3.5-3.7 - Wave speed, period and wave equations

The wave speed relationship

Frequency tells you how many complete waves pass a point each second. Wavelength tells you the distance from one matching point on a wave to the next, such as crest to crest or compression to compression. Multiplying them gives the distance the pattern travels in one second, which is the wave speed.

In words: wave speed = frequency $\times$ wavelength. Edexcel does not give you this equation in the exam, in this form or rearranged, so learn it.

In this equation, $v$ is wave speed in metres per second, $f$ is frequency in hertz, and $\lambda$ is wavelength in metres. One hertz means one complete wave per second, so Hz is the same as $\mathrm{s}^{-1}$.

The units show why the equation works:

A common exam trap is to multiply numbers before checking what each number represents. If a question gives the distance between two adjacent crests, that is one wavelength. If it gives the number of waves passing each second, that is the frequency.

Rearranging the wave speed equation

The relationship $v=f\lambda$ can be rearranged depending on the missing quantity:

Choose the form that puts the unknown on its own. Then substitute values with units, calculate, and give the final answer with the correct unit.

Worked example: water waves have a frequency of 5.0 Hz and a wavelength of 0.30 m. Calculate their wave speed.

Worked example: a sound wave in air travels at 340 m/s and has a frequency of 680 Hz. Calculate its wavelength.

The second example is a good check on the size of the answer. A higher frequency at the same speed gives a shorter wavelength, because more complete waves must fit into the same distance travelled each second.

Frequency and period

The time period is the time for one complete wave cycle. Frequency is the number of complete cycles each second. These two quantities are reciprocals: if each cycle takes a long time, fewer cycles fit into one second.

Here f is measured in hertz and T is measured in seconds. You can also rearrange it as:

If a detector records 15 complete cycles in 0.30 s, each cycle takes $T=0.30/15=0.020\,\mathrm{s}$, and $f=15/0.30=50\,\mathrm{Hz}$. Counting several cycles reduces the fractional effect of timing one cycle poorly.

Worked example: the frequency of a wave is 50 Hz. Calculate its time period.

Worked example: the time period of a wave is 0.0040 s. Calculate its frequency.

If a question gives a number of complete waves in a measured time, find the period first: it is the total time divided by the number of complete waves. For example, 10 complete waves in 2.0 s gives $T = 2.0 \div 10 = 0.20\ \text{s}$, so $f = \frac{1}{0.20} = 5.0\ \text{Hz}$.

Do not write $f=T$. They are linked, but they are not equal. Their values move in opposite directions: doubling the frequency halves the period.