Suppose a disturbance which is initially confined to a finite region is propagated in free space according to the wave equation. It spreads outward and eventually reaches every point in space. However, at any fixed point the disturbance eventually dies out, decaying at a rate that depends on the number of space dimensions. If there are three space dimensions it follows from Huyghen's principle that there is eventually no disturbance at all. In two dimensions the rate of decay with time t can be exactly l/t.For physical reasons, it has been conjecturedl that the same situation holds if there is a reflecting body somewhere in space. The wave from the initial disturbance reaches the body and is reflected from it. At any point the disturbance can be measured in terms of "energy". Eventually one expects all the energy, and thus the disturbance too, to be carried outward. Of course, if a body has a very complicated indented shape, the rate of decay will be slowed by the reverberations within the indentations. But one would still expect the energy to leak out in time.For a spherical body it has been shown by C . H.

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If a plane shock hits a wedge, a self-similar pattern of reflected shocks travels outward as the shock moves forward in time. The nature of the pattern is explored for weak incident shocks (strength b) and small wedge angles 20,. through potential theory, a number of different scalings, some study of mixed equations and matching asymptotics for the different scalings. The self-similar equations are of mixed type. A linearization gives a linear mixed flow valid away from a sonic curve. Near the sonic curve a shock solution is constructed in another scaling except near the zone of interaction between the incident shock and the wall where a special scaling is used. The parameter p = clO$/(y + 1)b ranges from 0 to co. Here y is the polytropic constant and CI is the sound speed behind the incident shock.For p > 2 regular reflection (weak or strong) can occur and the whole pattern is reconstructed to lowest order in shock strength. For p < Mach reflection occurs and the flow behind the reflection is subsonic and can be constructed in principle (with an open elliptic problem) and matched. The case 0 = 0 can be solved. For $ < 0 < 2 or even larger p the flow behind a Mach reflection may be transonic and further investigation must be made to determine what happens.The basic pattern of reflection is an almost semi-circular shock issuing, for regular reflection, from the reflection point on the wedge and for Mach reflection, matched with a local interaction flow.Assuming their nature, choosing the least entropy generation, the weak regular reflection will occur for p sufficiently large (von Neumann paradox). An accumulation point of vorticity occurs on the wedge above the leading point.

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