Because both masses suspended on a spring and waves at the beach exhibit periodic motion, we can use much of the same vocabulary and mathematical tools to describe both. However, there is a significant difference: waves are extended in space, while a mass on a spring just oscillates back and forth in one place.

A familiar and concrete example of wave motion is the “wave” spectators create at sporting events by standing up and sitting down at appropriate intervals. Each person stands up just as that person’s neighbor stands up, transmitting a form of energy all the way around the stadium. There are two things worth noting about how this works:

Think of waves as a means of transmitting energy over a distance. One object can transmit energy to another object without either object, or anything in between them, being permanently displaced. For instance, if a friend shouts to you across a room, the sound of your friend’s voice is carried as a wave of agitated air particles. However, no air particle has to travel the distance between your friend and your ear for you to hear the shout. The air is a medium, and it serves to propagate sound energy without itself having to move. Waves are so widespread and important because they transmit energy through matter without permanently displacing the matter through which they move.

Waves travel in crests and troughs, although, for reasons we will discuss shortly, we call them compressions and rarefactions when dealing with longitudinal waves. The terms crest and trough are used in physics just as you would use them to refer to waves on the sea: the crest of a wave is where the wave is at its maximum positive displacement from the equilibrium position, and the trough is where it is at its maximum negative displacement. Therefore, the displacement at the crest is the wave’s amplitude, while the displacement at the trough is the negative amplitude. There is one crest and one trough in every cycle of a wave. The wavelength, , of a traveling wave is the distance between two successive crests or two successive troughs.

The period of oscillation, T, is simply the time between the arrival of successive wave crests or wave troughs at a given point. In one period, then, the crests or troughs travel exactly one wavelength. Therefore, if we are given the period and wavelength, or the frequency and wavelength, of a particular wave, we can calculate the wave speed, v:

The oscillation of the mass on the spring determines the oscillation of the string, so the period and frequency of the mass’s oscillation are the same as those of the string. The period of oscillation of the string is T = 0.5 s, since the string oscillates up and down once every half second. The frequency is just the reciprocal of the period: f = 1/T = 2 Hz.

The maximum positive displacement of the mass’s oscillation signifies a wave crest. Since each crest is 1.25 m apart, the wavelength, , is 1.25 m.

Given the frequency and the wavelength, we can also calculate the wave speed: m/s.

Imagine placing a floating cork in the sea so that it bobs up and down in the waves. The up-and-down oscillation of the cork is just like that of a mass suspended from a spring: it oscillates with a particular frequency and amplitude.

Now imagine extending this experiment by placing a second cork in the water a small distance away from the first cork. The corks would both oscillate with the same frequency and amplitude, but they would have different phases: that is, they would each reach the highest points of their respective motions at different times. If, however, you separated the two corks by an integer multiple of the wavelength—that is, if the two corks arrived at their maximum and minimum displacements at the same time—they would oscillate up and down in perfect synchrony. They would both have the same frequency and the same phase.