Wave Basics

Overview

This page develops the core foundations needed for later wave topics:

• transverse versus longitudinal motion;
• displacement-distance graphs;
• displacement-time graphs;
• wave speed and $v=f\lambda$;
• basic electromagnetic spectrum facts.

Definition

A progressive wave transfers energy from one place to another. In a mechanical wave, particles of the medium oscillate about fixed equilibrium positions while neighbouring particles pass on the disturbance. Matter does not move forward overall.

The distinction is fundamental:

• the wave travels;
• the particles oscillate.

Why It Matters

Most wave errors come from mixing up particle motion and wave propagation, or from reading the wrong quantity from the wrong graph. These basics support superposition, sound, light, polarisation, and quantum wave ideas.

Key Representations

Transverse Waves

Particle vibration is perpendicular to the direction of wave propagation.

Examples:

• waves on a rope;
• electromagnetic waves;
• surface water waves as an approximate model.

Features:

• crests and troughs;
• polarisation possible.

If a transverse wave travels right, particles may move up and down.

Longitudinal Waves

Particle vibration is parallel to the direction of wave propagation.

Examples:

• sound in air;
• compression waves in springs.

Features:

• compressions;
• rarefactions;
• polarisation not possible.

If a longitudinal wave travels right, particles move left-right about equilibrium.

Feature	Transverse	Longitudinal
Oscillation direction	Perpendicular to propagation	Parallel to propagation
Shape words	Crest and trough	Compression and rarefaction
Polarisation possible	Yes	No
Example	Light	Sound

Basic Quantities

Displacement $y$ is the signed displacement from equilibrium for a one-dimensional wave graph. In a full vector description, particle displacement is a vector, but JC wave graphs usually show signed components.

Amplitude $A_0$ is the maximum magnitude of displacement.

Period $T$ is the time for one complete oscillation.

Frequency $f$ is the number of oscillations per second:

$$f=\frac{1}{T}$$

Wavelength $\lambda$ is the shortest distance between adjacent points in phase, such as crest to crest, trough to trough, or compression to compression.

Displacement-Distance Graph

A displacement-distance graph shows displacement of all particles at one instant in time.

Horizontal axis:

• position $x$.

Vertical axis:

• displacement $y$.

You can read amplitude, wavelength, and the instantaneous shape of the wave.

If crest-to-crest distance is $0.80\mathrm{m}$:

$$\lambda =0.80\mathrm{m}$$

Common mistake:

This graph does not show one particle moving through space. It is a snapshot of many particles simultaneously.

Displacement-Time Graph

A displacement-time graph shows motion of one particle as time passes.

Horizontal axis:

• time $t$.

Vertical axis:

• displacement $y$.

You can read amplitude, period, and frequency.

If one full cycle takes $0.20\mathrm{s}$:

$$T=0.20\mathrm{s}$$ $$f=\frac{1}{0.20}=5.0\mathrm{Hz}$$

Graph Type	Horizontal Axis	Represents	Gives
Displacement-distance	Position	Wave shape at one instant	$\lambda$, $A_0$
Displacement-time	Time	Motion of one particle	$T$, $f$, $A_0$

Wave Speed

Wave speed is the speed at which the disturbance propagates:

$$v=f\lambda$$

One wavelength passes a point in one period, so:

$$v=\frac{\lambda }{T}$$

Since:

$$f=\frac{1}{T}$$

therefore:

$$v=f\lambda$$

For a wave in the same medium, wave speed usually stays constant. If $f$ increases, $\lambda$ decreases.

Electromagnetic Waves

Electromagnetic waves are transverse waves made of oscillating electric and magnetic field vectors, $\vec{E}$ and $\vec{B}$. The two field vectors are perpendicular to each other and to the direction of propagation. They do not require a medium.

Speed in vacuum:

$$c=3.0\times 10^8\mathrm{m}{\mathrm{s}}^{-1}$$

All obey:

$$c=f\lambda$$

From low frequency to high frequency:

1. Radio
2. Microwaves
3. Infrared
4. Visible light
5. Ultraviolet
6. X-rays
7. Gamma rays

As you move down the list, frequency increases, wavelength decreases, and photon energy increases.

Common Mistakes

• Reading wavelength from a time graph.
• Reading period from a distance graph.
• Thinking particles move forward with the wave.
• Saying sound is transverse.
• Saying electromagnetic waves need a medium.
• Mixing amplitude with wavelength.

Links

• Main topic: Waves
• Related concept: Phase Difference
• Related concept: Intensity and Energy
• Related concept: Polarisation
• Related topic: Superposition of Waves