Unit 4 AoS 1 – Mechanical Waves & Wave Phenomena

Wave Model Overview

• Waves are described by the wave model, which assumes:
  • Periodic variation of a physical quantity (pressure, height, displacement, field strength).
  • Transmission of energy without net transfer of matter.
  • Particles of a medium oscillate about fixed equilibrium positions.
  • Two primary categories: transverse (particle motion ⟂ direction of energy propagation) and longitudinal (particle motion ∥ direction of propagation).
• All wave analysis in this unit assumes the medium is homogeneous and isotropic unless otherwise specified.

Types of Waves: Transverse vs Longitudinal

• Transverse waves
  • Particle displacement is perpendicular to wave travel.
  • Typical examples: water-surface ripples, waves on strings/ropes, electromagnetic (light) waves.
  • Characteristic points:
  • Crest = maximum positive displacement.
  • Trough = maximum negative displacement.
• Longitudinal waves
  • Particle displacement is parallel to wave travel.
  • Examples: sound in air, compression waves in a slinky/spring.
  • Compressions (high-pressure regions) and rarefactions (low-pressure regions) replace crests/troughs.
  • Observational note: a chosen medium particle oscillates but exhibits no net translation after the wave passes.
• Mechanical waves
  • Require a material medium (air, water, rope, string).
  • Energy transferred depends on medium properties; matter itself does not translate.

Key Wave Parameters (measurable properties)

• Amplitude (A)
  • Maximum displacement from equilibrium.
• Wavelength (λ)
  • Spatial period of the wave: distance between successive identical phase points (crest-to-crest, compression-to-compression).
  • Unit: metres (m).
• Period (T)
  • Time for one complete oscillation or for a phase point to travel one wavelength.
  • Unit: seconds (s).
• Frequency (f)
  • Number of cycles per second.
  • Unit: hertz (Hz = s⁻¹).
  • Relation: $f = \frac{1}{T}$
• Wave speed (v)
  • Rate at which energy/phase propagates through the medium.
  • Fundamental wave equation: $v = f\lambda$
  • Alternate form: $v = \frac{\lambda}{T}$
  • For a given medium, v is independent of A, f, λ, T for mechanical waves (e.g.
    sound ≈ $340\,\text{m s}^{-1}$ in air at room temperature).

Graphing Waves

• Displacement–distance graph
  • Snapshot of the entire medium at an instant; reveals λ and A directly.
• Displacement–time graph
  • Follows a single particle; reveals A and T (→ f) directly.
• Interpretation tips
  • Sinusoidal graphs imply simple harmonic motion of particles.
  • For travelling waves the shape translates; for standing waves the envelope is stationary.

Mechanical Wave Speed & Wave-Equation Applications

• Example calculations
  • Hummingbird (70 Hz) → $\lambda = \frac{350}{70} = 5.0\,\text{m}$ (v = $350\,\text{m s}^{-1}$ assumed).
  • Boats 4 m apart; one up when other down, T = 3 s; since there is half a wavelength between boats (no crest between them) $\lambda = 8\,\text{m}$; v = $\frac{8}{3}=2.67\,\text{m s}^{-1}$.
  • 2019 exam: point P moves from max to zero in $0.120\,\text{s}$ → quarter of a period; hence $T = 4(0.120)=0.480\,\text{s}$, $f=2.08\,\text{Hz}$, $v = f\lambda = 2.08(1.40)=2.91\,\text{m s}^{-1}$ (3 s.f.).

Wave Behaviours: Reflection & Refraction

• Reflection
  • Occurs at a boundary; obeys law $\theta<em>i = \theta</em>r$ measured to the normal.
  • Frequency and wavelength remain unchanged (only direction flips).
  • Fixed-end reflection of a string introduces a $\frac{\lambda}{2}$ phase reversal (crest → trough).
• Refraction (brief link)
  • Transmission into a second medium changes v and λ but leaves f unchanged, bending the direction.

Superposition Principle & Interference

• Principle: net displacement = algebraic sum of individual displacements at every point in space/time.
• After interaction waves re-emerge unaltered.
• Constructive interference
  • Displacements reinforce (crest + crest or trough + trough) → larger amplitude.
• Destructive interference
  • Opposite displacements cancel (crest + trough) → reduced or zero amplitude.
• Real-world illustrations: musical timbre, noise-cancelling headphones, water-ripple patterns, Slinky demos.

Path Difference & Coherent Sources

• Coherent sources: identical f & λ with constant phase relationship → stable interference pattern.
• Path difference (pd) for a point X: $|S<em>1X - S</em>2X|$
  • Constructive (antinodes): $pd = n\lambda\quad(n = 0,1,2,…)$
  • Destructive (nodes): $pd = (n+\tfrac12)\lambda$
• Terminology
  • Nodes: persistent zero-amplitude points.
  • Antinodes: persistent maximum-amplitude points.
• Sample exam (2018): sound at 340 Hz between two facing speakers → λ=1 m; centre is loud (pd=0). Moving toward B: first node at $\tfrac{1}{4}λ =0.25\,\text{m}$, second node additional $\tfrac{1}{2}λ=0.50\,\text{m}$ → total 0.75 m.

Standing Waves

• Formation
  • Travelling wave reflects and interferes with itself.
  • Requires waves of same f, A, v moving in opposite directions.
• Fixed-end strings
  • Ends are nodes; only wavelengths satisfying integer half-wavelength fit: $L = n\frac{\lambda}{2}$ (n=1,2,3…)
  • Fundamental (n=1): $\lambda<em>1 = 2L$, $f</em>1 = \frac{v}{2L}$.
  • Higher harmonics: $f<em>n = nf</em>1$.
• Resonance
  • Driving frequency matches natural frequency → large A; energy transfer maximised.
  • Illustrations: swing pumping, shattered wine glass, tuning forks, spaghetti resonance.
• Worked examples
  • 60 cm string showing 1.5 loops (n=3) → $\lambda = \frac{2L}{3}=0.40\,\text{m}$.
  • 0.8 m guitar string, n=2 displayed envelope, f=250 Hz → $\lambda = \frac{2L}{n}=0.8\,\text{m}$, $v=200\,\text{m s}^{-1}$. Tripling f (750 Hz) forces n=6 pattern; draw 3 full wavelengths in 0.8 m (i.e.
    six half-waves).
  • 6.0 m string under tension v=40 m s⁻¹, f=7.5 Hz → $\lambda = \frac{v}{f}=5.33\,\text{m}$. Required n from $L = n\frac{\lambda}{2} ⇒ n = \frac{2L}{\lambda}=2.25$ non-integer → standing wave will not form.

Diffraction

• Definition: spreading/bending of waves when encountering obstacles or apertures.
• Qualitative dependence on ratio $\frac{\lambda}{w}$ (w = gap/obstacle size)
  • $\frac{\lambda}{w} \gtrsim 1$ → pronounced diffraction; circular wavefronts.
  • $\frac{\lambda}{w} \ll 1$ → minimal diffraction; wavefronts remain plane beyond aperture.
• No change to v, λ, or f across the aperture; only energy distribution in space is affected.
• Huygens’ principle: every point on a wavefront acts as a secondary source of spherical wavelets → explains diffraction edges.
• Imaging limitations
  • Optical microscopes limited by visible λ (380–700 nm). Objects ≪ λ produce blurred, diffracted images.
  • Electron microscopes use de Broglie wavelengths ≪ 1 nm → higher resolution.
  • Example: virus length $2.0\times10^{-7}\,\text{m}$ vs visible light λ (380–700 nm): λ comparable → significant diffraction, image unclear (option A from example 20).
• Worked diffraction example
  • Gap $6.50\times10^{-7}\,\text{m}$, blue light λ $4.80\times10^{-7}\,\text{m}$ → $\frac{\lambda}{w}=0.74$ → substantial diffraction expected.

Sample Concept-Check & Quiz Answers

• Universal property of waves → energy transfer without mass transfer (Quick Quiz Q1: option A).
• Wave types → Transverse & Longitudinal (Q2: D).
• In transverse waves particle motion ⟂ wave direction (Q3: B).
• Interference statement correct: constructive → larger crest/trough (Example 5: option A).
• Reflection statement correct: waves hitting along normal reflect back along normal (Student Q8: option D).
• Two positive pulses superposing → constructive with doubled amplitude (Student Q9: option A).
• Resonant driving increases amplitude (Q10 answer A) and phenomenon is resonance (Q11 answer C).

Connections & Real-World Relevance

• Sound engineering: placement of speakers avoids destructive nodes in auditoria.
• Medical imaging: ultrasound (λ~mm) diffracts around small structures; limits vs X-ray.
• Communications: long-wave radio bends over hills (large λ), whereas high-frequency microwaves are line-of-sight.
• Structural safety: bridges & buildings designed to avoid resonance with wind or seismic forcing frequencies (Tacoma Narrows failure).
• Musical instruments exploit harmonics and standing waves to create timbre; adjusting string length/tension changes f via $f = \frac{v}{2L}$.

Ethical & Philosophical Reflections

• Advances in wave-based measurement (microscopy, spectroscopy) expand human capacity to observe the micro-world, raising questions about responsible use (e.g.
  nano-tech, privacy in imaging, environmental noise).
• Understanding resonance informs safety standards, preventing disasters yet also enabling artistic endeavours (concert-hall acoustics, instrument crafting).

Formula Summary (quick reference)

• $v = f\lambda$
• $f = \frac{1}{T}$
• Standing waves on string fixed both ends: $L = n\frac{\lambda}{2}$, $f_n = n\frac{v}{2L}$
• Constructive interference: $pd = n\lambda$
• Destructive interference: $pd = (n+\tfrac12)\lambda$
• Diffraction criterion (qualitative): significant when $\frac{\lambda}{w} \gtrsim 1$