THE RESPONSE OF AN ELASTIC PLATE SUBMERGED IN A LIQUID HALF‐SPACE TO EXPLOSIVE SOUND

Abstract

A mathematical analysis is presented for the case of a point‐source explosion in the liquid layer above an elastic plate of infinite horizontal extent immersed in a liquid half‐space parallel to the free surface of the liquid. An asymptotic solution, valid for long times after the explosion, is derived; it expresses the pressure response in the liquid layer in terms of characteristic vibrations of the layered medium. Trapped and exponentially decaying modes have been investigated numerically for the Lucite plate in water. Special emphasis is placed on the description of sustained reverberations (singing). This phenomenon is described in terms of complex modes, where some energy travels back radially towards the source. At long times, singing can be described in terms of “standing” waves of nonvanishing horizontal wave number. It is also closely connected with a type of trapped wave in the liquid layer‐plate combination whose horizontal phase velocity is greater than the velocity of sound in the fluid, but which is completely decoupled from the liquid half‐space below the plate. At very long times, however, the strongest signal is associated with an almost completely decoupled shear motion of the plate, and the horizontal wave number approaches zero. A brief discussion of the total transmission of plane harmonic sound waves through a Lucite plate in water is given. The total transmission curves are used to show qualitatively that singing often may not be observed in connection with the above‐mentioned trapped waves.