Wave Properties

Wave Properties

Info: Board Coverage AQA Paper 2 | Edexcel CP2 | OCR (A) Paper 2 | CIE P2

Explore the simulation above to develop intuition for this topic.

Intuition

Waves are disturbances that travel without carrying matter: Think of a stadium wave — people stand up and sit down, but nobody moves from their seat. The wave pattern travels around the stadium, but the people stay put. This is exactly how sound waves, water waves, and light work — the disturbance propagates, but the medium doesn’t travel with it.

Why it matters: Waves are how energy moves through the universe — from sunlight reaching Earth to Wi-Fi signals reaching your phone. Understanding wave properties explains why you can hear around corners but not see around them, why fibre optics work, and how musical instruments produce different notes.

The key insight: All waves obey the same fundamental equation $v = f\lambda$, but the speed $v$ depends on the medium. Changing frequency changes wavelength, not speed. This is why a sound sounds the same whether it’s loud or quiet — the frequency (pitch) doesn’t change, only the amplitude (volume).

1. Progressive Waves

A progressive wave is a disturbance that transfers energy through a medium (or vacuum) without Transferring matter. Each particle in the medium oscillates about its equilibrium position.

Key Definitions

• Displacement $y$: the distance of a point on the wave from its equilibrium position (m)
• Amplitude $A$: the maximum displacement from equilibrium (m)
• Wavelength $\lambda$: the distance between two consecutive points in phase — e.g., crest-to-crest (m)
• Period $T$: the time for one complete oscillation (s)
• Frequency $f$: the number of complete oscillations per unit time (Hz)
• Phase $\phi$: a measure of the position in the cycle of oscillation (rad)

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

The Wave Equation

$\boxed{v = f\lambda}$

Derivation. Consider a wave travelling at speed $v$. In one period $T$The wave advances by Exactly one wavelength $\lambda$ (the source completes one full oscillation, producing one complete Wave cycle):

$v = \frac{\mathrm{distance travelled}}{\mathrm{time taken}} = \frac{\lambda}{T} = \lambda \cdot \frac{1}{T} = f\lambda$

$\square$

Intuition. If you shake a rope at 2 Hz with a wavelength of 1.5 m, the crests travel at 3 m/s. Faster shaking or longer waves both make the wave move faster.

Real-World Application: Seismic Waves

When an earthquake occurs, seismic waves carry energy through the Earth. P-waves (primary waves) Are longitudinal compressional waves that travel fastest (5 to 8 km/s through the crust) and can Pass through solids and liquids. S-waves (secondary waves) are transverse waves that travel Slower (3 to 4.5 km/s) and cannot pass through liquids — this is how geophysicists determined that The Earth’s outer core is liquid.

Seismographs at different stations detect P-waves and S-waves arriving at different times. By Measuring the time difference $\Delta t$ and knowing the approximate speeds $v_P$ and $v_S$The Distance to the epicentre can be estimated:

$d \approx \frac{v_P \cdot v_S}{v_P - v_S} \cdot \Delta t$

Three or more stations are needed to triangulate the epicentre location.

Mathematical Description of a Progressive Wave

A sinusoidal progressive wave travelling in the $+x$ direction is described by:

$y(x,t) = A\sin(kx - \omega t + \phi_0)$

Where:

• $k = \frac{2\pi}{\lambda}$ is the wave number (rad m$^{-1}$)
• $\omega = 2\pi f = \frac{2\pi}{T}$ is the angular frequency (rad s$^{-1}$)
• $\phi_0$ is the initial phase

Proof that $v = f\lambda$ from the wave function. A point of constant phase satisfies $kx - \omega t = \mathrm{const}$. Differentiating: $k\frac{dx}{dt} - \omega = 0$So:

$v = \frac{dx}{dt} = \frac{\omega}{k} = \frac{2\pi f}{2\pi/\lambda} = f\lambda$

$\square$

2. Transverse and Longitudinal Waves

Transverse Waves

The oscillations are perpendicular to the direction of energy propagation.

• Electromagnetic waves (light, radio, X-rays)
• Waves on a string
• Water waves (approximately, for small amplitudes)

Longitudinal Waves

The oscillations are parallel to the direction of energy propagation.

• Sound waves
• Pressure waves in springs

Longitudinal waves have compressions (regions of high pressure, where particles bunch together) And rarefactions (regions of low pressure, where particles spread apart).

Real-World Application: Musical Instruments

Sound production in musical instruments relies on creating standing waves (covered in the next Topic), but the initial wave type depends on the instrument:

• Stringed instruments (guitar, violin): A plucked or bowed string vibrates transversely, creating a transverse wave on the string. The vibrating string then pushes air molecules back and forth, producing a longitudinal sound wave. This is a key exam point: the wave on the string is transverse, but the sound wave in air is longitudinal.
• Wind instruments (flute, clarinet, trumpet): A column of air vibrates longitudinally. The player’s lips or a reed create compressions and rarefactions that travel along the air column.
• Percussion (drums): The drum membrane vibrates transversely, and the resulting pressure changes in the surrounding air produce longitudinal sound waves.

The speed of sound in air at 20°C is approximately 343 m/s. For a concert A (440 Hz), the wavelength In air is $\lambda = 343/440 = 0.78$ m. Higher notes have shorter wavelengths; lower notes have Longer wavelengths.

3. Polarisation

Definition. Polarisation is the restriction of the oscillation direction of a transverse Wave to a single plane.

Proof of Malus’s Law

Consider unpolarised light of intensity $I_0$ incident on a polarising filter (analyser) whose Transmission axis makes an angle $\theta$ with the polarisation direction of the incoming light.

The electric field of the incoming polarised wave can be resolved into components parallel and Perpendicular to the analyser’s transmission axis:

$E_{\parallel} = E_0 \cos\theta, \qquad E_{\perp} = E_0 \sin\theta$

Only the parallel component is transmitted. The transmitted intensity is:

$I = \frac{1}{2}\varepsilon_0 c E_{\parallel}^2 = \frac{1}{2}\varepsilon_0 c E_0^2 \cos^2\theta = I_0 \cos^2\theta$

$\boxed{I = I_0 \cos^2\theta}$

Where $I_0 = \frac{1}{2}\varepsilon_0 c E_0^2$ is the intensity of the polarised light incident on The analyser.

Intuition. When $\theta = 0$All light passes through. When $\theta = 90^\circ$No light Passes. At $\theta = 45^\circ$The intensity is halved. This is a direct consequence of the vector Nature of the electric field.

Polarisation by Reflection (Brewster’s Angle)

When light reflects from a dielectric surface, the reflected ray is partially polarised. At Brewster’s angle $\theta_B$The reflected light is fully polarised perpendicular to the plane Of incidence:

$\tan\theta_B = n_2/n_1$

Polarisation in Real Life

Polarising filters have many practical applications:

• Polaroid sunglasses reduce glare by blocking horizontally polarised light reflected from horizontal surfaces (water, roads). Reflected light is predominantly horizontally polarised, so vertically oriented filters absorb most of the glare.
• LCD screens use two crossed polarisers with liquid crystals between them. The crystals rotate the polarisation of light passing through, allowing controlled transmission of each pixel.
• Stress analysis in engineering: transparent plastic models under stress become birefringent, rotating the polarisation of transmitted light. Viewed between crossed polarisers, stress concentrations appear as coloured fringes (photoelasticity).

Polarisation and Microwaves

Microwave ovens use polarised EM waves (wavelength ~12 cm). The metal mesh on the oven door acts as A polarising grid — the holes are much smaller than the wavelength, so the microwaves are reflected While visible light (with a much shorter wavelength) passes through, allowing you to see inside.

4. The Electromagnetic Spectrum

All EM waves travel at $c = 3.00 \times 10^8$ m s$^{-1}$ in vacuum and are transverse. They are Distinguished by wavelength (equivalently, frequency).

All EM waves are produced by accelerating charges. In order of increasing frequency: radio (oscillating currents in aerials), microwave (klystrons/magnetrons), infrared/visible/UV (atomic Electron transitions), X-rays (electron deceleration), gamma rays (nuclear transitions).

5. Intensity and Amplitude

Definition. The intensity $I$ of a wave is the power per unit area:

$I = \frac{P}{A}$

With SI units W m$^{-2}$.

Proof that $I \propto A^2$ for a 2D Wave

Consider a sinusoidal wave on a string. The power transmitted past a point is the rate at which the Transverse force does work:

$P = F_{\mathrm{transverse}} \times v_{\mathrm{transverse}}$

For a wave $y = A\sin(kx - \omega t)$The transverse velocity is:

$v_y = \frac{\partial y}{\partial t} = -A\omega\cos(kx - \omega t)$

The transverse component of tension is (for small amplitudes):

$F_y \approx -T\frac{\partial y}{\partial x} = -TAk\cos(kx - \omega t)$

The instantaneous power is:

$P = F_y \cdot v_y = TAk \cdot A\omega \cos^2(kx - \omega t) = TA^2 k\omega \cos^2(kx - \omega t)$

The average power over one cycle (since $\langle\cos^2\rangle = 1/2$):

$\langle P \rangle = \frac{1}{2}TA^2 k\omega$

Since $v = \omega/k$We have $k\omega = \omega^2/v$. Also, for a string, $v = \sqrt{T/\mu}$ where $\mu$ is the mass per unit length, so $T = \mu v^2$:

$\langle P \rangle = \frac{1}{2}\mu v^2 A^2 \cdot \frac{\omega^2}{v} = \frac{1}{2}\mu v \omega^2 A^2$

The intensity is power per unit… But for a string wave, the “intensity” concept maps to power Directly. The key result is:

$\boxed{I \propto A^2}$

For electromagnetic waves, the intensity is:

$I = \frac{1}{2}\varepsilon_0 c E_0^2$

Which also shows $I \propto A^2$ (where $A$ is the electric field amplitude $E_0$).

Intuition. Double the amplitude and you quadruple the energy carried. This is analogous to Kinetic energy ($E_k = \frac{1}{2}mv^2$): double the speed and you quadruple the energy. The wave Amplitude plays the same role as “speed” for the oscillating particles.

Inverse Square Law for Point Sources

For a wave spreading uniformly from a point source in 3D:

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

At distance $r$The power is spread over a sphere of area $4\pi r^2$. Doubling the distance Quarters the intensity.

$I_1 r_1^2 = I_2 r_2^2$

Intensity and the Decibel Scale

In practice, intensity ratios are often expressed using the decibel scale:

$\mathrm{Intensity level (dB)} = 10 \log_{10}\left(\frac{I}{I_0}\right)$

Where $I_0 = 1.0 \times 10^{-12}$ W m$^{-2}$ is the threshold of hearing. This logarithmic scale Reflects how human perception of loudness works: a 3 dB increase represents a doubling of intensity, While a 10 dB increase sounds roughly “twice as loud” to human ears.

Example. Normal conversation is approximately 60 dB. A whisper is about 30 dB. The intensity Ratio is $10^{(60-30)/10} = 10^3 = 1000$Meaning conversation is 1000 times more intense than a Whisper.

Info: Board Coverage AQA Paper 2 | Edexcel CP2 | OCR (A) Paper 2 | CIE P2 All boards cover the inverse square law. The decibel scale is explicitly on CIE and sometimes Appears in AQA context questions. Edexcel and OCR (A) focus more on $I \propto A^2$ and $I \propto 1/r^2$ calculations.

Amplitude and the Inverse Square Law

Since $I \propto A^2$ and $I \propto 1/r^2$It follows that:

$A \propto \frac{1}{r}$

The amplitude decreases inversely with distance (not as $1/r^2$). This is an important distinction In exam questions that ask about amplitude at different distances.

6. Phase and Phase Difference

Two points on a wave are in phase if they have the same displacement and are moving in the same Direction. They are in antiphase (completely out of phase) if their displacements are equal and Opposite.

The phase difference between two points separated by distance $\Delta x$ is:

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

Special cases:

• In phase: $\Delta\phi = 0, 2\pi, 4\pi, \ldots$ (integer multiples of $2\pi$)
• Antiphase: $\Delta\phi = \pi, 3\pi, 5\pi, \ldots$ (odd multiples of $\pi$)

Phase Difference from Wave Equations

If two waves are described by:

$y_1 = A_1\sin(kx - \omega t + \phi_1), \qquad y_2 = A_2\sin(kx - \omega t + \phi_2)$

Then the phase difference is $\Delta\phi = \phi_2 - \phi_1$. This constant phase difference Is what determines whether two waves interfere constructively ($\Delta\phi = 2n\pi$) or Destructively ($\Delta\phi = (2n+1)\pi$).

Phase Difference from Graphs

When given a displacement—position graph:

1. Identify the positions of the two points: $x_1$ and $x_2$
2. Measure the wavelength $\lambda$ from the graph (peak to peak)
3. Calculate $\Delta x = |x_2 - x_1|$
4. Apply $\Delta\phi = \frac{\Delta x}{\lambda} \times 360^\circ$ (or $\times 2\pi$ rad)

When given a displacement—time graph at a single point, the phase difference between two instants $t_1$ and $t_2$ is:

$\Delta\phi = \frac{\Delta t}{T} \times 360° = \frac{\Delta t}{T} \times 2\pi \mathrm{ rad}$

Info: Board Coverage AQA Paper 2 | Edexcel CP2 | OCR (A) Paper 2 | CIE P2 AQA and OCR (A) frequently test phase difference from displacement—position graphs. CIE often Combines phase difference with path difference in interference questions. Edexcel emphasises the Connection between phase difference and the condition for constructive/destructive interference.

Real-World Application: Doppler Ultrasound

The Doppler effect causes a change in observed frequency when there is relative motion between Source and observer. The phase difference between transmitted and received waves is used in Doppler ultrasound to measure blood flow velocity.

When ultrasound of frequency $f_0$ is reflected off blood cells moving with velocity $v$ away from The transducer, the received frequency is:

$f' = f_0 \cdot \frac{c - v}{c}$

Where $c$ is the speed of ultrasound in tissue (~1540 m/s). The frequency shift $\Delta f = f_0 - f'$ is proportional to blood velocity. By measuring this shift, clinicians can Detect narrowed arteries, heart valve problems, and foetal heartbeats.

Problem Set

Cross-References

• Superposition and Interference: Expands on how multiple waves combine, building on the wave properties defined here.
• Refraction and Total Internal Reflection: Applies wave speed and wavelength concepts to explain how waves change direction at boundaries.
• Electromagnetic Waves: Connects the EM spectrum covered here with detailed properties and applications of electromagnetic radiation.
• Sound Waves: Explores longitudinal wave behaviour in more depth, including resonance and musical applications.