Driving forces and resonance

A glass may shatter if they it is immersed in a sound wave having just the right frequency. This is demonstrated by the movie immediately following this paragraph. The sound wave seems to have no affect on the glass at the start of the movie. The frequency of the sound wave is changed as the movie proceeds. Finally, the glass shatters once the sound wave is tuned to a particular frequency. It must be stressed that it was not the amplitude (i.e., the intensity or "loudness") of the sound wave which was changed, but the frequency of the sound wave.

The behavior of many systems depend sensitively on the frequency on the external forces acting on them. An excellent model for these systems is a mass, m, attached to a spring which made to oscillate through the action of some external agent. A piston will be the external agent in the example which we shall consider. The spring has a stiffness k so that Newton's Second Law tells us that

where a is the acceleration of the mass. The term -γv results from the frictional forces acting on the mass. The variable v is the speed of the mass while γ is a constant which depends on the nature of the frictional forces acting on the mass. The quantity Δx is equal to the displacement of the spring (i.e., the amount by which the spring is stretched or compressed). Thus, Δx is equal to the difference between the length of the string at a given instant of time and the length of the spring when it is unstretched. Let L be the length of the spring when it is unstretched, x_m be the x coordinate of the mass, and x_p be the x coordinate of the piston. The instantaneous length of the mass is equal to x_m-x_p, so that

Let A be amplitude of the piston's oscillation (i.e., the maximum displacement of the piston relative to the piston's initial location) and ν be the piston's frequency of oscillation. Then

where ω=2πν. Putting the last three equations together yields

The three terms on the left hand side of the previous relation involve quantities that describe the motion of the mass. In particular, these terms involve the mass's position, speed, and acceleration. The first term on the right hand side is of no importance to the present discussion. It simply states that L is the equilibrium length of the spring. The second term, on the right hand side, describes the influence of the piston on the spring. It is the influence of the piston which causes, or drives, the spring to start oscillating. Thus, this second term is an example of what is known as a driving force (or external force). Sound plays the role of the driving force in the above movie of a breaking glass.

It is the parameters of the driving force which can be manipulated by the experimenter (in contrast, the stiffness of the spring and the frictional coefficient γ can not be easily changed by the experimenter). Thus, in the present example, the experimenter is able to control the amplitude and frequency of the piston; whereas the experimenter is able to control the volume (i.e., intensity) and frequency in the case of the movie of the breaking glass.