compare transverse and longitudinal waves

Progressive Waves – Transverse vs Longitudinal (Cambridge 9702 – Topic 7)

1. Definition and Fundamental Relations (AO1)

• Progressive (travelling) wave: a disturbance that moves through a medium, carrying energy without a permanent displacement of the medium’s particles.
• Wave‑speed relation (required by the syllabus): $$v = f\lambda$$ where v is the speed, f the frequency and λ the wavelength.
• One‑dimensional wave equation: $$\frac{\partial^{2}y}{\partial x^{2}}=\frac{1}{v^{2}}\frac{\partial^{2}y}{\partial t^{2}}$$ (for a longitudinal displacement s the same form holds). This equation mathematically expresses that the shape of the disturbance propagates unchanged at speed v; the speed relation v = fλ follows directly by substituting a sinusoidal solution into the wave equation.

2. Wave‑speed Expressions and Restoring Forces (AO1)

• Transverse wave on a stretched string
  • Restoring force: tension T acting on a displaced element.
  • Derivation (outline): balance the transverse components of tension on a small element of length Δx → $$T\frac{\partial^{2}y}{\partial x^{2}} = \mu \frac{\partial^{2}y}{\partial t^{2}}$$ where μ is the linear mass density. Rearranging gives the speed $$\boxed{v = \sqrt{\dfrac{T}{\mu}}}$$
• Longitudinal wave in a fluid or gas
  • Restoring force: compression/expansion characterised by the bulk modulus B.
  • Derivation (outline): a small element of length Δx experiences a pressure difference Δp = B(ΔV/V). Using Newton’s second law leads to $$B\frac{\partial^{2}s}{\partial x^{2}} = \rho \frac{\partial^{2}s}{\partial t^{2}}$$ where ρ is the density of the medium. Hence $$\boxed{v = \sqrt{\dfrac{B}{\rho}}}$$

3. Simple Experimental Investigation (AO3)

4. Comparison of Transverse and Longitudinal Waves (AO1)

Feature	Transverse Wave	Longitudinal Wave
Particle motion	Perpendicular to the direction of propagation (up‑and‑down or side‑to‑side)	Parallel to the direction of propagation (back‑and‑forth)
Typical examples	Light in vacuum, waves on a string, surface water waves	Sound in air, pressure waves in a spring‑mass system, seismic P‑waves
Restoring force	Tension  (or shear) in the medium	Compression/expansion characterised by bulk modulus
Speed formula (for the simplest case)	\(v = \sqrt{T/\mu}\) (string)	\(v = \sqrt{B/\rho}\) (fluid/gas)
Polarisation	Can be polarised – the oscillation direction is defined.	Cannot be polarised – oscillation is along the travel direction.
Intensity (energy transport)	\(I = \tfrac12 \rho v \,\omega^{2} A^{2}\) (∝ A²)	\(I = \tfrac12 \rho v \,\omega^{2} A^{2}\) (same form, ρ is the medium density)
Typical diagram	Sinusoidal curve drawn perpendicular to the travel direction (crests & troughs).	Series of compressions and rarefactions drawn along the travel direction.

5. Mathematical Description of a Sinusoidal Progressive Wave (AO1)

For a wave travelling in the +x direction:

$$y(x,t)=A\sin\bigl(kx-\omega t+\phi\bigr)$$

• y – transverse displacement (use s for longitudinal displacement).
• A – amplitude (maximum displacement).
• k = 2\pi/\lambda – wave‑number.
• ω = 2\pi f – angular frequency.
• φ – phase constant.

The same functional form applies to longitudinal displacement s(x,t), with the particle motion now parallel to x.

6. Polarisation of Electromagnetic Waves (AO1 & AO2)

• Only transverse electromagnetic (EM) waves can be polarised because the electric‑field vector defines a unique direction.
• Malus’s law: when linearly polarised light passes through an ideal analyser whose transmission axis makes an angle θ with the incident polarisation, the transmitted intensity is $$I = I_{0}\cos^{2}\theta$$ where I₀ is the intensity before the analyser.
• Worked example: Unpolarised light of intensity 200 W m⁻² passes through two ideal polarisers whose axes differ by 45°.
  1. First polariser reduces intensity to half: \(I_{1}=100\ \text{W m}^{-2}\).
  2. Second polariser: \(I_{2}=I_{1}\cos^{2}45^{\circ}=100\times(0.707)^{2}=50\ \text{W m}^{-2}\).

7. Doppler Effect for Sound (AO1 & AO2)

8. Electromagnetic Spectrum – Quick Reference (AO1)

Region	Wavelength (λ)	Frequency (f)	Typical uses / examples
Radio	≥ 1 m	≤ 3 × 10⁸ Hz	Broadcasting, radar
Microwave	1 mm – 1 m	3 × 10⁸ – 3 × 10¹¹ Hz	Cooking, satellite communication
Infrared (IR)	700 nm – 1 mm	3 × 10¹¹ – 4 × 10¹⁴ Hz	Thermal imaging, remote sensing
Visible	400 nm – 700 nm	4 × 10¹⁴ – 7.5 × 10¹⁴ Hz	Human vision
Ultraviolet (UV)	10 nm – 400 nm	7.5 × 10¹⁴ – 3 × 10¹⁶ Hz	Sterilisation, fluorescence
X‑ray	0.01 nm – 10 nm	3 × 10¹⁶ – 3 × 10¹⁹ Hz	Medical imaging, crystallography
γ‑ray	< 0.01 nm	> 3 × 10¹⁹ Hz	Radioactive decay, astrophysics

All EM waves are transverse, travel at speed c ≈ 3.00 × 10⁸ m s⁻¹ in vacuum, and can be polarised.

9. Key Points for A‑Level Exams (Aligned with AO1–AO3)

1. Definitions & equations (AO1): state the definition of a progressive wave, write \(v = f\lambda\), the wave equation, and the appropriate speed formulas for strings and fluids.
2. Direction of particle motion (AO1): identify perpendicular motion for transverse waves and parallel motion for longitudinal waves.
3. Polarisation (AO2): recall that only transverse waves can be polarised and apply Malus’s law when required.
4. Intensity (AO2): use \(I = \tfrac12 \rho v \omega^{2} A^{2}\); note that intensity ∝ A² for both wave types.
5. Doppler effect (AO2): select the correct formula for moving source, moving observer, or both, and solve for the observed frequency.
6. Sketches (AO2): be able to draw one complete cycle showing crests/troughs (transverse) or compressions/rarefactions (longitudinal).
7. Experimental planning (AO3): outline a simple investigation (see §3) that distinguishes transverse from longitudinal waves and links measured speed to the theoretical formulas.
8. EM spectrum identification (AO2): given a wavelength or frequency, locate the correct region of the spectrum using the table in §8.

10. Sample Exam Questions (AO1–AO3)

Q1 – Sound (longitudinal) A sound wave of frequency 500 Hz travels through air where the speed of sound is 340 m s⁻¹.

• Calculate the wavelength.
• Describe the particle motion.

Solution \[ \lambda = \frac{v}{f} = \frac{340}{500}=0.68\ \text{m} \] Air particles oscillate back‑and‑forth along the direction of propagation, producing alternating compressions and rarefactions.

Q2 – String (transverse) A string under a tension of 80 N has a linear mass density of 0.02 kg m⁻¹.

• Determine the speed of a transverse wave on the string.
• Find the wavelength of a 250 Hz wave travelling on it.
• State the direction of particle motion.

Solution \[ v = \sqrt{\frac{T}{\mu}} = \sqrt{\frac{80}{0.02}} = \sqrt{4000}=63.2\ \text{m s}^{-1} \] \[ \lambda = \frac{v}{f} = \frac{63.2}{250}=0.253\ \text{m} \] Particle motion is perpendicular to the string’s length (up‑and‑down).

Q3 – Polarisation (EM wave) Unpolarised light of intensity 200 W m⁻² passes through two ideal polarisers whose transmission axes differ by 45°. Find the intensity after the second polariser.

Solution First polariser reduces intensity to half: \(I_{1}=100\ \text{W m}^{-2}\). Second polariser: \(I_{2}=I_{1}\cos^{2}45^{\circ}=100\times(0.707)^{2}=50\ \text{W m}^{-2}\).

Q4 – Doppler (sound) A source emitting 600 Hz moves towards a stationary observer at 20 m s⁻¹. The speed of sound in air is 340 m s⁻¹. Calculate the frequency heard by the observer.

Solution \[ f' = 600\frac{340}{340-20}=600\frac{340}{320}=637.5\ \text{Hz} \] (The observer hears a higher pitch because the source approaches.)

11. Summary (AO1)

Transverse and longitudinal progressive waves share the fundamental relationships \(v = f\lambda\), the one‑dimensional wave equation, and an intensity that varies as the square of the amplitude. They differ in:

• Direction of particle motion (perpendicular vs. parallel).
• Nature of the restoring force (tension vs. bulk modulus).
• Ability to be polarised (only transverse).
• Specific speed formulas for the simplest media.

Understanding these differences, together with the Doppler effect for sound and the polarisation properties of electromagnetic waves, equips students to answer the full range of wave‑related questions in the Cambridge 9702 syllabus.