Wave Features and the Wave Equation

Every wave, whether it is light from a star, sound from a guitar or a ripple on a pond, can be described using just four measurements. Once you can read these features off a diagram, you can describe and compare any wave in the universe.

Amplitude is the maximum displacement of a particle from its rest position. On a transverse wave this is the height from the centre line up to a crest, or down to a trough. It is measured in metres (m). A common trap: amplitude is measured from the middle to the top, not from crest to trough. The full crest-to-trough distance is twice the amplitude. Amplitude tells you how much energy the wave carries, a louder sound or a brighter light has a larger amplitude.

Wavelength, given the symbol λ (the Greek letter lambda), is the distance between two consecutive matching points on a wave. The easiest points to use are crest to crest, or trough to trough, but any two identical points one full cycle apart will do. Wavelength is also measured in metres (m).

Frequency, symbol f, is the number of complete waves that pass a fixed point every second. It is measured in hertz (Hz), where 1 Hz means one cycle per second. Middle C on a piano has a frequency of 261.6 Hz, meaning 261.6 complete sound waves reach your ear every second. A higher frequency sounds higher in pitch.

Period, symbol T, is simply the time for one complete wave to pass, measured in seconds (s). Frequency and period are two ways of describing the same thing, so they are reciprocals of one another: $T = \dfrac{1}{f}$ and $f = \dfrac{1}{T}$. If 4 waves pass each second (f = 4 Hz), then each wave takes a quarter of a second to pass (T = 0.25 s).

Here is the idea that ties the four features together. Imagine standing on a jetty counting waves. If 3 waves pass you each second (f = 3 Hz) and each wave is 2 m long (λ = 2 m), then in one second a length of 3 × 2 = 6 m of wave has gone past. That distance per second is the wave speed. This reasoning gives us the most important formula in this topic, the wave equation:

$v = f\lambda$

where v is the wave speed in metres per second (m/s), f is the frequency in hertz (Hz), and λ is the wavelength in metres (m). Like any equation with three quantities, you can rearrange it to find whichever one is missing:

• To find frequency: $f = \dfrac{v}{\lambda}$
• To find wavelength: $\lambda = \dfrac{v}{f}$

A key insight: in a given medium, the wave speed is fixed. Sound in air always travels at about 340 m/s no matter what note you play. So if you increase the frequency, the wavelength must decrease to keep the product v = fλ constant. High notes have short wavelengths, low notes have long wavelengths, but both travel at the same speed. This is exactly why the bass and the whistle from your Think First answer reach you at the same instant.

The period T and the frequency f are simply two views of the same rhythm, connected by $T = \dfrac{1}{f}$. A high-frequency wave has many cycles per second, so each one takes a very short time, giving a tiny period. Middle C at 261.6 Hz has a period of T = 1 ÷ 261.6 = 0.0038 s, less than four thousandths of a second per wave, yet your ear separates every one.

Combining the wave equation with the period relationship lets you solve almost any wave problem. If you know any two features, you can usually find the rest. For instance, given the speed of sound (v = 340 m/s) and a tuning fork frequency (f = 440 Hz), you can find both the wavelength and the period of that note:

• Wavelength: λ = v ÷ f = 340 ÷ 440 = 0.77 m
• Period: T = 1 ÷ f = 1 ÷ 440 = 0.0023 s

This kind of reasoning is exactly how engineers tune instruments, design speakers, and even calibrate the medical and radar technologies you will meet in later lessons.