We shall now study the transmission of ultrasonic plane waves through a layer with plane-parallel boundaries. We denote the acoustic impedance of the layer by z = ρc and the acoustic impedance of the medium on both sides of the layer by z1 = ρ1c1. We orient the x-axis perpendicular to the boundaries of the layer, to which we assign the coordinates x = 0 and x = d (d is the thickness of the layer), and we consider at the outset the general case of oblique incidence of ultrasonic waves at an arbitrary angle θ to the x-axis (Fig. 48). Reflected and refracted waves will appear at each boundary; in addition, because of the symmetry, the wave passing through the layer will emerge from the layer at the angle of incidence θ1. By direct analogy with Eqs. (VII.29)—(VII.31) we have the following expressions for the potentials of these waves: for the incident wave, (φ1=φ1maxexp{i[ωt−k1(x cos θ1+γ cos θ1)]}; https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9780429332227/64e5e3df-c95f-4a2a-a02e-98ef4c2d440f/content/unequ8_1.tif"/>