THE INTERACTION OF FINITE AMPLITUDE DEFLECTION AND STRETCHING WAVES IN ELASTIC MEMBRANES AND STRINGS

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Abstract
Disturbances produced by the motion of a driver which is rigidly bonded to the edge of a plate are used to motivate parameter expansion techniques which, when applied to the equations of finite elasticity, generate approximating equations which describe low-frequeney deflection and stretching waves travelling along stretched elastic plates and rods in the limit when bending forces are negligible compared with membrane forces. The structure of the boundary layer at the driven edge and shock layers, where the low-frequency or filament approximations are locally invalid, are also discussed. The low-frequency equations are used to discuss the interaction between progressing finite amplitude deflection and stretch waves in the limit when the stretch rate is small compared with the angular speed of the plate. The disturbance is locally that of a pure deflection simple wave whose amplitude and frequency are modulated by slow variations in the stretch. As the stretch increases the frequency increases while the amplitude decreases. The stretch wave is also modified by deflection of the plate: the speeds of wavelets carrying constant values of stretch are always less than their values in the pure stretch simple wave.

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