Stability of self-similar plane shocks with Hertzian nonlinearity

The nonlinear progressive wave equation [ McDonald and Kuperman , J. Acoust. Soc. Am. 81 , 1406-1417 ( 1987 ) ] is expressed in a form to accommodate Hertzian nonlinearity of order n = 3 ∕ 2 typical of granular media. Stability of self-similar plane weak shocks against perturbations in three dimensions is demonstrated for arbitrary order nonlinearity with 1 < n < 3 . Investigation of stability in the presence of Hertzian nonlinearity is motivated by the nonlinearity coefficient being arbitrarily large in the unstressed state. Energy in the perturbation field is shown to fall off at least as fast as 1 ∕ t , while energy in the self-similar wave falls off slower than 1 ∕ t . For an initial wave profile increasing in the direction of propagation, it is shown that where the quadratic nonlinearity coefficient diverges, the slope of the wave goes to zero.