Consider the equation of motion of the th mass. (See Figure 30.) The extensions of the springs to the immediate left and right of the mass are and , respectively. Thus, the -directed forces that these springs exert on the mass are and , respectively. The mass's equation of motion therefore becomes
We can interpret the quantities and , that appear in the equations of motion for and , respectively, as the longitudinal displacements of the left and right extremities of springs attached to the outermost masses in such a manner as to form the left and right boundaries of the array. The respective equilibrium positions of these extremities are and . The end displacements, and , must be prescribed, otherwise Equations (294) do not constitute a complete set of equations. In other words, there are more unknowns than equations. The particular choice of and depends on the nature of the physical boundary conditions at the two ends of the array. Suppose that the left extremity of the leftmost spring is anchored in an immovable wall. This implies that : that is, the left extremity of the spring cannot move. Suppose, on the other hand, that the left extremity of the leftmost spring is not attached to anything. In this case, there is no reason for the spring to become extended, which implies that . In other words, if the left end of the array is fixed (i.e., attached to an immovable object) then , and if the left end is free (i.e., not attached to anything) then . Likewise, if the right end of the array is fixed then , and if the right end is free then .
Suppose, for the sake of argument, that the left end of the array is free, and the right end is fixed. It follows that , and . Let us search for normal modes of the general form
It follows, from the preceding analysis, that the longitudinal normal modes of a linear array of spring-coupled masses, the left end of which is free, and the right end fixed, are associated with the following characteristic displacement patterns,
Figures 31 and 295 display the normal modes and normal frequencies of a linear array of eight spring-coupled masses, the left end of which is free, and the right end fixed. The data shown in these figures is obtained from Equations (299) and (300), respectively, with . The modes in Figure 31 are all plotted at the instances in time when they attain their maximum amplitudes: namely, when . It can be seen that normal modes with small wavenumbers--that is, , so that --have displacements that vary in a fairly smooth sinusoidal manner from mass to mass, and oscillations frequencies that increase approximately linearly with increasing wavenumber. On the other hand, normal modes with large wavenumbers--that is, , so that --have displacements that exhibit large variations from mass to mass, and oscillation frequencies that do not depend linearly on wavenumber. We conclude that the longitudinal normal modes of an array of spring-coupled masses have analogous properties to the transverse normal modes of a beaded string. (See Section 5.1.)
The dynamical system pictured in Figure 30 can be used to model the effect of a planar sound wave (i.e., a longitudinal oscillation in position that is periodic in space) on a crystal lattice. In this application, the masses represent parallel planes of atoms, the springs represent the interatomic forces acting between these planes, and the longitudinal oscillations represent the sound wave. Of course, a macroscopic crystal contains a great many atomic planes, so we would expect to be very large. However, according to Equations (297) and (300), no matter how large becomes, cannot exceed (because cannot exceed ), and cannot exceed . In other words, there is a minimum wavelength that a sound wave in a crystal lattice can have, which turns out to be twice the interplane spacing, and a corresponding maximum oscillation frequency. For waves whose wavelengths are much greater than the interplane spacing (i.e., ), the dispersion relation (298) reduces to