Wave Motion

Introduction

Wave motion: periodic disturbance transfer through medium or space without net mass transport. Medium oscillates about equilibrium. Classical mechanics studies mechanical waves: energy transfer via oscillations in solids, liquids, gases. Wave phenomena underpin acoustics, optics, seismology, fluid dynamics.

"A wave is a disturbance that travels through space and matter transferring energy from one point to another without the transport of matter." -- R. Resnick, D. Halliday

Definition and Classification

Definition

Wave: repetitive oscillatory disturbance propagating in space and time transferring energy. Medium particles oscillate about fixed points; energy moves forward. Wave characterized by wavelength, frequency, amplitude, speed.

Classification by Medium

Mechanical waves: require medium (solid, liquid, gas). Electromagnetic waves: no medium required (not covered here).

Classification by Particle Motion

Transverse waves: particle displacement perpendicular to wave direction. Longitudinal waves: particle displacement parallel to wave direction. Surface waves: combination of transverse and longitudinal, propagate along interface.

Classification by Waveform

Periodic waves: continuous, repetitive oscillations (sinusoidal). Non-periodic waves: single pulses, transient disturbances.

Basic Properties of Waves

Wavelength (λ)

Distance between adjacent points in phase (crest to crest, trough to trough). Measured in meters (m).

Frequency (f)

Number of oscillations per second. Measured in hertz (Hz). Inverse of period.

Period (T)

Time for one complete oscillation. T = 1/f. Measured in seconds (s).

Amplitude (A)

Maximum displacement from equilibrium position. Related to wave energy.

Wave Speed (v)

Speed at which wave propagates through medium. v = f λ.

Types of Waves

Transverse Waves

Particle displacement perpendicular to propagation. Examples: waves on strings, electromagnetic waves (not mechanical), surface water waves.

Longitudinal Waves

Particle displacement parallel to propagation. Examples: sound waves in air, compression waves in solids.

Surface Waves

Combination of transverse and longitudinal oscillations. Examples: water waves at air-water interface.

Progressive and Standing Waves

Progressive: travel energy transfer. Standing: formed by interference, fixed nodes and antinodes.

Wave Equation

Mathematical Form

Describes wave displacement y(x,t) as function of position and time. For one-dimensional wave:

∂²y/∂x² = (1/v²) ∂²y/∂t²

Solution: Harmonic Wave

y(x,t) = A sin(kx - ωt + φ), where k = wave number, ω = angular frequency, φ = phase.

Parameters

Wave number k = 2π/λ, angular frequency ω = 2πf, velocity v = ω/k.

Wave Speed and Factors Affecting It

General Relation

v = f λ, depends on medium properties.

Speed in Strings

v = √(T/μ), where T = tension, μ = linear mass density.

Speed in Air (Sound)

v = √(γRT/M), depends on temperature, gas constants.

Speed in Solids

v = √(E/ρ), E = Young's modulus, ρ = density.

Superposition and Interference

Principle of Superposition

When two or more waves meet, resultant displacement = algebraic sum of individual displacements at each point.

Constructive Interference

Waves in phase add amplitudes, produce maxima.

Destructive Interference

Waves out of phase subtract amplitudes, produce minima.

Applications

Diffraction patterns, noise cancellation, signal processing.

Reflection and Transmission

Reflection of Waves

Wavefront reverses direction on boundary. Fixed end: inverted reflection. Free end: upright reflection.

Transmission and Refraction

Part energy passes through boundary, changes speed and wavelength in new medium.

Boundary Conditions

Continuity of displacement and slope at interface govern reflection and transmission coefficients.

Energy Considerations

Energy conserved; partitioned between reflected and transmitted waves.

Standing Waves and Resonance

Formation

Superposition of two waves traveling in opposite directions with same frequency and amplitude.

Nodes and Antinodes

Nodes: points of zero displacement. Antinodes: points of maximum displacement.

Resonance

Natural frequencies cause large amplitude oscillations; depends on boundary conditions and medium properties.

Examples

Vibrating strings, air columns in pipes, microwave cavities.

Energy Transport and Intensity

Energy in Waves

Energy proportional to square of amplitude. Transported through medium without mass transport.

Intensity

Power transmitted per unit area perpendicular to propagation direction. I = P/A.

Relation to Amplitude

Intensity ∝ A².

Energy Density

Energy per unit volume stored in wave motion.

Doppler Effect

Definition

Change in observed frequency due to relative motion between source and observer.

Source Moving Towards Observer

Observed frequency increases.

Source Moving Away from Observer

Observed frequency decreases.

General Formula

f' = f (v ± v_o) / (v ∓ v_s)

where f' = observed frequency, f = source frequency, v = wave speed, v_o = observer velocity, v_s = source velocity. Signs depend on direction.

Applications of Wave Motion

Acoustics

Sound wave propagation, musical instruments, architectural acoustics.

Seismology

Earthquake wave analysis, P-waves and S-waves.

Medical Imaging

Ultrasound uses mechanical wave propagation in tissues.

Communication

Signal transmission via waves, antennas, sonar.

Engineering

Vibration analysis, nondestructive testing, structural health monitoring.

References

• Resnick, R., Halliday, D., "Physics," Wiley, Vol. 1, 2013, pp. 250-310.
• Tipler, P.A., Mosca, G., "Physics for Scientists and Engineers," W.H. Freeman, 6th Ed., 2007, pp. 650-700.
• Feynman, R.P., Leighton, R.B., Sands, M., "The Feynman Lectures on Physics," Addison-Wesley, Vol. 1, 1964, pp. 45-90.
• French, A.P., "Vibrations and Waves," CRC Press, 1971, pp. 10-120.
• Halliday, D., Resnick, R., Walker, J., "Fundamentals of Physics," Wiley, 10th Ed., 2014, pp. 650-720.