### ASA 128th Meeting - Austin, Texas - 1994 Nov 28 .. Dec 02

## 4pPAa6. A study of the role of diffraction in the behavior of focusing
step shocks using NPE-type equations and a finite difference program.

**Andrew A. Piacsek
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*Graduate Prog. in Acoust., Penn State Univ., P.O. Box 30, State College,
PA 16804
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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 l[sub sh]; the wavefront has a
single ripple of length scale L[sub wf], concave towards the direction of
propagation. When (epsilon)=l[sub sh]/L[sub wf] is much less than unity, the
linear numerical simulation showed that the shock behaves nearly according to
geometric theory. Several different values of (epsilon) were investigated,
verifying the linear theory predictions of normalized amplitude ~(epsilon)[sup
-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 (epsilon), 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.