A monopole source of volume-flow amplitude oscillating at angular frequency in a…

PHYSICS · Multi-part · 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

A monopole source of volume-flow amplitude Q0Q_0 oscillating at angular frequency ω\omega in a fluid of density ρ0\rho_0 and sound speed c0c_0 produces the far-field pressure p′=ρ0ωQ04πrsin⁡(ωt−kr)p' = \dfrac{\rho_0\omega Q_0}{4\pi r}\sin(\omega t - kr), with the associated radial fluid velocity u′=p′/(ρ0c0)u' = p'/(\rho_0c_0) in the far field, k=ω/c0k=\omega/c_0. ((a)) Using the intensity I=⟨p′u′⟩I=\langle p'u'\rangle and integrating over a sphere of radius rr, derive the radiated power P=ρ0ω2Q028πc0P = \dfrac{\rho_0\omega^2Q_0^2}{8\pi c_0}, and confirm it does not depend on rr. ((b)) A small sphere of radius a=4.0 cma=4.0\,\mathrm{cm} pulsates radially with surface velocity amplitude u0=3.0 cm s−1u_0=3.0\,\mathrm{cm\,s^{-1}} at frequency f=300 Hzf=300\,\mathrm{Hz} in air at 20∘C20^\circ\mathrm{C} (ρ0=1.204 kg m−3\rho_0=1.204\,\mathrm{kg\,m^{-3}}, c0=343.2 m s−1c_0=343.2\,\mathrm{m\,s^{-1}}). Taking Q0=4πa2u0Q_0=4\pi a^2u_0, check that ka≪1ka\ll1 and evaluate the radiated power.

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