The Wave Equation

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All waves obey the wave speed equation
- This is the relationship between the wave speed, frequency and wavelength of a wave

$v = f \times λ$

Where:
- v = wave speed in metres per second (m/s)
- f = frequency in hertz (Hz)
- λ = wavelength in metres (m)

Formula triangle for the wave speed equation

Wave speed triangle (2), IGCSE & GCSE Physics revision notes

A formula triangle can be used to help rearrange the wave speed equation

For more information on how to use a formula triangle refer to the revision note on Speed

Frequency & time period

Frequency and time period are defined in Describing wave motion
The following equation relates time period and frequency:

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

Where:
- $f$ is frequency measured in Hertz (Hz)
- $T$ is time period, measured in seconds (s)

Worked example

Visible light has a frequency of about 6 × 10¹⁴ Hz.

How long does it take for one complete cycle of visible light to enter our eyes?

Answer:

Step 1: List the known values

Frequency, f = 6 × 10¹⁴ Hz

Step 2: State the relationship between frequency and time period

This question involves quantities of time and frequency, so the equation which relates time period and frequency of a wave is:

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

Step 3: Substitute the known values into the equation and calculate the time period

$T = 1 \div (6 \times 10^{14}) = 1.67 \times 10^{- 15} s$

Therefore, it takes 1.67 × 10^-15 s (to 2 decimal places) for one wave of visible light to pass our eyes

Calculations in different contexts

The wave equation can be applied and rearranged to calculate the properties of Transverse and longitudinal waves
The wave equation applies to all types of waves, including Sound waves and Electromagnetic waves

Worked example

A certain sound wave moves at about 330 m/s and has a time period of 0.0001 seconds.

Calculate:

a) The frequency of the sound wave

b) The wavelength of the sound wave

Answer:

Part (a)

Step 1: List the known quantities

Time period, T = 0.0001 s

Step 2: Write out the equation relating time period and frequency

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

Step 3: Rearrange for frequency, f, and calculate the answer

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

$f = \frac{1}{0.0001}$

$f = 10 000 Hz = 1 \times 10^{4} Hz$

Part (b)

Step 1: List the known quantities

Wave speed, v = 330 m/s
Frequency, f = 1 × 10⁴ Hz

Step 2: Write out the wave speed equation

$v = f \times λ$

Step 3: Rearrange the equation to calculate the wavelength

$λ = \frac{v}{f}$

Step 4: Use the frequency you calculated in part (a) and put the values into the equation

$λ = \frac{330}{(1 \times 10^{4})}$

$λ = 0.033 m$

Worked example

A local radio station broadcasts at a frequency of 200 kHz. The wavelength of these radio waves is 1500 m.

Calculate the speed of these radio waves and state an appropriate unit.

Answer:

Step 1: List the known quantities

Frequency, f = 200 kHz = 200 000 Hz
Wavelength, λ = 1500 m

Step 2: Write out the wave speed equation

This question requires wave speed, so state the equation linking wave speed, wavelength and frequency:

$v = f \times λ$

Step 3: Substitute the known values to calculate the wave speed

$v = 200 000 \times 1500$

$v = 300 000 000 = 3 \times 10^{8}$

Step 4: State the unit with the answer

The units for speed are m/s
Therefore, the speed of these radio waves is 3 × 10⁸m/s

Examiner Tip

When stating equations make sure you use the right letters:

For example, use λ for wavelength, not L or W

If you can’t remember the correct letters, then state the word equations required

Be careful with units: wavelength is usually measured in metres and speed in m/s, but if the wavelength is given in cm you might have to give the speed in cm/s

Likewise, watch out for the frequency given in kHz: 1 kHz = 1000 Hz

The Wave Equation (Edexcel IGCSE Physics: Double Science)

Revision Note