Reflection of Spherical Sound Wave at a Rigid Sphere

Abstract

In this paper, a calculation method for the amplitude of reflected sound waves from a spherical body at a short distance from the sound source is described. This method is expressed in the terms of the "directional sound reflectivity of the spherical body". This directional sound reflectivity is the ratio of the sound pressure of the reflected wave from a spherical body at the position of the receiving point to the particular pressure which would be calculated by the theory of geometrical optics. As shown in Fig. 1, a case is considered in which the spherical body is put on the origin of spherical coordinate. V_N, that is the component of the particle velocity of the incident wave in the radial direction at the surface of the body, is given by Eqs. (4) and (5). Reflected waves can be expressed by Eq. (6), which is the solution of Laplace's equation. The particle velocity of the reflected wave at the surface V_R must be -V_N. So, the coefficients C_n are given by Eq. (8). The velocity potential of the reflected wave from a spherical body can be written by Eqs. (9) and (10). If the sound wave can be treated by the theory of geometrical optics, the approximate value of the value potential of the reflected wave is expressed by Eq. (15). From the above relations, the absolute value of the directional sound reflectivity of a spherical body, R_&ltps&gt, is given by Eqs. (16) and (17). This relation becomes Stenzel's relation when L=M and M is increased up to infinity. Figs. 3, 4 and 5 show examples of the numerical values of |R_&ltps&gt| when L=M. Some experiments were performed in the air at a frequency of 40. 00 kHz so as to verify the validity of above theories. Fig. 6 shows the arrangement of the apparatus in the experiment. Two piezoelectric transducers were put toward a steel ball. The pulse width of the sound wave from the transmitter was 0. 4 ms. The amplitude of the reflected pulse was measured on the screen of the C. R. O. Fig. 7 and 8 show examples of these experiments. In these experiments, the results qualitatively agree with the theory when the angle between the direction of incidence and that of reflection is comparatively small. From the results of above experiments, it is found that the calculation method for the reflection of sound waves from a spherical body and its numerical results are useful when the sound source is at a short distance from the spherical body.