The wave traveling in one direction is now expressed by the complex exponential

An additional parameter, θ0, called the phase of the wave has been added. This is determined by the initial conditions and represents the fact that one might not have placed their coordinates exactly on a crest at t = 0. Attenuation is an exponential damping of the amplitude that usually occurs in the direction of propagation:

where it is assumed that the source is at n x = 0. Combining the exponents gives

with κ ≡k + iγ. For the most part the basic laws of optics hold for acoustics as well. This includes the law of reflection from a smooth hard barrier, that is, the angle of reflection equals the angle of incidence, and the law of refraction at a flat interface between two media, Snell’s law. Up to this point the presentation of wave phenomenon has been phenomenological, based on observation (pseudo-observation) and an attempt to categorize the basic properties observed. Now that a clear vocabulary for the description of wave phenomenon is in place, and some guidance as to how the observed parameters are related to a source versus the medium, the presentation will be less descriptive and more mathematical.

3.3 One-Dimensional Waves on a String

To derive a linear wave equation for one-dimensional waves, consider a line of identical coupled masses, mi = m connected to nearest neighbors by identical springs, with spring constant ki = k. The number of masses is finite and the first and last springs are connected to rigid boundaries. Each mass has a coordinate position, xi measured relative to its equilibrium position, and velocity, xi = dxidt. The kinetic energy of the system may be written as

The total potential energy of the springs is a function of the position of each mass. Except for the end springs, each is stretched or compressed relative to its original length by an amount xi − xi − 1:

The first and last terms account for the end springs that are attached to large infinitely massive walls. The equations of motion for the masses are derived from the Lagrangian L = K −V:

Applying (3.15) to the Lagrangian associated with (3.13) and (3.14) gives a system of coupled equations for the motion of the masses:

A solution of the form xit = ui cosωt is assumed, which leads to the characteristic equation

This is an eigenvalue equation. For nontrivial solutions to exist, the determinant of the matrix must be equal zero, leading to an eigenvalue equation for ω2. For each eigenvalue ωn, there is a corresponding eigenvector un. A system of N-coupled masses will yield N eigenvalues:

The somewhat awkward notation for the eigenvector references the normal mode via n and the particle by i. The natural frequency of the single mass–spring system is introduced in (3.20):

k

ω0 = (3.22) m

The aforementioned treatment provides a complete description of the discrete mass–spring system that is used as a starting point for a continuum equation. Passing to the continuum involves letting the number of masses go to infinity as the distance between them goes to zero under constraints that the mass density and other physical quantities remain finite. To develop a suitable equation in the continuum limit, attention should be focused on an arbitrary mass, not coupled to either boundary, for example, the ith mass. The equilibrium distance between two adjacent masses, L0, is the quantity that will go to zero in the limiting process. As the number of masses goes to infinity, the individual mass will tend to zero and the local mass density,

ρ = mL0, will be held constant. Young’s modulus is defined via Y = kL0 in this limit. Rearranging terms in the equation of motion and using these definitions gives the following equation for the i-th mass:

A spatial parameter, denoted σ, is defined and used to locate a position on the x-axis independent of the mass locations, labeled xi. The location of the mass element, as the limit is taken, is a function of time and this new spatial parameter, x(t, σ). The continuous parameter σ acts as, or replaces, the discrete index i, for example, xt,iL0 = xit. The following approximations to the first and second partial derivative of, x(t, σ) with respect to σ are made:

With this, the discrete equation for the motion of the i-th mass becomes

The end points, infinite massive walls, become boundary conditions on the field x(t, σ). Defining the walls to be at σ = 0, and σ = N + 1L0 ≈ NL0 = L, leads to the conditions xt,0 = xt,L0 = 0. The discrete system of N masses has N normal modes of vibration. Passing the limit to a continuum model, the spectrum becomes

which can be derived by taking the small argument approximation of (3.20) and using the definitions of the macroscopic quantities previously defined. Similarly, for the amplitude vector:

The discrete set of normal modes becomes a field:

The eigenfrequencies can be written in terms of the lowest frequency, n = 1, ωn = nω1, with ω1 derived from (3.27). This process is meant to demonstrate how the two paradigms, discrete versus continuous, are related and impress upon the reader the value of reductionism. Building models of bulk behavior from particle models is a common approach to understanding how discrete properties of the micro system lead to properties of the macro system [2].

3.4 Waves in Elastic Solids

In the study of elasticity, the medium is assumed to be a continuum. The laws governing the deformation of the continuum under application of a force are derived by applying Newton’s laws to an infinitesimal volume of the solid. The stress tensor is defined as the force per unit area acting in each direction for any planar slice through the medium. Conversely, projecting the stress tensor onto different directions will give the internal forces acting at any point in the solid medium. For any plane through the material, the projection of the stress tensor in the direction normal to the plane is the normal stress or tensile stress, while the projection along the direction tangent to the plane defines the shear stress. Also defined is a strain tensor, which is related to the differential deformation of the material relative to its original state. In three dimensions, each may be expressed as 3-by-3 matrix:

The individual components of the strain are defined as follows:

The set of variables {uk} are the infinitesimal displacements along the axes defined by {xk}, where the following notation has been adopted for position and displacement of an element of the solid, x1, x2, x3 = x,y,z. It is worth noting that the stress and strain tensor components are defined relative to different systems. Stress is defined by looking at the equilibrium conditions applied to a deformed body, referred to as Eulerian coordinates. Strain is defined relative to the undeformed set of axes, a Lagrangian coordinate system. For small deformations, it can be assumed that these are approximately the same coordinates. Based on symmetry and physical arguments, one can prove that each of these is a symmetric matrix, σij = σji. In the study of

elasticity, there is a generalization of Hooke’s law that relates local stress and strain of a body at each point. This can be expressed easily as a 6-by-6 matrix:

Equation (3.33) can be written more succinctly as Σ = CE. This relationship is a threedimensional generalization of the law used to describe the mass–spring system, F = −kx. It is worth noting that (3.33) could be written in terms of a fourth rank tensor:

c1212 = C44, and so on. The 36 values of the matrix C are called the elastic constants of the material. For homogeneous materials, these are constant, that is, the same value at every point inside the material. In general, a solid material will exhibit anisotropic behavior. There exist a wide variety of elastic matrices depending on the symmetries of the material structure. For an isotropic material, there are only two independent elastic constants. The isotropic elastic tensor is given as follows:

The three parameters are related to each other:

These two independent parameters can be related to either Lamé’s parameters or Young’s modulus and Poisson’s ratio. Another type of material is one with cubic symmetry. For such materials, the stiffness matrix is identical in form to that of a purely isotropic material but without (3.36) relating the three variables. Cubic materials have three independent parameters.