On the reflexion of a spherical sound pulse by a parabolic mirror

Extract

This paper is an attempt to apply a simple series solution of the wave equation to the reflexion of a sound pulse. The reflector is a paraboloid of revolution, and the incident pulse is spherically symmetrical and comes from the focus of the reflector, so that the wave fronts of the reflected pulse are planes at right angles to the axis of symmetry of the reflector. If the incident pulse consists, at any point, of a discontinuous pressure rise followed by constant excess pressure, the series reduces to a power series in the time counted from the onset of the reflected pulse; for other forms of the incident pulse it can be interpreted as the result of the superposition of elementary pressure pulses which are constant in a small time interval and vanish outside it. No convergence proof is given, so that the interest of the investigation is physical rather than mathematical; but the numerical results indicate that the convergence of the series is unsatisfactory except near the vertex of the reflecting paraboloid. The coefficients of the series are obtained with the aid of recurrence formulae, and the first seven coefficients have been calculated. The calculations become more laborious at each successive stage. A detailed numerical investigation of the reflexion of the ‘simple rectangular pulse’ referred to already (for which the series reduces to a power series) reveals that initially the maximum pressure on any plane section at right angles to the axis of the paraboloid, due to the reflected pulse alone, occurs at the axis and has the same value everywhere on it; but after some time a secondary pressure maximum is established over a circular ring at some distance from the axis. A consideration of the initial pressure gradient of the reflected pulse suggests that a similar state of affairs exists at all distances from the vertex, but the actual calculations only extend to a plane section whose distance from the vertex is four times the focal length. The unsatisfactory convergence of the series precludes the investigation of subsequent changes in the distribution of pressure. It is finally pointed out that these results apply to a certain extent to a finite parabolic mirror.