Wave Properties and Calculations

The wavelength, symbol $\lambda$ (lambda), is the distance from a point on one wave to the equivalent point on the adjacent wave. For a transverse wave, this can be measured from one crest to the next crest, or one trough to the next trough. For a longitudinal wave, it can be measured from the centre of one compression to the centre of the next. Wavelength is also measured in metres (m).

The upper graph is a snapshot of a wave at one instant. Its horizontal axis shows distance, so it describes the shape of the wave across different positions. The crests are 0.02 m above the undisturbed position: the amplitude is therefore 0.02 m. Adjacent crests are at 1 m and 5 m, so the wavelength is $5-1=4\text{ m}$.

Frequency and period: counting and timing

The frequency, $f$, is the number of waves passing a point each second. It is measured in hertz (Hz). A frequency of 5 Hz means five complete waves pass a fixed point every second; a point in the medium makes five complete oscillations each second.

The period, $T$, is the time taken for one complete wave to pass a point, or for one complete oscillation. It is measured in seconds (s).

The lower graph follows the displacement of one point over time. Adjacent peaks occur at 0.05 s and 0.25 s, giving a period of $0.25-0.05=0.20\text{ s}$. Unlike the upper graph, its horizontal spacing represents time, not wavelength.

Frequency and period are reciprocals:

$$T=\frac{1}{f}$$

For the wave with frequency 5 Hz:

$$T=\frac{1}{5}=0.20\text{ s}$$

Five cycles fit into one second because each takes one-fifth of a second. A higher frequency therefore means a shorter period. To calculate frequency from period, use $f=\frac{1}{T}$.

Wave speed links spacing to timing

Wave speed, $v$, is the speed at which energy is transferred, or the wave moves, through the medium. It is not the speed of an individual particle oscillating about its undisturbed position.

During one period, the wave pattern advances by one wavelength. If $f$ waves pass a point each second and each wave occupies a distance $\lambda$, the wave travels $f\lambda$ metres each second. All waves obey:

$$v=f\lambda$$

Use $v$ in metres per second (m/s), $f$ in hertz (Hz) and $\lambda$ in metres (m).

For the wave, the frequency is 5 Hz and the wavelength is 4 m:

$$v=5\times4=20\text{ m/s}$$

This means the wave pattern, carrying energy, advances 20 metres each second.

Calculating an unknown frequency or wavelength

Rearrange the same equation according to the quantity needed:

$$f=\frac{v}{\lambda} \qquad \lambda=\frac{v}{f}$$

As another example, suppose a wave has speed 12 m/s and wavelength 3 m. Its frequency is $f=12/3=4\text{ Hz}$: four waves pass a point each second.

If instead a wave has speed 12 m/s and frequency 8 Hz, its wavelength is $\lambda=12/8=1.5\text{ m}$. These calculations connect the distance travelled by the wave with the number of cycles passing each second.

$$T=\frac{1}{f} \qquad f=\frac{1}{T}$$ $$v=f\lambda \qquad f=\frac{v}{\lambda} \qquad \lambda=\frac{v}{f}$$

Higher frequency means shorter period.

Reading wave graphs

• Measure amplitude vertically from the undisturbed position to a crest or trough.
• On a distance axis, adjacent crests are one wavelength apart.
• On a time axis, adjacent peaks are one period apart.
• For a longitudinal wave, adjacent compression centres are one wavelength apart.