The scattering of a scalar wave by a semi-infinite rod of circular cross section

Abstract

The form of the exact solution for the scattering of a plane harmonic scalar wave by a semi-infinite circular cylindrical rod of diameter 2 a is found when the boundary condition is u — 0 or du/dv — 0,where u represents the scalar field and v is the normal to the rod. When the angle of incidence is n, i.e. the angle between the direction of propagation of the incident wave and the normal (out of the rod) to the end is , the average pressure amplitude on the end of the rod and the scattering coefficient are found for the boundary condition 0. Graphs are given showing the behaviour of these quantities for the range 0 < < 10, where k is the wave-number. When ka reaches 10, the quantities have almost become constant. For small values of ka the scattering coefficient is shown to be {ka)2', it appears from the numerical results that this is, in fact, a fairly close approximation for ka <2. It is further shown that the average pressure amplitude on the end for other angles of incidence is approximately the product of the average pressure amplitude for an angle of incidence of n and the amplitude of the symmetric mode (ka < 3-83) which the incident field would produce inside a hollow semi-infinite cylinder occupying the same position as the rod. When the boundary condition is u — 0 and ka is small it is proved that the scattered field is the same as that due to a semi-infinite hollow cylinder longer by an amount 0-la approximately. A similar result does not hold for the boundary condition 0. The theory is extended to the case when a pressure pulse falls on a circular rod. It is found that the pressure on the end drops almost to its final value in the time taken for a wave to travel the diameter of the rod, and that the average pressure during this process is given, at time by (0-9154-0-745(2— a0tja)2}I approximately, where a0 is the speed of sound. Tables, in the range 0(0-25)10 of ka, of the ‘split’ functions which arise in connexion with a semi-infinite cylinder are given.