Big picture
A pulse travels along a rope although each rope segment mainly moves around its own equilibrium position. A wave transports a disturbance and energy without requiring bulk transport of the medium over the whole propagation distance. This distinction separates wave speed from the speed of a vibrating particle.
Sound, musical strings and water ripples share ideas such as frequency, wavelength and superposition, but their physical mechanisms differ. Start by identifying the medium and restoring force. Then choose a travelling-wave model, an interference model or a standing-wave model according to the actual boundary conditions.
Learning goals
Distinguish transverse, longitudinal, mechanical and electromagnetic waves.
Interpret displacement equations and calculate phase differences.
Apply wave-speed relations for strings and sound.
Explain reflection, superposition and standing-wave modes.
Calculate beats and elementary Doppler shifts.
Core ideas
1. Wave types and basic quantities
A transverse wave has particle displacement perpendicular to propagation, as on a stretched string. A longitudinal wave has displacement parallel to propagation, producing compressions and rarefactions, as in sound in air. Fluids do not support ordinary bulk shear waves because they lack static shear rigidity, but their surfaces can support more complicated wave motion.
Mechanical waves require a material medium. Electromagnetic waves can travel through vacuum. Amplitude, A, is maximum displacement; period, T, is time per oscillation; frequency, f = 1/T, is oscillations per second; wavelength, λ, is spatial separation between equivalent phases. The fundamental relation is v = fλ. For f = 250 Hz and λ = 1.36 m, v = 340 m s⁻¹. Frequency is set by the source and remains unchanged on entering a stationary different medium, while speed and wavelength can change.
2. Travelling-wave equation
A sinusoidal wave travelling in the positive x direction can be written y = A sin(kx − ωt + φ), with k = 2π/λ, ω = 2πf and v = ω/k. Replacing −ωt with +ωt gives propagation in the negative x direction for positive k and ω. A constant phase condition explains this: kx − ωt = constant gives increasing x as time increases.
For y = 0.020 sin(4πx − 80πt), using metres and seconds, A = 0.020 m, k = 4π rad m⁻¹, λ = 2π/(4π) = 0.50 m, f = 80π/(2π) = 40 Hz and v = fλ = 20 m s⁻¹. Particle velocity is ∂y/∂t, not v. Its maximum magnitude is Aω = 0.020 × 80π = about 5.03 m s⁻¹. A particle can momentarily be at rest while the wave continues propagating.
3. Phase and energy
The phase difference between points separated by Δx at the same time is Δφ = 2πΔx/λ. Separation λ/2 gives π radians, or opposite phase; separation λ gives 2π, or the same phase. At one point, a time separation Δt gives phase change magnitude 2πΔt/T.
For λ = 0.80 m and Δx = 0.20 m, Δφ = 2π × 0.20/0.80 = π/2 rad. For otherwise identical harmonic waves, average energy transport is proportional to amplitude squared and frequency squared in the usual linear string model. Doubling amplitude quadruples average power. This is not a claim that particle displacement or wave speed also quadruples.
4. Speed on strings and in media
For a stretched string, v = √(F/μ), where F is tension and μ mass per unit length in kg m⁻¹. The relation assumes small transverse disturbances and an effectively flexible string. With F = 100 N and μ = 0.010 kg m⁻¹, v = √10,000 = 100 m s⁻¹. Quadrupling tension doubles speed; doubling the driving frequency does not change speed if tension and density remain fixed.
Longitudinal-wave speed in a fluid is v = √(B/ρ), where B is the relevant bulk modulus. For sound in an ideal gas undergoing rapid approximately adiabatic compression, v = √(γP/ρ) = √(γRT/M). The earlier isothermal estimate underpredicts sound speed because pressure changes occur too rapidly for complete thermal exchange. For a fixed ideal-gas composition, v is proportional to √T. Heating from 300 K to 1,200 K therefore doubles the idealised sound speed.
5. Superposition and reflection
The principle of superposition states that overlapping disturbances add algebraically in a linear medium. Equal in-phase waves reinforce; equal opposite-phase waves can cancel displacement locally. The individual waves continue after passing through one another in the ideal model, rather than permanently destroying each other.
At a fixed string end, reflected displacement is inverted, giving a phase reversal of π. At a free end, reflected displacement is not inverted. Sketch incident and reflected pulses at successive times, showing the end condition. A fixed endpoint must remain at zero displacement, which explains why incident and reflected contributions cancel there. Reflection concerns displacement phase; statements about pressure reflection in sound require their own boundary interpretation.
6. Standing waves
Two equal waves travelling in opposite directions can form standing waves. A typical expression is y = 2A sin(kx) cos(ωt). Nodes remain at zero displacement; antinodes have maximum oscillation amplitude. Adjacent nodes are λ/2 apart, and a node and neighbouring antinode are λ/4 apart.
Standing waves have no net average energy transport along the ideal uniform system, although energy moves locally between kinetic and elastic forms. All particles between adjacent nodes oscillate in phase; neighbouring segments oscillate in opposite phase. A travelling wave instead has continuously changing spatial phase. Nodes are fixed locations, not particles moving along the string with the wave.
7. Normal modes of strings and pipes
A string fixed at both ends has nodes at both ends. Allowed wavelengths satisfy L = nλ/2, giving fₙ = nv/(2L), n = 1, 2, 3, …. The fundamental is the lowest mode; harmonics are integer multiples. For L = 0.75 m and v = 120 m s⁻¹, f₁ = 120/1.50 = 80 Hz and f₃ = 240 Hz.
System | Displacement boundary conditions | Ideal frequencies |
|---|---|---|
Fixed string | Node at each end | nv/(2L) |
Open pipe | Antinode at each open end | nv/(2L) |
Pipe closed at one end | Node at closed end, antinode at open end | (2n − 1)v/(4L) |
An ideal closed pipe has only odd harmonics. With L = 0.50 m and v = 340 m s⁻¹, fundamental = 340/(4 × 0.50) = 170 Hz, with next resonances 510 Hz and 850 Hz. The first overtone is therefore the third harmonic, not the second harmonic. For sound, a displacement node corresponds approximately to a pressure antinode. End corrections are neglected in these elementary pipe formulas.
8. Beats
Two waves of nearly equal frequencies can produce alternating loud and soft sound, called beats. The beat frequency is fb = |f₁ − f₂|. Tuning forks of 256 and 260 Hz produce 4 beats per second. The rapid audible oscillation is near their mean frequency, 258 Hz; the beat rate is the slower modulation.
A single beat-frequency measurement does not determine whether an unknown frequency lies above or below a known frequency. If an unknown gives 5 beats with a 300 Hz source, possibilities are 295 Hz or 305 Hz. A controlled change to one frequency can resolve the ambiguity. Do not select one possibility without additional evidence.
9. Doppler effect
The Doppler effect is a change in observed frequency due to source–observer motion relative to the medium. For collinear approach in stationary air, fobserved = fsource(v + vo)/(v − vs), where vo is observer speed toward the source and vs source speed toward the observer. Recession reverses the corresponding signs.
A stationary observer hears a 500 Hz source approaching at 20 m s⁻¹ with sound speed 340 m s⁻¹. Observed frequency = 500 × 340/(340 − 20) = 531.25 Hz. For recession at the same speed, it is 500 × 340/360 = about 472.22 Hz. Source and observer motion are not perfectly interchangeable in a material medium: source motion changes wavelength, while observer motion changes encounter rate. These elementary formulas exclude supersonic complications and moving-medium effects.
10. Observations and experimental care
A slinky shows longitudinal compression pulses; a rope shows transverse pulses. Mark one coil or rope point to observe local motion separately from propagation. To examine string modes, vary tension or driving frequency and note stable node patterns. A mode diagram should mark both endpoints and count loops, each spanning half a wavelength.
In a resonance-tube activity, identify successive loudness maxima and use their separation to infer wavelength. Keep temperature reasonably stable because sound speed changes with temperature. State whether end corrections are neglected. Beat experiments should use similar amplitudes and close frequencies so the modulation remains clear.
A reliable approach to questions
Identify wave type and distinguish particle motion from propagation.
Extract A, k and ω from the equation before calculating λ, f and v.
Apply boundary conditions to determine allowed wavelengths.
Distinguish harmonic number from overtone number.
For Doppler problems, sketch approach or recession before assigning signs.
Connections that matter
Oscillations supplies phase, angular frequency and harmonic motion.
Elasticity provides restoring forces and wave-speed moduli.
Kinetic theory connects temperature and adiabatic gas response with sound speed.
Common mistakes
Equating particle speed with wave speed ignores local oscillation.
Calling a closed pipe's first overtone its second harmonic ignores missing even harmonics.
Taking a beat rate as the sound frequency confuses modulation and oscillation.
Choosing Doppler signs without a motion sketch can reverse the shift.
Revision checklist
I can interpret a travelling-wave equation and phase.
I can calculate wave speeds and explain their dependencies.
I can draw nodes and antinodes for strings and pipes.
I can solve beat problems with ambiguity checks.
I can calculate simple Doppler shifts with justified signs.