compare transverse and longitudinal waves

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

Feature	Transverse Wave	Longitudinal Wave
Particle motion	Perpendicular to the direction of propagation (up‑and‑down or side‑to‑side)	Parallel to the direction of propagation (back‑and‑forth)
Typical examples	Light in vacuum, waves on a string, surface water waves	Sound in air, pressure waves in a spring‑mass system, seismic P‑waves
Restoring force	Usually tension or shear (e.g., tension in a string)	Usually compression/expansion of the medium (e.g., bulk modulus)
Displacement diagram	Shows a sinusoidal curve perpendicular to the travel direction	Shows 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)
Polarisation	Can be polarised because the oscillation direction is defined	Cannot be polarised (oscillation direction is along propagation)
Energy transport	Energy is carried by the kinetic and potential energy of the transverse displacement	Energy 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

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.