Wave Properties and the Wave Equation

Every wave, transverse or longitudinal, can be described by the same four measurements. Get these definitions exactly right, because the wording is marked closely.

• Amplitude — the maximum displacement of a point on the wave from its rest (undisturbed) position. It is a distance, measured in metres. Louder sounds and brighter light have larger amplitudes.
• Wavelength (λ) — the distance between a point on one wave and the same point on the next wave, for example crest to crest or compression to compression. Measured in metres, symbol λ (the Greek letter lambda).
• Frequency (f) — the number of complete waves that pass a point each second. Measured in hertz (Hz), where 1 Hz = 1 wave per second.
• Period (T) — the time taken for one complete wave to pass a point. Measured in seconds.

Amplitude is measured from the rest position to a crest, not from a trough to a crest. Measuring the full top-to-bottom height gives double the amplitude and is a common error.

You must be able to identify amplitude and wavelength directly from a wave diagram. The transverse wave below shows both.

Reading it correctly:

• Amplitude is measured vertically, from the central rest line up to a crest (or down to a trough). It is not the distance from a trough all the way to a crest.
• Wavelength is measured horizontally, from one crest to the next crest, or from any point to the matching point one full cycle later.

For a longitudinal wave the same ideas apply, but wavelength is measured from one compression to the next compression, and amplitude relates to how far the particles move from their rest positions.

Period and frequency describe the same thing from opposite directions. Frequency counts waves per second; period times how long one wave takes. They are reciprocals of each other.

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

where $T$ is the period in seconds (s) and $f$ is the frequency in hertz (Hz).

This equation is given on the Physics equation sheet, so you do not have to memorise it, but you must be able to use and rearrange it confidently.

Worked example — period from frequency. A sound wave has a frequency of 200 Hz. Find its period.

$T=\frac{1}{f}=\frac{1}{200}=0.005\text{ s}$

So one complete wave takes 0.005 seconds (5 milliseconds) to pass.

Worked example — frequency from period. A pendulum-like wave has a period of 0.02 s. Rearranging $T=1/f$ gives $f=1/T$:

$f=\frac{1}{T}=\frac{1}{0.02}=50\text{ Hz}$

The wave speed is the speed at which the wave (and its energy) moves through the medium. It links frequency and wavelength through the wave equation.

$v=f\lambda$

where $v$ is wave speed in metres per second (m/s), $f$ is frequency in hertz (Hz) and $\lambda$ is wavelength in metres (m).

You must recall and apply this equation. It is not given on the equation sheet.

Worked example — finding speed. Water ripples have a frequency of 5 Hz and a wavelength of 0.04 m. Find their speed.

$v=f\lambda =5\times 0.04=0.2\text{ m/s}$

Worked example — rearranging for wavelength. A sound wave travels at 340 m/s with a frequency of 170 Hz. Find its wavelength. Rearranging $v=f\lambda$ gives $\lambda =v/f$:

$\lambda =\frac{v}{f}=\frac{340}{170}=2\text{ m}$

A key consequence: when a wave passes into a different medium, its frequency stays the same but its speed changes, so its wavelength must change to keep $v=f\lambda$ balanced. (Physics only) this is why velocity, frequency and wavelength are all inter-related when a wave passes between media.

Required practical 8 asks you to select suitable apparatus to measure the frequency, wavelength and speed of waves, using a ripple tank for water waves and a stretched string or spring for waves in a solid.

Ripple tank (water waves):

• Apparatus: a shallow tank of water, a motorised bar (ripple generator) with an adjustable frequency, a lamp above and a white screen below so the ripples cast shadows.
• Method: switch on the generator to make straight ripples. Read the frequency from the generator's setting. Measure the wavelength by using a strobe light (or a dimmed image) to freeze the ripples, then measure across several wavelengths with a ruler and divide by the number of gaps — measuring 10 wavelengths and dividing by 10 reduces the percentage error. Calculate speed with $v=f\lambda$.
• Why measure several wavelengths? A single wavelength is small and hard to read precisely; measuring ten and dividing shrinks the effect of the ruler's uncertainty.

Waves in a solid (stretched string):

• Apparatus: a string fixed at one end and attached to a vibration generator driven by a signal generator, with a hanging mass keeping it taut.
• Method: adjust the signal generator's frequency until a clear standing-wave pattern forms. Read the frequency directly from the signal generator. Measure the wavelength with a ruler (for a standing wave, the distance between adjacent still points is half a wavelength, so measure across several loops). Calculate speed with $v=f\lambda$.

Variables: the independent variable is the frequency you set; the dependent variable is the wavelength you observe; controls include the water depth (ripple tank) and the tension/mass and string type (solid), which must be kept constant for a fair test.

Beyond the required practical, you should be able to describe two everyday methods.

Speed of sound in air (echo or two-observer method). Two students stand a measured distance apart, say 100 m. One bangs two blocks together; the other starts a stopwatch on seeing the blocks meet and stops it on hearing the bang. Because light reaches the eye almost instantly, the timed gap is the sound's travel time. Repeating and averaging reduces reaction-time error.

Worked example. The sound covers 100 m in an average of 0.30 s. Using $v=\text{distance}÷\text{time}$:

$v=\frac{100}{0.30}=333\text{ m/s}$

This is close to the accepted value of about 330–340 m/s for sound in air, confirming the method.

Speed of ripples on water. Time how long a single ripple takes to travel a measured length of the tank, then use speed = distance ÷ time. Alternatively, measure the frequency and wavelength as in the required practical and apply $v=f\lambda$. Both routes should give the same answer.

1. Measuring amplitude from trough to crest

Amplitude is measured from the rest position to a crest, so the full trough-to-crest height is twice the amplitude. Halve it before quoting an amplitude.

2. Forgetting to convert units before using v = fλ

The wave equation needs wavelength in metres. If a wavelength is given in centimetres or millimetres, convert first: 4 cm becomes 0.04 m. Mixing units gives an answer wrong by a factor of 100 or 1000.

3. Confusing frequency and period

Frequency is waves per second (Hz); period is seconds per wave. They are reciprocals, so a 200 Hz wave has a period of 0.005 s, not 200 s. Check which one the question wants.

4. Thinking frequency changes when a wave enters a new medium

When a wave crosses into a new material its speed and wavelength change, but its frequency stays the same. Assuming the frequency changes leads to wrong wavelength calculations.

5. Measuring only one wavelength in the ripple tank

Single-wavelength measurements carry a large percentage error. Measure across ten wavelengths and divide by ten to reduce the uncertainty; examiners reward this technique.