Two point monopoles of equal amplitude and angular frequency

PHYSICS · Multi-part · Fermi University Physics · Oscillations, Waves & Acoustics · Wave motion, the wave equation and wave energy · Acoustics of fluids: generation, propagation and attenuation

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

Problem

Two point monopoles of equal amplitude Q0Q_0 and angular frequency ω\omega, oscillating in antiphase, sit a small distance d≪λd\ll\lambda apart along the xx-axis in a fluid of density ρ0\rho_0 and sound speed c0c_0. ((a)) By subtracting the two monopole fields and expanding the path difference to first order in dd, show that in the far field this pair radiates as a dipole with directional pattern cos⁡θ\cos\theta (θ\theta measured from the xx-axis), equivalent to a point force of amplitude F0=ρ0ωQ0dF_0=\rho_0\omega Q_0 d acting on the fluid, and that its total radiated power is P=ω2F0224πρ0c03.P=\frac{\omega^2F_0^2}{24\pi\rho_0c_0^3}. ((b)) One tine of a tuning fork vibrates as such a dipole, exerting a force of amplitude F0=0.050 NF_0=0.050\,\mathrm{N} on the surrounding air at f=440 Hzf=440\,\mathrm{Hz} (20∘C20^\circ\mathrm{C} air: 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}}). Find the acoustic power it radiates.

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