Spring-Coupled Masses

(293) |

where represents longitudinal displacement from equilibrium. It is assumed that all of the displacements are relatively small: that is, , for .

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

where . Because there is nothing special about the th mass, the preceding equation is assumed to hold for all masses: that is, for . Equation (294), which governs the longitudinal oscillations of a linear array of spring-coupled masses, is analogous in form to Equation (232), which governs the transverse oscillations of a beaded string. This observation suggests that longitudinal and transverse waves in discrete dynamical systems (i.e., systems with a finite number of degrees of freedom) can be described using the same mathematical equations.

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

where , , , and are constants. The preceding expression automatically satisfies the boundary condition . This follows because and , and, consequently, . The other boundary condition, , is satisfied provided

(296) |

which yields [cf., Equation (242)]

where is an integer. As before, the imposition of the boundary conditions causes a quantization of the possible mode wavenumbers. (See Section 5.1.) Finally, substitution of Equation (295) into Equation (294) gives the dispersion relation [cf., Equation (240)]

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,

where

and the and are arbitrary constants determined by the initial conditions. Here, the integer indexes the masses, and the mode number indexes the normal modes. It can be demonstrated that there are only unique normal modes, corresponding to mode numbers in the range 1 to .

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

where is a constant that has the dimensions of velocity. It seems plausible that Equation (301) is the dispersion relation for sound waves in a continuous elastic medium. Let us investigate such waves.