Topic 10. Reflections, part 1. Waves on a rope

Model a rope as a series of small masses connected by rubber bands. When one of the masses is displaced from its equilibrium position, it exerts forces on its neighbors. These forces cause the adjacent masses to move and also act to bring the initial mass back to its equilibrium position. If all of the masses and rubber bands are the same then the displacement propagates from one mass to the next one. In the absence of friction or other loss mechanisms, the resulting wave propagates without any decrease in its amplitude – each mass transmits its energy to the next mass in the chain through the rubber-band link between them.

The propagation of the wave consists of an exchange between two types of energy: the elastic potential energy that is stored in the stretched rubber bands is converted to kinetic energy of the moving masses, and this kinetic energy in turn is converted back into elastic potential energy when the next rubber band is stretched. This is analogous to a light wave which propagates by an exchange between electric and magnetic potential energies.

The velocity of propagation in this case depends on two quantities: (1) the stiffness of the rubber bands, which determines the magnitude of the force that is transmitted to an adjacent mass for a given displacement, and (2) the size of each mass, which determines how quickly the mass will respond to the transmitted force. The velocity of propagation varies directly with the stiffness of the rubber bands and inversely with the mass at each junction. (A more detailed analysis shows that the velocity of propagation varies as the square root of the ratio of these two quantities.)

When the disturbance reaches a fixed object (or a mass that is much greater than any of the others), the same force produces a much smaller displacement (possibly no displacement at all if the object is fixed), so that the restoring force is greater than at any other point on the rope. The result of the larger restoring force is to generate a reflected wave with a displacement that has the opposite sign. Using the previous paragraph, a larger mass implies a smaller propagation velocity.

When the disturbance reaches the free end of the rope (or a mass that is much smaller than any of the others), the same force produces a much larger displacement than at other points so that a reflected wave of the same sign is generated.

In the more general case, a reflection is generated whenever a wave reaches a point where its velocity of propagation changes. The reflected wave is reversed when the velocity decreases at the boundary and is not reversed when the velocity increases. Using the various definitions of the phase of a wave, the reversal at the reflection can be thought of as a phase change of one-half of a wavelength or as 180° in a sine-wave cycle or as a time delay of one-half of the period of the wave.

Our eyes and many other optical detectors are sensitive only to the amplitude of the light wave (or to the intensity, which is proportional to the square of the amplitude) and not to its phase, so that these reversals produce no visible change in the light and can be ignored. However, this is not always the case, and the phase-reversal on reflection plays an important role in interference between two light waves.

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