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Starting mechanics of an evanescent wave field

Published online by Cambridge University Press:  21 April 2006

D. C. Hill
D. C. Hill
Affiliation:
Cambridge University Engineering Department, Trumpington Street, Cambridge CB2 1PZ
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Abstract

The linear field induced by the sudden starting of a wave on an infinite plane surface is found exactly. An acoustic wavefront, on which the pressure remains constant and finite, moves outwards from the surface at the sound speed c. Behind this front the pressure field consists of two distinct components. The first is recognizable as the field due to steady motion over a wavy wall, while the second is an acoustic transient which propagates through the fluid to accelerate it into its steady asymptotic state. We find that whenever the surface phase speed is subsonic, there is an equipartition of the energy between the steady evanescent field attached to the surface and the outward-travelling sound. If the surface has sonic phase speed then the pressure on the surface grows like t½, and becomes unbounded. For subsonic waves the final momentum of the fluid parallel to the surface is shown to be equal to the total energy radiated divided by the surface phase speed. In the asymptotic state the ratio of momentum in the far field to that in the near field is ½m2/(1 − ½m2), where m is the ratio of the surface phase speed to the sound speed. The transient field on the surface can be identified as consisting of two travelling sound waves. One travels in the same direction as the surface wave, and the other in the opposite direction. They have amplitudes which are proportional to (1 − m)−l and (1 + m)−1 respectively, and decay in time as t−½. Their wavelength parallel to the surface is, of course, the same as that of the surface wave that induced them. We show that when the surface wave is very subsonic, the evanescent field is established very slowly, settling down only after the surface wave has travelled about m−1 wavelengths.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 165 , April 1986 , pp. 319 - 333
Copyright
© 1986 Cambridge University Press

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