A progressive (or travelling) wave transports energy and information from one point to another without the net transport of matter. The disturbance moves through the medium while the individual particles of the medium oscillate about their equilibrium positions.
Key Characteristics
Direction of propagation: the direction in which the wave travels.
Amplitude (A): maximum displacement of particles from equilibrium.
Wavelength (λ): distance between two successive points in phase (e.g., crest‑to‑crest).
Frequency (f): number of complete cycles that pass a fixed point per second.
Period (T): time for one complete cycle, $T = \frac{1}{f}$.
Wave speed (v): rate at which the wave pattern moves through the medium.
Fundamental Relationship
The speed of a progressive wave is directly related to its frequency and wavelength:
$$v = f\lambda$$
where
$v$ is the wave speed (m s⁻¹),
$f$ is the frequency (Hz),
$\lambda$ is the wavelength (m).
Suggested diagram: A sinusoidal progressive wave travelling to the right, showing one wavelength and one period.
Using $v = f\lambda$ – Worked Examples
Example 1: A sound wave in air has a frequency of 500 Hz and a wavelength of 0.68 m. Find its speed.
$$v = f\lambda = (500\ \text{Hz})(0.68\ \text{m}) = 340\ \text{m s}^{-1}$$
Example 2: A transverse wave on a string moves at 12 m s⁻¹ and has a wavelength of 0.30 m. Determine its frequency.
$$f = \frac{v}{\lambda} = \frac{12\ \text{m s}^{-1}}{0.30\ \text{m}} = 40\ \text{Hz}$$
Example 3: A wave on a rope has a period of 0.025 s and a speed of 5 m s⁻¹. Find its wavelength.
First find the frequency: $f = \frac{1}{T} = \frac{1}{0.025\ \text{s}} = 40\ \text{Hz}$.
Then use $v = f\lambda$: $\lambda = \frac{v}{f} = \frac{5\ \text{m s}^{-1}}{40\ \text{Hz}} = 0.125\ \text{m}$.
Typical Wave Speeds in Different Media
Medium
Typical Wave Speed (m s⁻¹)
Wave Type
Air (at 20 °C)
≈ 343
Longitudinal (sound)
Water (surface)
≈ 0.3 – 1.5
Transverse (water waves)
Steel rod (longitudinal)
≈ 5 200
Longitudinal (sound)
String (under tension)
Varies with tension and linear density
Transverse
Common Mistakes to Avoid
Confusing wavelength with amplitude – they are independent quantities.
Using the period $T$ directly in $v = f\lambda$ without converting to frequency first.
Mixing units (e.g., using cm for wavelength and m s⁻¹ for speed).
Assuming the wave speed changes when only frequency or wavelength changes; in a given medium $v$ is constant.
Summary
For any progressive wave travelling in a uniform medium, the speed, frequency and wavelength are linked by the simple equation $v = f\lambda$. By knowing any two of these quantities you can determine the third, which is a fundamental skill required for the Cambridge A‑Level Physics 9702 examination.