The Wave Equation (Cambridge International IGCSE Physics)

Spec point — 3.1.2 · Recall and use the equation for wave speed v = f λ

The wave equation

• The equation used to calculate wave speed is:

$v=f\timesλ$

• Where:

  • $v$ = wave speed, measured in metres per second (m/s)
  • $f$ = wave frequency, measured in hertz (Hz)
  • $λ$ = wavelength, measured in metres (m)

• Wave speed is defined as:

The distance travelled by a wave each second

• Wave speed is the speed at which energy is transferred through a medium
• Transverse and longitudinal waves both obey the wave equation

Worked Example A wave has a wavelength of 5 cm and a frequency of 6.6 kHz. Calculate its wave speed.

[4]

Answer:

Step 1: List the known quantities

• Wavelength, $λ$ = 5 cm
• Frequency, $f$ = 6.6 kHz

Step 2: Write out the wave speed equation

$v=f\timesλ$ [1 mark]

Step 3: Convert the wavelength into metres

$λ=\frac{5}{100}=0.05m$ [1 mark]

Step 4: Convert the frequency into hertz

$f=6.6\times1000=6600\mathrm{Hz}$ [1 mark]

Step 4: Calculate the wave speed

$v=6600\times0.05$

$v=330m/s$ [1 mark]

Exam Hint Be careful with units: speed is usually measured in m/s, so make sure to convert wavelength into metres and frequency into hertz before substituting.

Watch for wavelengths in cm, speeds in km/s, and frequencies in kHz or MHz — 1 kHz = 1000 Hz and 1 MHz = 1 000 000 Hz.

Frequency and period

• The period of a wave is defined as:

The time taken for one complete wave to pass a fixed point

• The frequency of a wave is related to its period by the following equation:

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

• And therefore,

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

• Where:

  • $f$ = frequency, measured in hertz (Hz)
  • $T$ = period, measured in seconds (s)

Worked Example A wave has a speed of 0.27 m/s and the distance from a crest to the nearest trough is 0.75 cm. Calculate:

a) The frequency of the wave [3]

b) The period of the wave [2]

Answer:

Part (a)

Step 1: List the known quantities

• Wave speed, $v$ = 0.27 m/s
• Distance from a crest to the nearest trough, $d$ = 0.75 cm

Step 2: Calculate the wavelength in metres

• The wavelength can be measured from one crest to the next crest, so

$λ=2\times0.75=1.5\mathrm{cm}$

$λ=\frac{1.5}{100}=0.015m$ [1 mark]

Step 3: Write out the wave speed equation

$v=f\timesλ$

Step 4: Rearrange the equation to calculate the frequency

$f=\frac{v}{λ}$

Step 5: Put the values into the equation

$f=\frac{0.27}{0.015}$ [1 mark]

$f=18\mathrm{Hz}$ [1 mark]

Part (b)

Step 1: List the known quantities

• Frequency, $f$ = 18 Hz (from part (a))

Step 2: State the equation relating time period and frequency

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

Step 3: Calculate the answer

$T=\frac{1}{18}$ [1 mark]

$T=0.056s (2 \mathrm{s} . \mathrm{f} .)$ [1 mark]

Exam Hint When stating equations make sure you use the right letters. For example, use $λ$ for wavelength, not L or W.

If you cannot recall the symbols under pressure, a word equation will still earn the first mark — but $v$, $f$ and $λ$ are the forms used in the question papers, so learn them.