The theory of singular surfaces yields a set of coupled evolution equations for the shock amplitude and the amplitudes of the higher order discontinuities which accompany the shock. To solve these equations, we use perturbation methods with a perturbation parameter ε characterising the initial shock amplitude. It is shown that for decaying shock waves, if the accompanying second order discontinuity is of order one, the straightforward perturbation procedure yields uniformly valid solutions, but if the accompanying second order discontinuity is of order ε, the method of multiple scales is needed in order to render the perturbation solutions uniformly valid with respect to the distance of travel. We also construct shock wave solutions from modulated simple wave solutions which are obtained with the aid of Hunter & Keller's “Weakly Nonlinear Geometrical Optics” method. The two approaches give exactly the same results within their common range of validity. The explicit evolution laws thus obtained enable us to see clearly how weak nonlinear curved shock waves are attenuated because of the effects of geometry and material nonlinearity, and on what length scale these effects are most pronounced.
|Number of pages||24|
|Journal||Archive for Rational Mechanics and Analysis|
|Publication status||Published - 1989|