ABSTRACT
The propagation of ultrasonic waves in different media, which we shall regard as continuous, is accompanied by a periodic displacement of particles of the medium from their equilibrium positions under the action of elastic forces. Here, “particle” refers to an infinitesimal volume element which itself contains a very large number of molecules so that the medium within it can be considered to be continuous. In the normal, unperturbed state all particles in the medium are in certain equilibrium positions, determined by the balance of intermolecular forces. We shall denote the equilibrium position of a particle by a radius vector r (position vector), drawn from the origin of some system of coordinates (laboratory system) that is stationary relative to the given medium. For such a system we shall most often choose a Cartesian rectangular coordinate system x, y, z. In many cases, it will be more convenient to use a spherical coordinate system r, θ, ψ, which is related to the rectangular system by the relations x = r sin θ cos ψ, y = r sin θ sin ψ, z = r cos θ, or a cylindrical system r, θ, z in which x = r cos θ, y = r sin θ, z = z. We shall describe the displacement of a particle from its position of equilibrium with the help of the displacement vector u. Thus the new position of the particle after displacement will be determined by the vector r + u. We shall denote the components of the displacement vector u along the coordinate axes by the symbols ξ, η, and ζ, respectively. The magnitude of the displacement depends on the position of the particle and, in the general dynamic case, the displacement can vary with time. Thus the components of the displacement ξ, η, and ζ are, in general, functions of the coordinates and time: ξ = ξ (x, y, z, t), η = η (x, y, z, t) and ζ = ζ(x, y, z, t).
