Waves move over time which makes it hard to draw on a piece of paper. There are two possible representations. We can take a snapshot of the wave and then plot the height (amplitude) versus position. This gives us the following picture of a wave frozen in time:
The amplitude (vertical height from the equilibrium position) will be measured in meters (or centimeters, etc.). We can also measure the horizontal distance from one peak to the next in meters. This distance is called the wavelength, λ. In the graph the peak to peak distance is from 2.5 m to 4.6 m which gives a wavelength of 4.6 m – 2.5 m = 2.1 m (this is also the bottom to bottom distance or distance from where it is increasing and has an amplitude of zero to the next place where it is increasing and has an amplitude of zero, etc.).
There is second way to represent waves in a graph. Suppose there is a cork floating in the water that is fixed at a certain location and we record the amplitude (how high and low it is from equilibrium) at different times. If we plot the amplitude versus time we have a frozen position graph:
The amplitude (height from the equilibrium position) will again be measured in meters but now the horizontal peak to peak distance is a time measurement. In fact the cork is undergoing simple harmonic motion and the horizontal peak to peak distance is the period, T of the wave; this is the same graph we saw in Chapter 3. According to the graph the first peak is at 1.5 s when the second is at 7.8 s so the period is 7.8 s – 1.5 s = 6.3 s. As in the last two chapters, the wave frequency is given by f = 1/T in Hertz so this oscillation has a frequency of 1/6.3 s = 0.16 Hz.
Imagine a long line of corks floating on the surface of a lake. As a wave passes by, the closer end of the row of corks starts moving up and down and then the rest of the row. Each cork is undergoing simple harmonic motion (which we studied in Chapter 3) but at a slightly different phase. For this reason the equations describing a wave are sine and cosine functions, just as for the simple harmonic motion we saw in Chapter 3 (the simulation exercise below explains this in detail).
The speed of the wave, its frequency and its wavelength are related. If two waves are traveling at the same speed but have different wavelengths, a cork floating on each will bob up and down at different rates and so have different frequencies. A shorter wavelength will make the cork bob more often while a longer wavelength will make the cork bob less often. Mathematically this relationship is expressed as v = λ f where v is the speed of the wave in meters per second, λ is the wavelength in meters and f is the frequency of the wave in Hertz. This equation applies to all types of waves and we will use it many more times in this book.
Simulation exercise 5A (turn in answers on a separate sheet of paper): Transverse Waves.
Transverse and longitudinal wave lecture.