derive, using the definitions of speed, frequency and wavelength, the wave equation v = f λ

Published by Patrick Mutisya · 14 days ago

Cambridge A-Level Physics 9702 – Progressive Waves

Progressive Waves

Learning Objective

Derive, using the definitions of speed, frequency and wavelength, the wave equation

\$v = f \lambda\$

Key Definitions

  • Wave speed (\$v\$): the distance a point of constant phase travels per unit time.

  • Frequency (\$f\$): the number of complete cycles that pass a fixed point each second (units: Hz).

  • Wavelength (\$\lambda\$): the spatial distance between two consecutive points that are in phase (e.g., crest to crest).

Derivation of the Wave Equation

Consider a sinusoidal progressive wave travelling along the \$x\$‑axis. Let a particular crest be observed at position \$x1\$ at time \$t1\$ and the same crest be observed at position \$x2\$ at a later time \$t2\$.

  1. By definition of speed,

    \$v = \frac{\text{distance travelled}}{\text{time taken}} = \frac{x2 - x1}{t2 - t1}.\$

  2. The distance \$x2 - x1\$ between two successive occurrences of the same phase is one wavelength, \$\lambda\$.

  3. The time \$t2 - t1\$ for one complete cycle to pass a fixed point is the period \$T\$ of the wave.

Substituting \$\lambda\$ for the distance and \$T\$ for the time gives

\$v = \frac{\lambda}{T}.\$

Frequency \$f\$ is the reciprocal of the period:

\$f = \frac{1}{T} \quad \Longrightarrow \quad T = \frac{1}{f}.\$

Replacing \$T\$ in the speed expression:

\$v = \frac{\lambda}{1/f} = f\lambda.\$

Thus the fundamental relationship for a progressive wave is

\$\boxed{v = f\lambda}\$

Units and Typical \cdot alues

QuantitySymbolSI UnitTypical A‑Level Example
Wave speed\$v\$metre per second (m·s⁻¹)Sound in air ≈ 340 m·s⁻¹
Frequency\$f\$hertz (Hz)Middle C note ≈ 261 Hz
Wavelength\$\lambda\$metre (m)Visible light ≈ 5×10⁻⁷ m

Worked Example

Find the wavelength of a sound wave travelling at \$v = 340\ \text{m·s}^{-1}\$ with a frequency of \$f = 500\ \text{Hz}\$.

  1. Start from the wave equation \$v = f\lambda\$.

  2. Re‑arrange for \$\lambda\$: \$\displaystyle \lambda = \frac{v}{f}\$.

  3. Insert the numbers: \$\displaystyle \lambda = \frac{340\ \text{m·s}^{-1}}{500\ \text{Hz}} = 0.68\ \text{m}\$.

Therefore the wavelength is \$0.68\ \text{m}\$.

Conceptual Check

  • If the frequency of a wave is doubled while the speed remains constant, what happens to the wavelength?

  • Explain why the wave equation holds for both transverse and longitudinal waves.

Suggested diagram: A sinusoidal wave showing one full wavelength \$\lambda\$, the distance a crest travels in one period \$T\$, and the corresponding speed \$v\$.

Summary

The wave equation \$v = f\lambda\$ links three fundamental properties of a progressive wave. It follows directly from the definitions of speed (distance per time), frequency (cycles per time), and wavelength (distance per cycle). Mastery of this relationship enables analysis of a wide range of wave phenomena encountered in A‑Level physics.