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 describes how far a particle is from its equilibrium position at a given instant. Phase describes the stage of oscillation of that particle within its cycle.

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

For a wave:

$$\text{phase}=\omega t-kx$$

where $\omega$ is angular frequency and $k$ is wave number. This compact wave-equation form is useful enrichment; most Topic 10 questions can be answered with wavelength, period, and phase fractions.

Phase Difference

The phase difference between two points is:

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

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

• $\Delta \varphi =0$ → in phase
• $\Delta \varphi =\pi$ → completely out of phase
• $\Delta \varphi =2\pi$ → one full cycle difference

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 displacement
• Displacement: instantaneous position
• Phase: position within the cycle

Key Insight

Displacement tells you where the particle is, while phase tells you where it is in the cycle.

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