Know that for a longitudinal wave, the direction of vibration is parallel to the direction of propagation and understand that sound waves and seismic P-waves (primary) can be modelled as longitudinal

3.1 General Properties of Waves

Learning Objective

Students will be able to:

  • Define a wave and explain that it transports energy without a permanent displacement of matter (including a brief note on electromagnetic waves).

  • Distinguish transverse and longitudinal waves, giving a correct everyday example of each (including seismic S‑waves as a transverse example).

  • State that in a longitudinal wave the particle vibration is parallel to the direction of propagation and model sound waves and seismic P‑waves accordingly.

  • Use the required terminology, relationships and formulae to solve typical IGCSE/A‑Level questions (AO1–AO2).

1. What Is a Wave?

  • A wave is a disturbance that transfers energy from one location to another without causing a permanent displacement of the material.

    • The same definition also applies to electromagnetic waves (light, radio), although they do not require a material medium.

  • While the wave travels, particles of the medium oscillate about their equilibrium positions and then return.

2. Types of Mechanical Waves

  • Transverse waves – particle displacement is perpendicular to the direction of travel.

    • Examples: a rope flicked up‑and‑down, water‑surface ripples, seismic S‑waves (shear waves).

  • Longitudinal waves – particle displacement is parallel to the direction of travel.

    • Examples: sound in air, seismic P‑waves (primary waves), compression waves in a slinky or spring.

3. Key Wave Terminology

TermDefinition (mechanical wave)
Wavelength λDistance between two consecutive points that are in the same phase (e.g., compression‑to‑compression or crest‑to‑crest).
Frequency fNumber of complete cycles that pass a fixed point each second (unit Hz).
Period TTime taken for one complete cycle; T = 1/f.
AmplitudeMaximum displacement of a particle from its equilibrium position.
Crest / TroughHighest and lowest points of a transverse wave.
Compression / RarefactionRegions of higher and lower pressure in a longitudinal wave.
PhaseRelative position in the cycle of two points; points can be “in phase” (same displacement) or “out of phase”.
Wave‑frontImaginary surface joining points that are in the same phase of vibration.

4. Fundamental Wave Relationship

For any mechanical wave travelling at speed v:

\( v = f\lambda = \dfrac{\lambda}{T} \)

The same equation is used for both transverse and longitudinal waves; the syllabus expects students to apply it in either context.

5. Speed of Longitudinal Waves in Different Media

  • Gases (e.g., air)

    \( v = \sqrt{\dfrac{\gamma \, R \, T}{M}} \)

    • \(\gamma\) – adiabatic index (ratio of specific heats, ≈ 1.4 for air).

    • \(R\) – universal gas constant (8.314 J mol⁻¹ K⁻¹).

    • \(T\) – absolute temperature (K).

    • \(M\) – molar mass of the gas (kg mol⁻¹).

  • Liquids

    \( v \approx \sqrt{\dfrac{B}{\rho}} \)

    • \(B\) – bulk modulus (measure of incompressibility).

    • \(\rho\) – density of the liquid.

  • Solids

    \( v \approx \sqrt{\dfrac{E}{\rho}} \)

    • \(E\) – Young’s modulus for longitudinal (compressional) waves.

    • For shear (transverse) waves the analogous expression uses the shear modulus \(G\): \(v_{\text{shear}} = \sqrt{G/\rho}\).

    • \(\rho\) – density of the solid.

Because solids have a much larger elastic modulus than liquids or gases, sound travels fastest in solids, slower in liquids, and slowest in gases.

6. Energy Transport & Intensity

  • The energy carried by a wave is proportional to the square of its amplitude: Energy ∝ Amplitude².

  • Intensity (I) is the power transmitted per unit area of the wave‑front and also varies as amplitude².

  • Energy moves with the wave‑front; the medium itself does not travel with the wave.

Quantitative example (AO2): If the amplitude of a sound wave is doubled, the intensity becomes four times larger because \(I \propto A^{2}\).

7. Wave Phenomena (Reflection, Refraction, Diffraction)

  • Reflection – the wave changes direction when it meets a barrier. The angle of incidence equals the angle of reflection.

  • Refraction – the wave‑front bends when it passes from one medium to another with a different speed.

    • In a ripple‑tank experiment, a wave travelling from shallow to deep water bends away from the normal because its speed increases.

    • For light, the syllabus also mentions the critical angle at which total internal reflection occurs (relevant to optics).

  • Diffraction – spreading of a wave as it passes through a narrow opening or around an obstacle; most noticeable when the opening size is comparable to the wavelength.

Suggested ripple‑tank description for exam preparation: “A wave striking a sloping side of the tank is refracted, a wave striking a barrier is reflected, and a wave passing through a narrow slit spreads out, demonstrating diffraction.”

8. Longitudinal‑Wave Illustration

Diagram (to be drawn): a series of coils of a slinky showing alternating compressions (closely spaced coils) and rarefactions (widely spaced coils) moving to the right. Small arrows on each coil point left‑right, indicating particle motion parallel to the direction of travel.

9. Sound Waves as Longitudinal Waves

  • A loud‑speaker diaphragm moves back and forth, pushing adjacent air molecules together (compression) and then pulling them apart (rarefaction).

  • The alternating high‑ and low‑pressure regions travel outward while each air molecule oscillates along the same line as the wave propagates.

  • Pitch is determined by the frequency of the wave (higher f → higher pitch).

  • Loudness is related to the amplitude (larger A → louder sound).

10. Seismic Waves

  • P‑waves (Primary waves) – longitudinal compressional waves that travel fastest through the Earth’s interior. They can move through solids, liquids and gases, compressing and expanding the material in the same direction as the wave travels.

  • S‑waves (Secondary or shear waves) – transverse waves where particle motion is perpendicular to propagation. They cannot travel through fluids, only through solids.

  • Because P‑waves arrive first on a seismograph, they are the first indication of an earthquake.

11. Worked Example (AO2)

Question: A tuning‑fork vibrates at 500 Hz. Calculate the wavelength of the sound it produces in air at 20 °C (speed of sound ≈ 343 m s⁻¹).

Solution:

  1. Use the wave‑speed relationship \(v = f\lambda\).

  2. Re‑arrange for wavelength: \(\lambda = \dfrac{v}{f}\).

  3. Substitute the known values: \(\lambda = \dfrac{343\ \text{m s}^{-1}}{500\ \text{Hz}} = 0.686\ \text{m}\).

Answer: \(\lambda = 0.69\ \text{m}\) (to two significant figures).

12. Summary Checklist (AO1–AO2)

  1. Define a wave and state that it transports energy without permanent displacement (mention EM waves briefly).

  2. Distinguish transverse and longitudinal waves; give one everyday example of each (include S‑wave as a transverse example).

  3. Identify the direction of particle vibration for a longitudinal wave (parallel to propagation).

  4. Describe compressions and rarefactions and link them to sound and seismic P‑waves.

  5. Recall the wave‑speed formulae \(v = f\lambda = \lambda/T\) and the specific expressions for the speed of sound in gases, liquids and solids (explain symbols γ, B, E, G).

  6. State why sound travels fastest in solids, slower in liquids, and slowest in gases.

  7. Explain reflection, refraction (including the ripple‑tank description) and diffraction; mention the critical angle for light.

  8. Remember that intensity ∝ amplitude²; be able to compare intensities for different amplitudes.

  9. Relate frequency to pitch and amplitude to loudness for sound waves.

  10. Apply the formulae to quantitative problems (e.g., calculate wavelength from frequency and speed).