Linear acoustic theory

The linear acoustic theory is concerned with the propagation of sound waves considered as small perturbations in a fluid or gas. Consequently the equations of acoustics are obtained by linearization of the equations for the motion of fluids. The two main media for the propagation and scattering of sound waves are air and water (underwater acoustics). A third important medium with properties close to those of water is the human body, i.e., biological tissue (ultrasound). We are herein interested in obtaining the differential equations that govern the acoustic wave propagation, whose linearization yields the scalar wave equation of acoustics. By considering simple-harmonic waves for the wave equation, we obtain finally the Helmholtz equation. When the frequency is zero, this equation turns into the Poisson or the Laplace equation. The corresponding boundary conditions are also developed, in particular the impedance boundary condition. A good and complete reference for the linear acoustic theory is the article by Morse & Ingard (1961), which is closely followed herein. Other references are DeSanto (1992), Elmore & Heald (1969), Howe (2007), Kinsler, Frey, Coppens & Sanders (1999), Kress (2002), and Strutt (1877). Acoustic motion is, almost by definition, a perturbation. The slow compressions and expansions of materials, studied in thermodynamics, are not thought of as acoustical phenomena, nor is the steady flow of air usually called sound. It is only when the compression is irregular enough so that overall thermodynamic equilibrium may not be maintained, or when the steady flow is deflected by some obstacles so that wave motion is produced, that we consider part of the motion to be acoustical. In other words, we think of sound as a byproduct, wanted or unwanted, of slower, more regular mechanical processes. And, whether the generating process be the motion of a violin bow or the rush of gas from a turbo-jet, the part of motion we call sound usually carries but a minute fraction of the energy present in the primary process, which is not considered to be acoustical. This definition of acoustical motion as being the small, irregular part of some larger, more regular motion of matter, gives rise to difficulties when we try to develop a consistent mathematical representation of its behavior. When the irregularities are large enough, for example, there is no clear-cut way of separating the acoustical from the non-acoustical part of the motion. In fact, only in the cases where the non-steady motions are first-order perturbations of some larger, steady-state motion can one hope to make a self-consistent definition which separates acoustic from non-acoustic motion and, even here, there are ambiguities in the case of some types of near field. Thus it is not surprising that the earliest work in acoustic theory, and even now a vast quantity, has to do with situations where the acoustical part of the motion is small enough so that linear approximations can be used. These are our cases of interest in this thesis. Strictly speaking, the equations to be discussed here are valid only when the acoustical component of the motion is ”sufficiently” small, but it is only in this limit that we can unequivocably separate the total motion into its acoustical and its non-acoustical parts.

Differential equations

Basic equations of motion

Considering the fluid as a continuous medium, two points of view can be adopted in describing its motion. In the first, the Lagrangian motion, the history of each individual fluid element, or particle, is recorded in terms of its position x as a function of the time t. Each particle is identified by means of a parameter, which is usually chosen to be the position vector x0 of the element at t = 0. The Lagrangian description of fluid motion is expressed by the set of functions x = x(x0, t). In the second, or Eulerian, description, on the other hand, the fluid motion is described in terms of a velocity field V(x, t) in which the position x and the time t are independent variables. The variation of V with time, or of any other fluid property in this description, refers thus to a fixed point in space rather than to a specific fluid element, as is the case with the Lagrangian description. If a field quantity is denoted by ΨL in the Lagrangian and by ΨE in the Eulerian description, then the relation between the time derivatives in the two descriptions is dΨL dt = ∂ΨE ∂t + (V · ∇)ΨE. (A.898) We remark that in the case of linear acoustics for a homogeneous medium at rest we need not be concerned about the difference between (dΨL/dt) and (∂ΨE/∂t), since the term (V · ∇)ΨE is then of second order. However, in a moving or inhomogeneous medium the distinction must be maintained even in the linear approximation. We shall ordinarily use the Eulerian description and, if we ever need the Lagrangian time derivative, we shall express it as the right-hand side of (A.898), omitting the subscripts. We express herein the fluid motion in terms of the three velocity components Vi of the velocity vector V. We denote further the velocity amplitude as V = |V |. In addition, the state of the fluid is described in terms of two independent thermodynamic variables such as pressure and temperature or density and entropy. We assume that thermodynamic equilibrium is maintained within each volume element. Thus in all we have five field variables: the three velocity components and the two independent thermodynamic variables. In order to determine these functions of x and t we need five equations. These turn out to be conservation laws: conservation of mass (one equation), conservation of momentum (three equations), and conservation of energy (one equation). If the density of the fluid is denoted by ̺ and i, j ∈ {1, 2, 3}, then the mass flow in the fluid can be expressed by the vector components and the total momentum flux by the tensor tij = Pij + ̺ViVj 

Boundary conditions

Reaction of the surface to sound

We discuss now the behavior of sound in the neighborhood of a boundary surface, and see whether we can express this behavior in terms of boundary conditions on the acoustic field. It turns out that in many cases the sorts of boundary conditions familiar in the classical theory of boundary-value problems, such as that the ratio of value to normal gradient of pressure is specified at every point on the boundary, is at least approximately valid. At first sight it may seem surprising that the ratio of pressure to its normal gradient, which to first order equals the ratio of pressure to normal velocity at the surface, could be specified, even approximately, at each point of the surface, independently of the configuration of the incident wave (vid. equation (A.936)). Of course, if the wall is perfectly rigid so that the value of the ratio is infinite everywhere, then the assumption that this ratio is independent of the nature of the incident wave is not so surprising. But many actual boundary surfaces are not very rigid, and in many problems in theoretical acoustics the effect of the yielding of the boundary to the sound pressure is the essential part of the problem. When the boundary does yield, for the classical boundary conditions to be valid would imply that the ratio of incident pressure to normal displacement of the boundary would be a characteristic of each point of the surface by itself, independent of what happens at any other point of the surface. To see what this implies, regarding the acoustic nature of the boundary surface, and when it is likely to be valid, let us discuss the simple case of the incidence of a plane wave of sound on a plane boundary surface. Suppose the boundary is the x2-x3 plane, with the boundary material occupying the region of positive x1 and the fluid carrying the incident sound wave occupying the region of negative x1, to the left of the boundary plane. Suppose also that the incident wave has frequency f = ω/2π and that its direction of propagation is at the angle of incidence φ to the x1 axis, the direction normal to the boundary.