Wave Properties

A wave is a disturbance that transfers energy from one place to another without transferring matter. Understanding the fundamental properties of waves — and being able to describe them precisely using mathematical language — is essential for everything that follows in this topic.

Transverse and Longitudinal Waves

Waves come in two fundamental types, defined by the relationship between the direction of oscillation and the direction of energy transfer.

In a transverse wave, the oscillations are perpendicular to the direction of energy transfer. If you flick a rope up and down, the wave travels horizontally along the rope while the particles of the rope move vertically. Light, water surface waves, and all electromagnetic waves are transverse.

In a longitudinal wave, the oscillations are parallel to the direction of energy transfer. Sound is the most important example. When a loudspeaker cone pushes forward, it compresses the air molecules in front of it, creating a region of compression (high pressure). As the cone pulls back, it creates a region of rarefaction (low pressure). These compressions and rarefactions propagate through the air, carrying energy to your ear.

A useful way to remember: in a longitudinal wave, the particles oscillate along the direction the wave travels (longitudinal — along).

graph LR
    A["Wave Type?"] -->|"Oscillation ⊥ energy transfer"| B["Transverse"]
    A -->|"Oscillation ∥ energy transfer"| C["Longitudinal"]
    B --> D["Examples: light, EM waves,\nwater surface waves,\nS-waves (seismic)"]
    C --> E["Examples: sound,\nultrasound,\nP-waves (seismic)"]
    B --> F["Can be polarised ✓"]
    C --> G["Cannot be polarised ✗"]

Key Wave Quantities

Every wave can be described using a small set of measurable quantities.

Amplitude (A) is the maximum displacement of a particle from its equilibrium (rest) position. It is measured in metres (m). For a sound wave, greater amplitude means a louder sound. For a light wave, greater amplitude means greater intensity. Crucially, the intensity of a wave is proportional to the square of its amplitude: I ∝ A². This means doubling the amplitude quadruples the intensity.

Wavelength (λ) is the distance between two consecutive points in phase — for example, from one crest to the next crest, or from one compression to the next compression. It is measured in metres (m).

Frequency (f) is the number of complete oscillations per second. It is measured in hertz (Hz), where 1 Hz = 1 oscillation per second. A tuning fork vibrating at 440 Hz completes 440 full cycles every second.

Period (T) is the time taken for one complete oscillation. It is measured in seconds (s). Frequency and period are reciprocals:

$T = \frac{1}{f} \quad \text{and} \quad f = \frac{1}{T}$

For example, if a wave has a frequency of 500 Hz, its period is T = 1/500 = 0.002 s = 2 ms.

Wave speed (v) is the distance travelled by the wave per unit time. It is measured in m s⁻¹. The fundamental wave equation links speed, frequency, and wavelength:

$v = f\lambda$

This equation applies to all waves. If you know any two of the three quantities, you can calculate the third.

Quantity	Symbol	SI Unit	Key relationship
Amplitude	A	m	I ∝ A²
Wavelength	λ	m	λ = v / f
Frequency	f	Hz (s⁻¹)	f = 1 / T
Period	T	s	T = 1 / f
Wave speed	v	m s⁻¹	v = fλ
Phase difference	Δφ	rad (or °)	Δφ = 2πΔx / λ

Worked Calculations

Worked example 1: A sound wave in air has a frequency of 680 Hz and a wavelength of 0.50 m. Calculate the speed of the wave.

v = fλ = 680 × 0.50 = 340 m s⁻¹

This is consistent with the known speed of sound in air at room temperature (~340 m s⁻¹ at 20 °C).

Worked example 2: A radio station broadcasts at 98.5 MHz. Calculate the wavelength of the broadcast signal (speed of EM waves = 3.00 × 10⁸ m s⁻¹).

First convert the frequency: f = 98.5 MHz = 98.5 × 10⁶ Hz

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

FM radio waves have wavelengths of the order of a few metres — this is why FM radio antennas are roughly 75 cm long (about λ/4).

Worked example 3: An ultrasound scanner emits pulses at 3.5 MHz. The speed of ultrasound in soft tissue is approximately 1540 m s⁻¹. Calculate the wavelength and explain why this frequency is chosen.

λ = v / f = 1540 / (3.5 × 10⁶) = 4.4 × 10⁻⁴ m ≈ 0.44 mm

This short wavelength provides good resolution in medical imaging because the ultrasound can distinguish structures that are about 1 mm or larger. Higher frequencies give shorter wavelengths and better resolution but are absorbed more quickly, reducing penetration depth.

Displacement–Distance and Displacement–Time Graphs

Waves are commonly represented using two types of graph, and it is vital to distinguish between them.

A displacement–distance graph is a snapshot of the wave at a single instant. The x-axis shows position along the wave (in metres), and the y-axis shows displacement. From this graph you can read off the amplitude (maximum displacement) and the wavelength (distance between consecutive points in phase).

A displacement–time graph shows how a single point on the wave oscillates over time. The x-axis shows time (in seconds), and the y-axis shows displacement. From this graph you can read off the amplitude and the period (time for one complete cycle). From the period, you can calculate the frequency using f = 1/T.

Common exam mistake: A very frequent error is reading a "wavelength" from a displacement–time graph. The repeat distance on a displacement–time graph gives the period, not the wavelength. Wavelength can only be determined from a displacement–distance graph. Similarly, you cannot read the period from a displacement–distance graph.

Phase and Phase Difference

Phase describes the stage a point on a wave has reached in its cycle. It is measured in degrees (0° to 360°) or radians (0 to 2π).

Two points on a wave are in phase if they have the same displacement and are moving in the same direction at all times. They have a phase difference of 0° (or 360°, or any multiple of 360°). For example, two adjacent crests are in phase, separated by exactly one wavelength.

Two points are in antiphase (or completely out of phase) if they have a phase difference of 180° (π radians). When one is at maximum positive displacement, the other is at maximum negative displacement.

The phase difference between two points separated by a distance Δx along a wave of wavelength λ is:

$\Delta\phi = \frac{2\pi \, \Delta x}{\lambda}$

Worked example 4: Two points on a wave of wavelength 0.40 m are separated by 0.10 m. Calculate the phase difference.

Δφ = (2π × 0.10) / 0.40 = π/2 radians = 90°

The points are a quarter of a cycle apart.

Worked example 5: On a wave of wavelength 12 cm, how far apart are two points with a phase difference of 120° (2π/3 radians)?

Rearranging: Δx = Δφ × λ / (2π) = (2π/3) × 0.12 / (2π) = 0.12/3 = 0.040 m = 4.0 cm

Intensity and the Inverse Square Law

For a point source radiating uniformly in all directions, the wave energy spreads out over ever-larger spheres. At a distance r from the source, the intensity (power per unit area) is:

$I = \frac{P}{4\pi r^2}$

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

Since I ∝ A², this also means A ∝ 1/r — the amplitude halves when you double the distance from the source.

Worked example 6: A loudspeaker emits sound at a power of 0.50 W. Calculate the intensity at a distance of 4.0 m, assuming the sound radiates uniformly.

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

Real-world application: The inverse square law explains why sound becomes quieter as you move away from a source, why we need more powerful transmitters to communicate over greater distances, and why the intensity of sunlight on Mars (~1.5 AU from the Sun) is only about 44% of that on Earth (~1 AU).

Wave Speed in Different Media

The speed of a wave depends on the medium through which it travels, not on its frequency or amplitude (for most waves at normal amplitudes).

Medium	Speed of sound / m s⁻¹	Notes
Air (20 °C)	343	Increases ~0.6 m s⁻¹ per °C
Water (25 °C)	1498	About 4× faster than air
Soft tissue	~1540	Used in medical ultrasound
Steel	~5960	Waves travel fastest in stiff materials
Vacuum	0	Sound cannot travel in a vacuum

For electromagnetic waves in a vacuum, v = c = 3.00 × 10⁸ m s⁻¹ regardless of frequency. In a medium, EM waves slow down (v = c/n, where n is the refractive index).

When a wave passes from one medium to another, its frequency stays constant (determined by the source), but its speed and wavelength change. Since v = fλ and f is fixed, a decrease in speed means a decrease in wavelength, and vice versa.

Summary of Key Equations

Quantity	Symbol	Unit	Equation
Period	T	s	T = 1/f
Frequency	f	Hz	f = 1/T
Wave speed	v	m s⁻¹	v = fλ
Phase difference	Δφ	rad	Δφ = 2πΔx / λ
Intensity	I	W m⁻²	I = P/(4πr²), I ∝ A²

These relationships form the foundation for every wave calculation in A-Level Physics. Make sure you can rearrange each equation confidently and select the correct one for a given problem.