Wave Displacement and Phase

Overview

Wave displacement describes where a particle is relative to equilibrium. Phase describes where that particle is within its cycle. The two ideas are related, but they are not the same quantity.

Definition

Displacement is a directed distance from equilibrium at a given instant; a graph normally shows one signed component. Phase describes the complete stage of oscillation within a cycle, including direction of motion.

Phase allows comparison between different particles or points on a wave.

This concept supports waves and later wave superposition.

Why It Matters

Understanding displacement and phase is essential for:

• interpreting wave graphs
• comparing motion of different points
• preparing for later phase reasoning in superposition

Confusion between displacement, amplitude, and phase leads to incorrect reasoning in wave problems.

Key Representations

Displacement

• Measured from equilibrium
• Can be positive or negative
• Varies with time and position

Phase

Phase is measured in:

• fractions of a cycle
• degrees
• radians

Enrichment — compact travelling-wave notation

For one common right-moving convention,

$$\text{phase}=kx-\omega t+{\varphi }_0$$

where $\omega$ is angular frequency, $k$ is wave number and ${\varphi }_0$ is an initial phase. An equivalent sign convention may be used if the whole wave equation is defined consistently. Most Topic 10 questions can be answered with wavelength, period and phase fractions.

Phase Difference

The signed phase difference depends on the order of subtraction. If only the separation is required, quote its magnitude or its equivalent value modulo $2\pi$:

$$\Delta \varphi ={\varphi }_1-{\varphi }_2$$

For two points separated by $\Delta x$ along a wave:

$$\Delta \varphi =\frac{2\pi \Delta x}{\lambda }$$

For a time separation $\Delta t$:

$$\Delta \varphi =\frac{2\pi \Delta t}{T}$$

Interpretation of Phase Difference

• $\Delta \varphi =2n\pi$ → in phase;
• $\Delta \varphi =(2n+1)\pi$ → antiphase;
• phase differences that differ by $2\pi$ describe the same relative cycle stage.

Graphical Representation

• Displacement–time graph: shows oscillation of a single point
• Wave profile (snapshot): shows displacement across space at an instant

Key Distinction

• Amplitude: maximum magnitude of displacement
• Displacement: instantaneous position
• Phase: position within the cycle

Key Insight

Displacement tells you one instantaneous coordinate; phase tells you the complete cycle state. Equal displacement does not guarantee equal phase because the directions of motion may differ.

Summary

Use displacement for instantaneous position relative to equilibrium. Use phase difference to compare cycle stage between two particles or waves. A separation of one wavelength gives $2\pi$, half a wavelength gives $\pi$, and a quarter wavelength gives $\pi /2$.

Links

• Main topic: Waves
• Related topic: Oscillations and Simple Harmonic Motion
• Later topic: Superposition of Waves