Wave Properties and Types

A wave is a disturbance that transfers energy from one place to another without transferring matter. Understanding the fundamental properties of waves — and the distinction between transverse and longitudinal types — is essential to every topic in this course.

Transverse and Longitudinal Waves

There are two fundamental categories of wave, classified by the direction of oscillation relative to the direction of energy transfer.

Transverse Waves

In a transverse wave, the oscillations are perpendicular to the direction of energy transfer (the direction of wave propagation).

Examples include:

All electromagnetic waves (light, radio, X-rays, etc.)
Waves on a string or rope
S-waves (secondary seismic waves)
Water surface waves (approximately transverse)

A key property of transverse waves is that they can be polarised — the oscillations can be restricted to a single plane. This is explored in detail in Lesson 9.

Longitudinal Waves

In a longitudinal wave, the oscillations are parallel to the direction of energy transfer. The medium undergoes alternating compressions (regions of high pressure/density) and rarefactions (regions of low pressure/density).

Examples include:

Sound waves in air, liquids, and solids
P-waves (primary seismic waves)
Ultrasound waves
Pressure waves in a spring (slinky)

Longitudinal waves cannot be polarised, because the oscillation direction is already constrained to be along the propagation direction.

Exam Tip: If asked how you can distinguish transverse from longitudinal waves experimentally, state that only transverse waves can be polarised. This is the definitive test.

Key Wave Quantities

Every periodic wave is described by a set of measurable quantities. These must be defined precisely for full marks in an exam.

Quantity	Symbol	Definition	SI Unit
Amplitude	A	Maximum displacement of a point on the wave from its equilibrium (rest) position	m
Wavelength	λ	Minimum distance between two points oscillating exactly in phase (e.g., crest to crest)	m
Frequency	f	Number of complete oscillations passing a given point per unit time	Hz (s⁻¹)
Period	T	Time taken for one complete oscillation	s
Wave speed	v	Distance travelled by the wave per unit time	m s⁻¹

The Relationship Between Frequency and Period

Frequency and period are reciprocals of each other:

f = 1/T and equivalently T = 1/f

Worked Example 1 — A sound wave has a period of 2.5 ms. Calculate its frequency.

T = 2.5 ms = 2.5 × 10⁻³ s

f = 1/T = 1/(2.5 × 10⁻³) = 400 Hz

The Wave Equation

The fundamental equation relating wave speed, frequency, and wavelength is:

v = f λ

This can be derived from first principles. In one complete period T, the wave advances by exactly one wavelength λ. Since speed = distance/time:

v = λ/T = λ × (1/T) = f λ

This equation applies to all waves — mechanical and electromagnetic, transverse and longitudinal.

Worked Example 2 — A radio wave has a frequency of 98.5 MHz. Calculate its wavelength.

v = c = 3.00 × 10⁸ m s⁻¹ (radio waves are electromagnetic)

f = 98.5 MHz = 98.5 × 10⁶ Hz

λ = v/f = (3.00 × 10⁸)/(98.5 × 10⁶) = 3.05 m

Worked Example 3 — A guitar string vibrates at 440 Hz and produces a sound wave in air with wavelength 0.773 m. Calculate the speed of sound in air.

v = f λ = 440 × 0.773 = 340 m s⁻¹ (to 3 s.f.)

Worked Example 4 — Ultrasound of frequency 2.0 MHz travels through soft tissue at 1540 m s⁻¹. Calculate the wavelength.

λ = v/f = 1540/(2.0 × 10⁶) = 7.7 × 10⁻⁴ m = 0.77 mm

Exam Tip: Always convert units before substituting into the wave equation. Frequencies in MHz or kHz must be converted to Hz; wavelengths in nm or mm must be converted to m.

Phase and Phase Difference

The phase of a point on a wave describes its position within the oscillation cycle. Phase is measured in radians (rad) or degrees (°), where one complete cycle = 2π rad = 360°.

Phase Difference

The phase difference between two points on a wave (or between two waves) describes how far one oscillation is ahead of or behind the other.

Phase difference	In degrees	In radians	Meaning
In phase	0° (or 360°)	0 (or 2π)	Points oscillate together — same displacement at all times
Antiphase	180°	π	Points have equal but opposite displacements at all times
Quarter cycle ahead	90°	π/2	One point leads the other by a quarter of a wavelength

Calculating Phase Difference from Path Difference

If two points on a wave are separated by a distance Δx along the direction of propagation, their phase difference Δφ is:

Δφ = (2π/λ) × Δx

Equivalently, the path difference in terms of wavelength determines the phase relationship:

Path difference = nλ (where n is an integer) → in phase (Δφ = 2nπ)
Path difference = (n + ½)λ → in antiphase (Δφ = (2n + 1)π)

Worked Example 5 — Two points on a wave of wavelength 0.60 m are separated by 0.45 m. What is their phase difference?

Δφ = (2π/λ) × Δx = (2π/0.60) × 0.45 = (2π × 0.75) = 1.5π rad

Convert: 1.5π rad = 270° (or equivalently 3π/2 rad)

The path difference is 0.75λ, which is neither a whole number nor a half-integer multiple of λ, so the points are neither in phase nor in antiphase.

Worked Example 6 — Two loudspeakers emit sound of wavelength 0.50 m in phase. A listener is 3.00 m from one speaker and 3.75 m from the other. What does the listener hear?

Path difference = 3.75 − 3.00 = 0.75 m

Path difference in wavelengths = 0.75/0.50 = 1.5λ

Since 1.5λ = (1 + ½)λ, this is a half-integer multiple of λ. The waves arrive in antiphase, producing destructive interference — the listener hears a quiet sound (a minimum).

Displacement-Distance and Displacement-Time Graphs

Waves can be represented graphically in two important ways:

Displacement-Distance Graph (at a fixed instant)

This is a snapshot of the wave at one moment in time, plotting displacement (y-axis) against position along the wave (x-axis). From this graph you can read:

Amplitude — the maximum displacement
Wavelength — the horizontal distance for one complete cycle (e.g., crest to crest)

Displacement-Time Graph (at a fixed position)

This shows how the displacement of one point varies with time. From this graph you can read:

Amplitude — the maximum displacement
Period — the horizontal distance for one complete cycle

Common Misconception: Students often confuse wavelength and period. Wavelength is a spatial quantity (metres) read from a displacement-distance graph; period is a temporal quantity (seconds) read from a displacement-time graph.

Wave Intensity

The intensity of a wave is the power transmitted per unit area, measured perpendicular to the direction of energy transfer:

I = P/A

where I is intensity (W m⁻²), P is power (W), and A is the area (m²) through which the wave passes.

For a point source radiating equally in all directions, the power spreads over the surface of a sphere of radius r:

I = P/(4πr²)

This gives the inverse square law: intensity is inversely proportional to the square of the distance from the source.

Intensity and Amplitude

Intensity is proportional to the square of the amplitude:

I ∝ A²

This means that doubling the amplitude quadruples the intensity.

Worked Example 7 — A point source emits sound with a power of 0.50 W. Calculate the intensity at a distance of 4.0 m from the source.

I = P/(4πr²) = 0.50/(4π × 4.0²) = 0.50/(4π × 16) = 0.50/201.1 = 2.5 × 10⁻³ W m⁻²

Worked Example 8 — At a distance of 2.0 m from a source, the intensity of a sound wave is 0.080 W m⁻². What is the intensity at 8.0 m?

Using the inverse square law: I₁r₁² = I₂r₂²

I₂ = I₁ × (r₁/r₂)² = 0.080 × (2.0/8.0)² = 0.080 × (0.25)² = 0.080 × 0.0625 = 5.0 × 10⁻³ W m⁻²

The distance quadrupled, so the intensity decreased by a factor of 16.

Summary

Transverse waves oscillate perpendicular to energy transfer; longitudinal waves oscillate parallel.
Key quantities: amplitude (A), wavelength (λ), frequency (f), period (T), wave speed (v).
The wave equation v = fλ applies to all waves.
Phase difference Δφ = (2π/λ) × Δx.
Intensity I = P/A and I ∝ A².
For a point source, I = P/(4πr²) (inverse square law).