The mechanics of edge-tones

Abstract By consideration of the frequency spectrum of the sound field of an oscillating linear jet, it is found that one may expect a continuous spectrum of frequencies from 0 to 0.08U0/d, where U0 = mean velocity of efflux of the jet, d = width of slit, the maximum amplitude occurring at the frequency 0.055U0/d which is found experimentally. To explain the tones produced when an edge is placed in such a flat jet, it is shown that the hydrodynamic effect of the edge is to amplify those oscillations of the jet which have a certain particular frequency in the band of frequencies to which the jet is unstable. This is due to the reciprocal influence of the maximum transverse velocity in the jet upon the shedding of vortices at the edge, and of the sudden production of free vorticity at the edge in deflecting the jet and so causing a slight extra production of vorticity on one side of the slit, which is in turn amplified as it moves downstream. From this one deduces that vortices of opposite circulation are produced simultaneously at the upper edges of the slit and the wedge, and similarly at the lower edges, and also that the jet wave-length λ and the edge to slit distance h are related by an equation h = (i + 1/4) λ, where i is an integer. It is suggested that the value of i is such that λ is near the jet-tone wave-length which gives maximum amplitude. Independently of this, an expression, partly theoretical and partly empirical, is obtained for the velocity with which the vortices move along. A formula for the edge-tone frequency is then deduced in the form f = 1/2 Uo {i + 1/4 / h – 1/30d}, where d = slit-width. This formula applies only when h/d > 10, i.e. when the edge to slit distance is at least 10 slit-widths. The predictions of the theory as regards the numerical values of the jumps in frequency, dependence of frequency upon slit-width, and absolute values of wave-length and frequency, are examined in the light of previous experimental results, and are found to be satisfactory.