Theory

Consider a system consisting of two gliders, each of mass m, connected by an ``inner'' spring of constant . The gliders are each attached to fixed supports by identical ``outer'' springs of constant (see Fig. 1). We assume, for the present, that there is no air resistance.

If we measure the displacement of each glider from its equilibrium position, taking displacement to the right as positive, Newton's second law gives
  
where the subscripts 1 and 2 refer to the left- and right-hand gliders, respectively.

These equations are non-trivial to solve since they are coupled differential equations: the equation for the acceleration of glider 1 depends on the positions of both glider 1 and 2, and similarly for glider 2. Detailed solutions are found in most intermediate-level mechanics books,[3] but for our purposes we need to know just a few simple results:

1. There are two ``normal'' modes of oscillation: an ``antisymmetric'' mode and a ``symmetric'' mode (see Fig. 2). In each case the gliders move equal distances,
   1. towards and away from each other (antisymmetric mode), or
   2. back and forth, keeping a constant distance between them (symmetric mode).
2. All other motions can be represented as linear combinations of these normal modes (superposition), and
3. The oscillations have the mathematical representation , where A is the amplitude of the motion (in meters), is the angular frequency (in radians/sec), and is a phase angle. The physical system determines , while A and are set by the initial conditions.

Thus we expect that the motion of, say, glider 1 to be represented by the sum of two cosine functions,
 
where the a and s subscripts refer to the antisymmetric and symmetric modes, respectively. We expect that the A's and 's are set by the initial conditions, and the two 's are set by the masses and spring constants.