Now that you understand the properties of waves, you can calculate how fast a wave travels and how its speed, frequency, and wavelength are connected. The wave equation v = fλ is one of the most important equations in waves.
2.1 Frequency and Period
Frequency (f) and period (T) both describe the 'pace' of a wave, but from opposite perspectives. Frequency asks 'how many cycles per second?'; period asks 'how many seconds per cycle?'. They are reciprocals of each other.
Formula
Frequency–period relationship
f = frequency (Hz), T = period (s)
SI unit: Hz
Exam Tip
MEMORY AID: f and T are inverses. If the frequency DOUBLES (wave is twice as fast), the period is HALVED. If T = 0,5 s, then f = 1/0,5 = 2 Hz.
Worked Example
A wave has a period of 0,25 s. Calculate its frequency.
Given
- T = 0,25 s
Find
f = ?
Solution
- 1f = 1/T
- 2f = 1 / 0,25
- 3f = 4 Hz
Worked Example
A wave has a frequency of 50 Hz. Calculate its period.
Given
- f = 50 Hz
Find
T = ?
Solution
- 1T = 1/f
- 2T = 1 / 50
- 3T = 0,02 s
Practice Question
A pendulum completes 15 full swings in 30 seconds. Calculate (a) the period and (b) the frequency of the pendulum's oscillation.
(4 marks)
2.2 Wave Speed and the Wave Equation
Definition
Wave speed (v)
Wave speed is the distance a wave travels per unit time.
Think of wave speed this way: if you watch a crest, it moves forward. The distance that crest travels in one second is the wave speed. Since in one full period T the crest moves forward exactly one wavelength λ, we can write: v = λ/T. Since f = 1/T, this becomes the wave equation.
Formula
Wave equation
v = wave speed (m·s⁻¹), f = frequency (Hz), λ = wavelength (m)
SI unit: m·s⁻¹
Exam Tip
EXAM TIP: You can rearrange the wave equation in three ways: v = fλ | f = v/λ | λ = v/f. Write the equation first, then substitute known values, then solve for the unknown. Always include units.
Worked Example
A wave on a string has a frequency of 10 Hz and a wavelength of 0,25 m. Calculate the wave speed.
Given
- f = 10 Hz
- λ = 0,25 m
Find
v = ?
Solution
- 1v = fλ
- 2v = (10 Hz)(0,25 m)
- 3v = 2,5 m·s⁻¹
Worked Example
Sound travels through steel at 5 100 m·s⁻¹. If the frequency of a sound wave in steel is 850 Hz, calculate the wavelength.
Given
- v = 5 100 m·s⁻¹
- f = 850 Hz
Find
λ = ?
Solution
- 1v = fλ → λ = v/f
- 2λ = 5 100 / 850
- 3λ = 6 m
Worked Example
Water waves in a ripple tank have a wavelength of 20 cm and a wave speed of 0,40 m·s⁻¹. (a) Calculate the frequency. (b) Calculate the period.
Given
- λ = 20 cm = 0,20 m
- v = 0,40 m·s⁻¹
Find
(a) f = ? (b) T = ?
Solution
- 1Convert λ to metres: λ = 20 cm = 0,20 m
- 2(a) f = v/λ = 0,40 / 0,20 = 2 Hz
- 3(b) T = 1/f = 1/2 = 0,5 s
Practice Question
A wave on a rope travels at 12 m·s⁻¹. The period of the wave is 0,04 s. Calculate (a) the frequency and (b) the wavelength of this wave.
(5 marks)
Practice Question
Two waves travel in the same medium at the same speed. Wave A has frequency 4 Hz and Wave B has frequency 8 Hz. (a) Which wave has the longer wavelength? (b) If Wave A has wavelength 3 m, calculate the wavelength of Wave B. (c) What is the speed of the waves in this medium?
(6 marks)