Stability of compressional waves in granular media

Nonlinear compressional waves in a granular medium such as the seafloor are subject to the Hertzian nonlinearity of order 3/2 in the strain rate. As a result, the coefficient of quadratic nonlinearity becomes arbitrarily large at low stress. Two theoretical problems result from the Hertzian nonlinearity: (1) lowest order nonlinearity in the equation of state is no longer quadratic and (2) questions of stability arise when nonlinear steepening occurs arbitrarily fast. A variant of the NPE [McDonald and Kuperman, J. Acoust. Soc. Am. 81, 1406–1417 (1987)] is used to derive a stability theorem for self‐similar waves in a granular medium subject to three‐dimensional perturbations. When the dominant nonlinearity is of order n between 1 and 3 (n is not necessarily an integer), the result suggests that wave stability is positively correlated with n. The method of characteristics is used to show that where the coefficient of nonlinearity diverges, the wave slope is forced to zero. [Work supported by the Office of ...