the radiated power, and find the ratio of the amplitude of the out-of-phase velocity to that of the in-phase velocity at

r=0.10\,\mathrm{m}

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