most recent revision 29 January 2000
If we can determine that a given mechanical medium is made up of simple harmonic oscillators, we immediately know that propagation of sound in the medium can be described by the Wave Equation. That, in turn, will mean that the medium has a number of very important characteristics, such as a well defined velocity of sound for the propagation of mechanical disturbances, and the support of multiple simultaneous travelling waves by linear superposition.
Either of the following attributes can be diagnostic of a simple harmonic oscillator: (1) a restoring force proportional to displacement from equilibrium, and (2) a potential energy proportional to the square of displacement from equilibrium. Which of those attributes is easier to observe will vary from one medium to another.
For each particle of a one-dimensional string or two-dimensional membrane, the nature of the restoring force can be observed directly. A very similar analysis applies to each tiny region of a three-dimensional gaseous medium like air. For a three-dimensional solid, on the other hand, the potential energy is more accessible to observation than the restoring force per se.
By "ideal" we mean that (1) string tension is the only source of the restoring force, i.e. that the string has negligible stiffness, and (2) the string has uniform density (mass per unit length), i.e. that any particle of the same size will have the same mass.
Consider a typical particle of the string. When the string is at its equilibrium position it will form a single straight line, all the particles that make it up will be at their equililbrium positions, and the two tension forces acting on each particle (TL to the left and TR to the right) will be equal in magnitude and opposite in direction, summing to a zero net force.
Now consider displacing a chosen particle some distance x along a line perpendicular to the axis of the string. [Initially it will be convenient to think in terms of a particle at the middle of the string, but our conclusions then will generalize to any location along the string.] The two segments of the displaced string will make equal small angles a with respect to the axis and the two tension forces TL and TR will no longer act in exactly oposite directions. For purposes of analysis, divide each of the tension forces into two components, one parallel to the axis of the string and the other perpendicular to it. Then the two parallel components will be of equal magnitudes and opposite directions, summing to zero, while the two perpendicular components will be of equal magnitudes in the same direction, i.e. they will sum to a net perpendicular restoring force. The perpendicular component of each tension force is proportional to the sine of the displacement angle [T sin(a)]. For small values of the angle a, the perpendicular component of each tension force is proportional to the perpendicular displacement. One way to demonstrate that is to note that the slope of sin(a) is 1 at a = 0: d(sin(a))/da = cos(a) and cos(0) = 1, so near a = 0 we find that sin(a) = a. Note that going to the limit of small values of the angle a will also remove any differences in our analysis among particles at different locations along the string.
So each particle on our ideal string is subject to a restoring force proportional to its displacement perpendicular to the string axis, and thus must function as a simple harmonic oscillator. Motions of the ensemble of all particles that make up the string, no matter how complex, must satisfy the Wave Equation.
Our text book approaches this analysis inductively, beginning with the single transverse vibrational mode of a single particle suspended between a pair of identical springs. Adding a second particle of identical mass, with another identical spring between particles, leads to a system with two vibrational modes. Similarly, a three particle "string" will have three modes, and in general an n-particle string n modes.
This case is exactly analogous to the one-dimensional ideal string, again assuming the absence of any stiffness in the membrane, and a uniform density (mass per unit area in this two-dimensional case). Again, the tension forces will sum to zero when the membrane, and thus all its particles, are in their equilibrium positions. When a typical particle is displaced along a direction perpendicular to the equilibrium plane of the membrane the tension components parallel to that plane again will sum to a zero net force in that plane. And again there will be a net restoring force perpendicular to the plane, one that is proportional to displacement from equilibrium for small displacement angles.
under construction -- check back later
under construction -- check back later