Transmission of Plane Waves Through Layers. Electroacoustical Analogies. Radiation of Plane Waves.

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"/>