Angular Distribution of the Acoustic Radiation from a Tuning Fork

Abstract

It is shown by elementary arguments that the familiar loudness distribution pattern in the immediate neighborhood of a tuning fork is not, as is often suggested in elementary texts, due to pathlength-dependent phase differences between the waves reaching the ear from the different faces of the prongs, but to pathlength-dependent amplitude differences. At great distances from the fork, however, amplitude variations become negligible and pathlength-dependent phase differences give rise to a quite different loudness distribution pattern. From an idealized model, which represents a tuning fork as four colinear sources of strengths +B, −B, −B, +B, the velocity potential is calculated and the mean-square-pressure and intensity distributions are derived. It is shown that the mean-square-pressure distribution depends on polar angle (θ) and distance (R) as (3 cos2θ − 1)2/R6 close to the fork—the easily audible (amplitude-dependent) nearfield pattern; and as cos4θ/R2 at great distances—the less familiar (phase-dependent) farfield case. The intensity distribution comprises a cos4θ/R2 radial outward flow with, superposed, a circulation of energy from equatorial (θ ≈ 90°) to polar (θ ≈ 0,thinsp;180°) regions, varying with distance as 1/R4. There are two zeroes of intensity: where R = 0.225λ, θ = 0 and 180°. The power radiated is estimated for a typical 520 cps fork, and is found to be 2.5 × 10−15 W when the prongs vibrate with an amplitude of 0.2 mm.