compare transverse and longitudinal waves

Published by Patrick Mutisya · 14 days ago

Cambridge A-Level Physics 9702 – Progressive Waves: Transverse vs Longitudinal

Progressive Waves – Comparison of Transverse and Longitudinal Waves

1. Definition of a Progressive Wave

A progressive (or travelling) wave is a disturbance that moves through a medium, carrying energy without permanent displacement of the medium’s particles. The wave propagates with a constant speed \$v\$, related to its frequency \$f\$ and wavelength \$\lambda\$ by

\$v = f\lambda\$

2. Basic Characteristics

Both transverse and longitudinal waves satisfy the same wave equation, but the direction of particle motion relative to the direction of wave propagation differs.

3. Comparison Table

FeatureTransverse WaveLongitudinal Wave
Particle motionPerpendicular to the direction of propagation (up‑and‑down or side‑to‑side)Parallel to the direction of propagation (back‑and‑forth)
Typical examplesLight in vacuum, waves on a string, surface water wavesSound in air, pressure waves in a spring‑mass system, seismic P‑waves
Restoring forceUsually tension or shear (e.g., tension in a string)Usually compression/expansion of the medium (e.g., bulk modulus)
Displacement diagramShows a sinusoidal curve perpendicular to the travel directionShows alternating compressions and rarefactions along the travel direction
Wave speed expression\$v = \sqrt{\dfrac{T}{\mu}}\$ for a string (where \$T\$ is tension, \$\mu\$ is linear mass density)\$v = \sqrt{\dfrac{B}{\rho}}\$ for a fluid (where \$B\$ is bulk modulus, \$\rho\$ is density)
PolarisationCan be polarised because the oscillation direction is definedCannot be polarised (oscillation direction is along propagation)
Energy transportEnergy is carried by the kinetic and potential energy of the transverse displacementEnergy is carried by the work done during compression and expansion of the medium

4. Mathematical Description

For a sinusoidal progressive wave travelling in the \$+x\$ direction, the displacement \$y\$ (or \$s\$ for longitudinal) can be written as

\$y(x,t) = A\sin\bigl(kx - \omega t + \phi\bigr)\$

where \$A\$ is amplitude, \$k = \dfrac{2\pi}{\lambda}\$ is the wave number, \$\omega = 2\pi f\$ is the angular frequency, and \$\phi\$ is the phase constant.

5. Visualising the Two Types

Suggested diagram: Side‑by‑side sketches of a transverse wave on a string (particles moving up/down) and a longitudinal sound wave in a tube (alternating compressions and rarefactions).

6. Key Points for A‑Level Exams

  1. Identify the direction of particle motion relative to wave travel.

  2. Recall the appropriate wave‑speed formula for the medium (tension/linear density for strings, bulk modulus/density for gases).

  3. Understand that only transverse waves can be polarised.

  4. Be able to sketch a single‑cycle diagram showing crests/troughs (transverse) or compressions/rarefactions (longitudinal).

  5. Apply \$v = f\lambda\$ and \$v = \omega/k\$ to relate frequency, wavelength and speed for both wave types.

7. Sample Exam Question

Question: A sound wave of frequency \$500\ \text{Hz}\$ travels through air where the speed of sound is \$340\ \text{m s}^{-1}\$. Calculate the wavelength and describe the particle motion.

Solution:

  1. Use \$v = f\lambda\$ → \$\lambda = \dfrac{v}{f} = \dfrac{340}{500} = 0.68\ \text{m}\$.

  2. Since it is a longitudinal wave, air particles oscillate back‑and‑forth along the direction of propagation, producing alternating compressions and rarefactions.

8. Summary

Transverse and longitudinal progressive waves share the same fundamental wave relationships but differ in particle motion, restoring forces, polarisation capability, and typical physical examples. Mastery of these distinctions is essential for solving A‑Level physics problems involving wave phenomena.