Abstract
A finite difference solution to an NPE-like equation is used to study the linear and nonlinear behavior of focusing step shocks in regions where caustics are formed. The initial condition is a nominally plane step pulse with a hyperbolic tangent rise profile of thickness lsh; the wavefront has a single ripple of length scale Lwf, concave towards the direction of propagation. When ε=lsh/Lwf is much less than unity, the linear numerical simulation showed that the shock behaves nearly according to geometric theory. Several different values of ε were investigated, verifying the linear theory predictions of normalized amplitude ∼ε−1/4 at the cusp [J. Hunter and J. Keller, Wave Motion 9, 429–443 (1987)]. Numerical solutions are also obtained for three cases of nonlinear propagation with dissipation. In addition to ε, the distinguishing parameters are shock amplitude and dissipation. It is observed that for cases in which nonlinearity is stronger relative to diffraction and focusing, the wavefront forms a cusp at a later time, caustics are less pronounced, and the (normalized) rise time is shorter. Implications of these results on the effect of caustics on sonic boom rise times is discussed.
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