Progressive Waves: Transverse and Longitudinal (1953)

Wave motion in its most familiar form occurs on the surface of water. Since the motion of the water particles on and near the surface is rather complicated, we will consider two simpler basic types of waves, transverse waves and longitudinal waves. First, transverse waves. Let's suppose we have a very long rope attached to some very distant support. If the hand is moved regularly up and down, a distortion in the shape of the rope will travel steadily away from the hand. This moving disturbance is called a progressive transverse wave. We can better understand steady wave production by studying the behavior of a single upward pulse. The displacement is transferred progressively to particles further along the rope. It is called a transverse disturbance because each rope particle travels in a direction perpendicular or transverse to that of pulse travel. If two identical ropes are used, but one whose tension is low and the other whose tension is high, the pulse will travel faster along the rope whose tension is high since in this case the restoring forces are larger. On the other hand, with tensions the same, a pulse will travel faster along a lighter rope since it has less mass per unit length and therefore has smaller inertia. The elastic factor and the inertial factor determine the velocity. Wave velocity equals the square root of the tension divided by the mass per unit length. To create a continuous train of waves, we have to have a vibrating source. Let's use a reed driven by an electromagnet. The simple harmonic motion of the reed is imparted to the string particles and a steady train of waves moves away from the reed. You'll notice that the separate string particles move in a direction transverse to that of wave travel just as in the case of a single pulse. The motion, like that of the reed, is simple harmonic. An important feature of wave motion is the wavelength, lambda. The wavelength is the distance between consecutive crests or consecutive troughs or between the closest two points where the phase of motion is identical. Frequency in the wave is the number of crests passing any stationary point in space per unit time. The frequency in the wave and the frequency of the separate particle motions are identical. Wave velocity equals frequency times wavelength or V = F * lambda. Let's see what this implies. If we have a vibrating source of fixed frequency and the wave velocity is low, the wavelength will be short. But if we use the same fixed frequency and the wave velocity is high, the wavelength will be long. Now let's see what happens if the wave velocities are the same. When the frequency at the source is low, the wavelength will be long. And when the frequency is high, the wavelength will be short. These effects are all a consequence of the relationship wave velocity equals frequency times wavelength. Let's suppose we have a very long string of identical beads connected by light springs. As in the case of the stretched rope, we can start a pulse or progressive wave disturbance by an abrupt displacement of the first particle. The motion of individual particles is along the line of the wave travel. That's why we call this type a longitudinal wave disturbance. The velocity is equal to the square root of an elastic factor divided by an inertial factor as in the case of transverse waves, although the exact expression under the square root sign is different. If we use a reed driven by an electromagnet to impart a simple harmonic motion to the first bead, a train of longitudinal waves will move out along the string. There are some regions where the beads are abnormally close together. There are other regions where the spacing is abnormally large. These regions move along with a common velocity. This velocity is the wave velocity. We can define the wavelength, lambda, as the distance between consecutive regions of similar spacing. The wave velocity equals the frequency times the wavelength. A sound wave is probably the most important example of longitudinal waves. Air molecules move about at random even with no sound waves present. If we start a plate vibrating, longitudinal waves travel out from its surface. In the presence of the wave, additional motions are imparted to the molecules which result in traveling regions of rarefaction condensation. A rarefaction consists of a region where the average spacing between molecules is greater than that at a condensation. Thus in effect, longitudinal waves travel in a gas in much the same manner as in any other elastic medium.