The Wave Equation (Edexcel IGCSE Physics)

Spec point — 5.5 · know and use the relationship between the speed, frequency and wavelength of a wave: wave speed = frequency × wavelength v = f × λ

The wave equation

• All waves obey the wave speed equation

  • This is the relationship between the wave speed, frequency and wavelength of a wave

$v=f\timesλ$

• 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

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

Spec point — 5.6 · use the relationship between frequency and time period: frequency = 1 / time period f = 1 / T

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 = frequency, measured in hertz (Hz)
  • T = 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

$T=1\div(6\times10^{14})=1.67\times10^{-15}s$

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

Calculations in different contexts

Spec point — 5.7 · use the above relationships in different contexts including sound waves and electromagnetic waves

• 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

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

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

$f=10000\mathrm{Hz}=1\times10^{4}\mathrm{Hz}$

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

$v=f\timesλ$

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

$\\lambda = \frac{330}{( 1 \times 10^{4} )}$

$λ=0.033m$

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:

$v=f\timesλ$

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

$v=200000\times1500$

$v=300000000=3\times10^{8}$

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

Exam Hint 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