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<div class="text-justify">The study of waves and their behavior may be more intuitive if we think of large mechanical models and very simple non-periodic waves, such as pulses. Consider, for example, a freight train with many wagons attached to a locomotive but stopped. If the locomotive starts abruptly, its traction on the first car sends a displacement wave that runs along the line of wagons.

The disturbance of the initial displacement comes from the locomotive that is making click the links one by one. In this example, the locomotive is the source of the disturbance, while the freight cars and their couplings are the means. The "stroke" that travels along the line of wagons is the wave. The disturbance moves from one end to the other of the train and with it goes the energy of displacement and movement. However, no particle of matter moves with the wave; each car moves only a little forward.

How long does it take the effect of the disturbance created at a point to reach a distant point? The time interval depends, of course, on the speed with which the disturbance propagates. This speed, in turn, depends on the type of wave and the characteristics of the medium. In any case, the effect of a disturbance is never transmitted instantaneously. Each component of the medium has inertia and each part of the medium is compressible. Therefore, it takes time to transfer energy from one part to another. The same applies equally to transverse waves.

The next series of images represents a wave, a pulse, on a string. Think of each image as a frame of a film, and that one is taken at equal intervals of time. We already know that the material of the rope does not travel together with the wave. But each part of the string passes moves up and down when the wave passes. Each piece undergoes exactly the same movement as the piece on its left, except that it does so a little later.</div>

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<div class="text-justify">Let's look at the small piece of string marked with an X in the first "frame". When the pulse traveling along the string reaches X what is happening is that the piece of string just to the left of X exerts an upward force on X. When X moves upward, the next piece exerts a restoring force (a downward force). The more X moves up, the greater the restorative forces. There comes a time when X stops moving up and begins to descend again. The section of the rope to the left of X now exerts a restoring force (downward), while the right hand section exerts an upward force. Therefore, the downward movement is similar, but opposite, to the upward movement. Finally, X returns to the equilibrium position when both forces have disappeared.

The time required for X to rise and fall, that is, the time required for the pulse to pass through that part of the string, depends on two factors. These factors are the magnitude of the forces in X and the mass of X. In other words and in more general terms: the speed with which a wave propagates depends on the stiffness and the density of the medium. The more rigid the medium, the greater the force that each section exerts on the neighboring sections and, therefore, the greater the speed of propagation. On the other hand, the greater the density of the medium, the less it will respond to the forces * and, therefore, the slower the propagation will be.

In fact, the speed of propagation depends on the relationship between the stiffness factor and the density factor. The exact meaning "stiffness factor" and "density factor" is different for each wave type and for different media. For example, for tense strings the stiffness factor is the tension T in the string, and the density factor is the mass per unit length, m / l, and the velocity of propagation v is given by v = [T / ( m / l)] ½</div>

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