The reflection coefficient of a surface of Rayleigh distributed impedance
Formulas are obtained for the probability distribution P(k) of the amplitude k of (z — c2)2 (z + c,)'2 in which c, and c2 are real constants, z is a complex number of the form c + zx where c is a complex constant and z, is a complex variable of Gaussian distribution of amplitude and uniform distribution of phase (see Fig. 1). Fig. 1. Geometrical configuration of the vectors z + Ci and z — c2. The parallelogram containing dxi and dx2 represents the area swept by the end point of Xi as ranges between Xi and Xj + dxi and x2 ranges between x2 and x2 + dx2 . The end point of z ranges over an equal area. The above relation between k and z arises when c, = c2 in connection with the reflection of a plane wave of sound from a plane surface and relates the power reflection coefficient k to the acoustic impedance ratio z. The present paper expresses P(k) in terms of the modified Bessel functions I0 and 11 if c is real, and in terms of the incomplete gamma function if c is complex. Asymptotic formulas are included to allow calculation of P(k) beyond the range of tabulation of the modified Bessel and incomplete gamma functions.