## 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 2a is found when the boundary condition is u=0 or $\partial $u/$\partial \nu $ = 0, where u represents the scalar field and $\nu $ is the normal to the rod. When the angle of incidence is $\pi $, i.e. the angle between the direction of propagation of the incident wave and the normal (out of the rod) to the end is $\pi $, the average pressure amplitude on the end of the rod and the scattering coefficient are found for the boundary condition $\partial $u/$\partial \nu $ = 0. Graphs are given showing the behaviour of these quantities for the range 0 $\leq $ ka $\leq $ 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 $\frac{1}{4}$(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 $\pi $ and the amplitude of the symmetric mode (ka < 3$\cdot $83) which the incident field would produce inside a hollow semi-infinite cylinder occupying the same position as the rod. When the boundary condition is u=0 and ka is small it is proved that the scattered field is the same as that due to a semi-infinite hollow cylinder longer by an amount 0$\cdot $1a approximately. A similar result does not hold for the boundary condition $\partial $u/$\partial \nu $ = 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 t, by {0$\cdot $915 + 0$\cdot $745(2 - a$_{0}$t/a)$^{2}$}$^{\frac{1}{2}}$ approximately, where a$_{0}$ is the speed of sound. Tables, in the range 0(0$\cdot $25)10 of ka, of the 'split' functions which arise in connexion with a semi-infinite cylinder are given.