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The speed of a wave in a tethered trolley system  

A theoretical argument
The pulse discussed is one in which the left-hand trolley in the diagram below is moved at a steady speed to the right. One by one, each trolley in turn acquires the same speed, u, to the right.

pulse on a row of trolleys

The speed with which the disturbance passes along the row depends on how quickly a trolley, initially at rest in equilibrium, accelerates when its neighbour is displaced from its equilibrium position. This in turn, of course, depends on the mass of the trolley, and the force acting on it. Now the force depends on the compression of the spring and its force constant. (One can show that adding mass to the trolleys slows the pulse down; and making the springs stiffer - by adding a second spring in parallel - speeds it up.)

By applying Newton's Laws to this event one can derive an expression for the speed of the pulse in terms of the masses of the trolleys, their spacing, and the stiffness of the springs linking them.

When the pulse travels along the row each trolley in turn begins to move, as the disturbance passes. The speed with which the disturbance moves along the trolley train depends on how quickly this movement is passed on from one trolley to the next.

The time it takes a trolley to respond to the push of its neighbour depends on the period of oscillation of the trolley: the longer the period the greater the time delay between one trolley starting to move and reaching the speed u and the next doing so.

It may help to think about other oscillating systems. The time it takes for a pendulum or child's swing to respond to a sudden impulse depends on its period. The response of a moving coil meter to a sudden change in current depends on the meter's response time, which is governed by 'mass' and stiffness' factors For a rapid response mass should be low and 'stiffness' high. A reasonable guess might be that the maximum deflection is reached after a time delay of a quarter of a period of oscillation.

at differnt times
The diagram above shows the line of trolleys at three successive short time intervals. In figure (b), trolley Q is just being set into motion. By the instant of diagram (c), the pulse has advanced by one section-length (x), and R is now in the same situation as Q was previously. The speed of advance of the pulse is thus the distance x divided by the time interval between diagrams (b) and (c).

If Q is considered as a tethered-trolley oscillator, then in figure (b) it is at the left-hand extreme of an oscillation; by figure (c) it has reached its equilibrium position between trolleys P and R. This is one-quarter of an oscillation. The time interval for the pulse to be handed on to the next oscillator thus appears to be one-quarter of the period of the oscillation. In fact, a more thorough analysis shows that the fraction is not  but  of an oscillation.

Thus the wave speed,

The argument has been presented in terms of a step impulse.
Experiment shows that waves travel with this speed, and that for low frequencies (that is, wave frequency << natural frequency of oscillator) the speed does not depend on wave frequency. The relationship

applies to continuous waves as well as the idealized step impulse just considered.
This qualitative argument is valid as long as the wave frequency is small compared with the natural frequency of oscillation of the particles of the medium. The argument breaks down at higher frequencies for which the wavelength begins to get as small as the spacing between particles. And by oscillating the end trolley rapidly one can show that high frequency waves are not transmitted at all.

A footnote to the argument
Someone who has followed this argument well might wonder why the natural oscillation frequency (fnat) should control the speed of propagation of a sinusoidal wave of much lower frequency (fwave). One might then argue as follows.
displacement or force on an individual particle
The displacement of an individual particle and therefore the force on it varies sinusoidally at frequency fwave. We could consider this continuous variation in force as being made up of many tiny step impulses, as shown below.
sinusoidal variation
The reaction of each particle to a step impulse is limited by its natural response time, that is, by fnat, not by the wave frequency.
It may be useful to show that an oscillating system can respond to a continuously varying input, if the frequency is low enough: a moving coil meter will show slow a.c. quite faithfully. But at high frequency it hardly responds at all.