Learning Objectives
- Understand the nature and types of waves
- Study the mathematical description of wave motion
- Learn about the speed of waves in different media
- Understand superposition principle, standing waves, and beats
- Study Doppler effect for sound waves
Key Concepts
Types of Waves
Transverse waves: Particles vibrate perpendicular to wave propagation (e.g., light, waves on strings). They can be polarized.
Longitudinal waves: Particles vibrate parallel to wave propagation (e.g., sound waves in air). They consist of compressions and rarefactions.
Both can be mechanical (need medium) or electromagnetic (no medium needed).
Wave Equation
Progressive wave: y(x,t) = A sin(kx - ωt + φ) (travelling in +x direction)
y(x,t) = A sin(kx + ωt + φ) (travelling in -x direction)
Wave number: k = 2π/λ. Angular frequency: ω = 2π/T = 2πf.
Wave speed: v = fλ = ω/k
Particle velocity: v_p = ∂y/∂t. Wave velocity: v = -v_p/(∂y/∂x).
Speed of Waves
Speed of transverse wave on a string: v = √(T/μ), where T is tension and μ is linear mass density (mass per unit length).
Speed of longitudinal wave in a medium: v = √(E/ρ), where E is the elastic modulus and ρ is density.
Speed of sound in a gas: v = √(γP/ρ) = √(γRT/M) (Newton-Laplace formula).
Speed of sound increases with temperature: v ∝ √T. At 0°C: v ≈ 331 m/s. At 20°C: v ≈ 343 m/s.
Superposition Principle
When two or more waves overlap, the resultant displacement is the algebraic sum of individual displacements: y = y₁ + y₂ + ...
Standing (Stationary) Waves
Formed by superposition of two identical waves travelling in opposite directions.
Equation: y = 2A sin(kx) cos(ωt) -- amplitude varies with position.
Nodes: Points of zero displacement (sin kx = 0, x = nλ/2). Antinodes: Points of maximum displacement (sin kx = ±1, x = (2n+1)λ/4).
Distance between consecutive nodes (or antinodes) = λ/2.
Vibrations of Strings (Fixed at Both Ends)
Allowed frequencies: fₙ = n × v/(2L) = (n/2L)√(T/μ), n = 1, 2, 3, ...
n = 1: fundamental (1st harmonic). n = 2: first overtone (2nd harmonic), etc.
Laws of vibrating strings:
- f ∝ 1/L (law of length)
- f ∝ √T (law of tension)
- f ∝ 1/√μ (law of linear density)
Vibrations in Pipes
Open pipe (open at both ends): fₙ = nv/(2L), n = 1, 2, 3, ... (all harmonics).
Closed pipe (closed at one end): fₙ = nv/(4L), n = 1, 3, 5, ... (odd harmonics only).
End correction: effective length = L + 0.6r (for open end).
Beats
Produced when two waves of slightly different frequencies superpose.
Beat frequency: f_beat = |f₁ - f₂|
Beats are heard as periodic variations in loudness.
Doppler Effect
The apparent change in frequency due to relative motion between source and observer.
General formula: f' = f × (v ± v₀)/(v ∓ vₛ)
Upper signs when approaching, lower signs when receding. v = speed of sound, v₀ = observer speed, vₛ = source speed.
Source approaching: f' = fv/(v - vₛ) (higher pitch). Source receding: f' = fv/(v + vₛ) (lower pitch).
Summary
Waves transfer energy without transferring matter. Transverse and longitudinal waves have different characteristics. The wave equation describes propagation. Speed depends on medium properties. Standing waves form from superposition of opposing waves, creating nodes and antinodes. Strings and pipes produce specific harmonic frequencies. Beats arise from slight frequency differences. The Doppler effect explains apparent frequency shifts due to relative motion.
Important Terms
- Wavelength (λ): Distance between two consecutive points in the same phase
- Frequency (f): Number of oscillations per second (Hz)
- Node: Point of zero displacement in a standing wave
- Antinode: Point of maximum displacement in a standing wave
- Harmonics: Integer multiples of the fundamental frequency
- Beats: Periodic intensity variation from two close frequencies
- Doppler Effect: Apparent frequency change due to relative motion
Quick Revision
- y = A sin(kx - ωt); v = fλ = ω/k
- String: v = √(T/μ); Sound in gas: v = √(γRT/M)
- String (both ends fixed): fₙ = nv/2L (all harmonics)
- Open pipe: fₙ = nv/2L (all harmonics); Closed pipe: fₙ = nv/4L (odd only)
- Beats: f_beat = |f₁ - f₂|
- Doppler: f' = f(v ± v₀)/(v ∓ vₛ)