analyse and interpret graphical representations of transverse and longitudinal waves

Progressive Waves

In this topic we consider waves that travel through a medium without the need for an external driver after they have been created. The aim is to develop the ability to analyse and interpret the graphical representations of both transverse and longitudinal progressive waves.

Key Definitions

• Progressive (traveling) wave: A disturbance that moves through a medium, carrying energy from one point to another.
• Transverse wave: The particle displacement is perpendicular to the direction of wave propagation.
• Longitudinal wave: The particle displacement is parallel to the direction of wave propagation.
• Amplitude (A): Maximum displacement from the equilibrium position.
• Wavelength (λ): Distance between two successive points in phase.
• Frequency (f): Number of cycles per second, measured in hertz (Hz).
• Angular frequency (ω): $ω = 2πf$.
• Wave number (k): $k = \frac{2π}{λ}$.
• Phase speed (v): $v = fλ = \frac{ω}{k}$.

Mathematical Description of a Progressive Wave

For a sinusoidal wave travelling in the +x direction the displacement $y$ as a function of position $x$ and time $t$ is given by

$$y(x,t) = A\sin(kx - ωt + φ)$$

where $φ$ is the initial phase. The same form can be used for longitudinal waves if $y$ is interpreted as the longitudinal displacement of the particles.

Graphical Representations

Two common types of graphs are used in examinations:

1. Displacement vs. Position (snapshot at a fixed time): Shows the shape of the wave along the medium at a particular instant.
2. Displacement vs. Time (at a fixed position): Shows how a single particle oscillates as the wave passes.

Interpreting a Displacement‑vs‑Position Graph

• The distance between successive peaks (or troughs) is the wavelength $λ$.
• The vertical distance from the equilibrium line to a peak is the amplitude $A$.
• If the graph is shifted to the right as time increases, the wave is travelling in the +x direction; a leftward shift indicates travel in the –x direction.
• The slope $dy/dx$ at any point is related to the local strain in the medium (important for longitudinal waves).

Interpreting a Displacement‑vs‑Time Graph

• The period $T$ is the time between successive identical points (e.g., peak to peak). Frequency $f = 1/T$.
• The maximum vertical displacement gives the amplitude $A$.
• The phase of the wave at that point can be read from the position of the curve relative to a reference sinusoid.
• For a travelling wave, the same shape appears at different positions with a time lag $Δt = Δx / v$.

Comparison of Transverse and Longitudinal Waves

Feature	Transverse Wave	Longitudinal Wave
Particle displacement direction	Perpendicular to propagation	Parallel to propagation
Typical examples	Light, waves on a string, water surface ripples	Sound in air, compression waves in a spring
Graphical representation (snapshot)	Sinusoidal curve $y$ vs $x$	Regions of compression and rarefaction; often plotted as $Δx$ or pressure vs $x$
Energy transport	Energy carried by both kinetic and potential energy of the medium	Energy carried mainly by pressure variations (potential) and particle motion (kinetic)
Speed formula in a uniform medium	$v = \sqrt{\frac{T}{μ}}$ (string tension $T$, linear density $μ$)	$v = \sqrt{\frac{B}{ρ}}$ (bulk modulus $B$, density $ρ$)

Worked Example – Interpreting a Graph

Question: A displacement‑vs‑position graph for a transverse wave on a string shows a wavelength of $0.40\ \text{m}$ and an amplitude of $2.0\ \text{mm}$. The same point on the string is observed to complete one oscillation in $0.025\ \text{s}$. Determine the wave speed.

1. Identify $λ = 0.40\ \text{m}$ from the graph.
2. Period $T = 0.025\ \text{s}$, so $f = 1/T = 40\ \text{Hz}$.
3. Use $v = fλ$: $$v = 40\ \text{Hz} \times 0.40\ \text{m} = 16\ \text{m s}^{-1}$$

The wave travels at $16\ \text{m s}^{-1}$ along the string.

Common Examination Tasks

• Given a snapshot graph, calculate $λ$, $A$, and deduce the direction of travel.
• Given a time‑history graph at a fixed point, determine $T$, $f$, and $A$.
• Combine information from two graphs (one snapshot, one time‑history) to find the wave speed $v$.
• Identify whether a given graph represents a transverse or longitudinal wave based on the nature of the plotted quantity.
• Explain how the phase difference between two points on the medium relates to the distance between them.

Key Points to Remember

1. The shape of a sinusoidal wave is the same for transverse and longitudinal waves; only the physical meaning of the vertical axis changes.
2. Wave speed is independent of amplitude for linear media.
3. Phase shift $Δφ = kΔx = \frac{2π}{λ}Δx$; a shift of $π$ corresponds to a half‑wavelength separation.
4. When interpreting graphs, always check the axes labels and units.
5. For a progressive wave, the entire waveform moves without changing shape; this is reflected by a constant translation of the snapshot graph over time.