Progressive Waves

Understanding wave properties, types, and phase relationships

SPEC 3.3.1.1 • WAVES

What is a Progressive Wave?

Definition

A progressive wave (or travelling wave) is a wave that transfers energy from one place to another without transferring matter.

Energy Transfer

Progressive waves carry energy from the source outwards. This energy can do work at a distance.

Matter Stays Put

Particles oscillate about fixed positions - they do not travel with the wave.

Amplitude & Wavelength

Amplitude (A)

The maximum displacement of a particle from its equilibrium position. Measured in metres (m).

Wavelength (λ)

The distance between any two successive corresponding points (e.g., crest to crest). Measured in metres (m).

Typical Mistake

Amplitude is NOT trough-to-crest distance. That would be 2A.

Typical Mistake

Wavelength is NOT crest-to-trough distance. That would be λ/2.

Frequency, Period & Wave Speed

QuantitySymbolDefinitionUnit

DisplacementyDistance from equilibrium position at any pointMetres (m)

FrequencyfNumber of complete oscillations per secondHertz (Hz)

PeriodTTime for one complete oscillationSeconds (s)

Wave Speedv (or c)Distance travelled by wave per secondm s⁻¹

Key Relationship

T = \frac{1}{f}

Period and frequency are reciprocals of each other

Frequency Units

1 kHz = 1000 Hz (kilohertz) | 1 MHz = 1,000,000 Hz (megahertz)

Example: 101.7 MHz = 101.7 × 10⁶ Hz = 1.017 × 10⁸ Hz

The Wave Equation

Wave Equation

v = f λ

Wave speed = frequency × wavelength

Derivation

In one period (T), the wave travels exactly one wavelength (λ):

Speed = Distance / Time = λ / T

Since T = 1/f:

v = λ × f = fλ

Important

The wave speed depends on the medium, not the source. Changing frequency changes wavelength, but wave speed remains constant in a given medium.

Worked Example01

A sound wave has a frequency of 440 Hz and travels at 340 m s⁻¹ in air. Calculate the wavelength of the sound wave.

1

Write down known values: f = 440 Hz, v = 340 m s⁻¹

2

Rearrange wave equation: v = fλ → λ = v/f

3

Substitute: λ = 340 / 440 = 0.773 m

Answer: λ = 0.77 m (2 s.f.)

Worked Example02

A radio wave has a wavelength of 3.0 m. Given that electromagnetic waves travel at 3.0 × 10⁸ m s⁻¹, calculate the frequency of the radio wave and its period.

1

Known: λ = 3.0 m, v = 3.0 × 10⁸ m s⁻¹

2

Rearrange for frequency: f = v/λ

3

Calculate: f = (3.0 × 10⁸) / 3.0 = 1.0 × 10⁸ Hz = 100 MHz

4

Calculate period: T = 1/f = 1/(1.0 × 10⁸) = 1.0 × 10⁻⁸ s

Answer: f = 1.0 × 10⁸ Hz (100 MHz), T = 1.0 × 10⁻⁸ s

Transverse Waves

Definition

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

Examples of Transverse Waves

All electromagnetic waves (light, radio, X-rays, microwaves, UV, infrared, gamma)

Water waves (ripples on the surface)

Waves on a string or rope

S-waves (seismic secondary waves)

Longitudinal Waves

Definition

In a longitudinal wave, the oscillations are parallel to the direction of energy transfer.

Examples of Longitudinal Waves

Sound waves in air, liquids, and solids

Ultrasound

P-waves (seismic primary waves)

Compression waves in a spring (slinky)

Comparing Wave Types

PropertyTransverseLongitudinal

Oscillation directionPerpendicular to wave travelParallel to wave travel

Can be polarised?YesNo

Travel through vacuum?EM waves: YesNo (need medium)

ExamplesLight, water waves, S-wavesSound, P-waves, ultrasound

Important

Only transverse waves can be polarised. This is because the oscillations occur in a plane perpendicular to the direction of travel, and polarisation restricts oscillations to a single plane.

Worked Example03

A seismic wave travels through the Earth. At one location, instruments detect that particles are oscillating in the same direction as the wave is travelling. Identify the type of wave and explain why this wave cannot be polarised.

1

Oscillations parallel to wave direction = longitudinal wave

2

This is a P-wave (primary seismic wave)

3

Polarisation restricts oscillations to one plane perpendicular to wave travel

4

In longitudinal waves, oscillations are along the direction of travel, so there is no perpendicular plane to restrict

Answer: P-wave (longitudinal). Cannot be polarised because oscillations are parallel to wave direction, not perpendicular.

Phase and Phase Difference

Phase Difference

Phase difference is the fraction of a cycle between two waves or two points, measured in degrees or radians.

IN PHASE (Δφ = 0° or 360°)

Crests align with crests. Waves reinforce.

ANTIPHASE (Δφ = 180° or π rad)

Crests align with troughs. Waves cancel.

Phase in Radians & Degrees

Conversions

360° = 2 π radians

One complete cycle = 360° = 2π rad

DegreesRadiansDescription

0° (or 360°)0 (or 2π)In phase

90°π/2Quarter cycle out of phase

180°πIn antiphase (half cycle)

270°3π/2Three-quarter cycle out

Tip

To convert: Radians = Degrees × (π/180). Degrees = Radians × (180/π).

Path Difference

Definition

Path difference (Δx) is the difference in distance travelled by two waves from their sources to a given point.

Key Relationship

Δ ϕ = \frac{2 π}{λ} \times Δ x

Quick Reference

Δx = λ → Δφ = 2π (360°) → in phase

Δx = λ/2 → Δφ = π (180°) → antiphase

Δx = λ/4 → Δφ = π/2 (90°)

Worked Example04

Two points on a wave are separated by a distance of 0.75 m. The wavelength of the wave is 1.5 m. Calculate the phase difference between the two points in both radians and degrees.

1

Known: Δx = 0.75 m, λ = 1.5 m

2

Use formula: Δφ = (2π/λ) × Δx

3

Δφ = (2π / 1.5) × 0.75 = π radians

4

Convert to degrees: π rad = 180°

Answer: Phase difference = π rad = 180° (points are in antiphase)

Worked Example05

Two speakers emit sound waves of frequency 680 Hz in phase. A listener stands at a point where the path difference from the two speakers is 0.25 m. If the speed of sound is 340 m s⁻¹, determine whether the listener experiences constructive or destructive interference.

1

Find wavelength: λ = v/f = 340/680 = 0.5 m

2

Path difference = 0.25 m = λ/2

3

Phase difference: Δφ = (2π/0.5) × 0.25 = π radians = 180°

4

Phase difference of π means waves arrive in antiphase

Answer: Destructive interference (waves cancel as they arrive 180° out of phase)

Practice

Question 1: A wave on a string has a frequency of 25 Hz and a wavelength of 0.8 m. Calculate the speed of the wave and the time period.

1

Wave speed: v = fλ = 25 × 0.8 = 20 m s⁻¹

2

Period: T = 1/f = 1/25 = 0.04 s

Answer: v = 20 m s⁻¹, T = 0.04 s

Practice

Question 2: Two points on a wave of wavelength 12 cm are separated by 9 cm. Calculate the phase difference in radians and state whether they are in phase, in antiphase, or neither.

1

Δφ = (2π/λ) × Δx = (2π/0.12) × 0.09

2

Δφ = (2π × 0.09) / 0.12 = 1.5π = 3π/2 radians

3

3π/2 rad = 270° (neither in phase nor in antiphase)

Answer: 3π/2 rad (270°) - neither in phase nor antiphase

Practice

Question 3: Explain why sound waves can travel through water but not through a vacuum, while light waves can travel through both.

1

Sound is a longitudinal mechanical wave - requires a medium to transfer energy through particle compressions and rarefactions

2

Water provides particles that can oscillate; vacuum has no particles, so no medium for sound

3

Light is an electromagnetic wave - oscillating electric and magnetic fields that do not require a medium

Answer: Sound requires particles to propagate (mechanical wave); light is electromagnetic and can propagate through oscillating fields without a medium.

Summary

Key Concepts

ConceptKey Point

Progressive WaveTransfers energy without transferring matter

Wave Equationv = fλ (also T = 1/f)

TransverseOscillations perpendicular to direction; can be polarised

LongitudinalOscillations parallel to direction; cannot be polarised

Phase DifferenceΔφ = (2π/λ) × Δx

Exam Skills

Use v = fλ and rearrange confidently

Convert between degrees and radians for phase

Identify wave types from oscillation direction

Calculate phase difference from path difference