Waves
Waves is one of the largest chapters in MDCAT Physics. The PMDC 2026 syllabus expects mastery of simple harmonic motion, transverse vs longitudinal waves, the speed of sound (Newton-Laplace correction), interference and stationary waves, and resonance in organ pipes and stretched strings. Expect 3-5 MCQs.
Wave Motion
A wave is a disturbance that propagates through a medium (or vacuum, in the case of EM waves) transferring energy and momentum without a net transfer of matter. Mechanical waves require a material medium with elasticity and inertia.
Transverse and Longitudinal Waves
| Property | Transverse wave | Longitudinal wave |
|---|---|---|
| Particle motion | Perpendicular to propagation | Parallel to propagation |
| Visible features | Crests and troughs | Compressions and rarefactions |
| Polarisable? | Yes — can be plane-polarised | No |
| Medium needed? | Mechanical: yes; EM: no | Yes (any state — solid, liquid, gas) |
| Travels through liquids / gases? | Mostly no (except surface ripples and EM) | Yes |
| Wavelength | Crest to next crest | Compression centre to next compression centre |
| Examples | Waves on a string, water surface ripples, light, all EM waves | Sound in air, ultrasound, P-waves in earthquakes, spring compressions |
Wave Characteristics
- Amplitude (A): maximum displacement from equilibrium. SI: m.
- Wavelength (λ): distance between two successive identical points (e.g. crest-to-crest). SI: m.
- Frequency (f): number of complete oscillations per second. SI: Hz.
- Time period (T): time for one complete oscillation; T = 1/f.
- Angular frequency (ω): ω = 2πf.
- Wave number (k): k = 2π/λ.
Wave Speed
For any wave: v = fλ. Wave speed depends on the medium, not on the source. For example, the speed of sound in air at 0°C is about 331 m/s; in water about 1500 m/s; in steel about 5000 m/s.
Progressive Waves
A progressive (travelling) wave continually transfers energy from one place to another. Mathematical form (sinusoidal):
y(x, t) = A·sin(kx − ωt)
The "−" sign denotes a wave travelling in the +x direction; "+" denotes −x direction. All particles oscillate with the same amplitude but with a phase that varies with x.
Simple Harmonic Motion
SHM is oscillation in which acceleration is directly proportional to displacement and always directed toward the equilibrium position: a = −ω²x.
Solution: x(t) = A·sin(ωt + φ), where φ is the phase constant.
Examples and time periods
- Mass-spring system: T = 2π√(m/k).
- Simple pendulum (small amplitude): T = 2π√(L/g).
- Total energy: E = ½kA² (constant); KE and PE oscillate, exchanging completely twice per period.
Circular Motion and SHM
SHM can be regarded as the projection of uniform circular motion onto a diameter. A particle moving in a circle of radius A with angular velocity ω produces, on its diameter, x = A cos(ωt) — the SHM equation. This duality is why ω (rad/s) is called the angular frequency for SHM even though no rotation occurs.
Superposition of Waves
Principle of superposition: when two or more waves meet at the same point, the resultant displacement equals the algebraic sum of the individual displacements.
Consequences: interference, beats, stationary waves — all follow from this single principle.
Interference of Sound Waves
Two coherent sources (same frequency, constant phase) produce a stable interference pattern.
- Constructive interference: path difference = nλ (n = 0, 1, 2…); amplitudes add.
- Destructive interference: path difference = (n + ½)λ; amplitudes subtract.
- Beats: superposition of two sound waves of slightly different frequencies f1 and f2 produces a periodic loud-soft variation at the beat frequency |f1 − f2|.
Stationary Waves
When two progressive waves of equal amplitude and frequency travel in opposite directions, they superpose to form a stationary (standing) wave.
- Nodes: points of zero amplitude (destructive interference); spacing = λ/2.
- Antinodes: points of maximum amplitude (constructive interference); midway between nodes.
- Energy is not transferred along the medium — only stored locally.
- All particles between two adjacent nodes oscillate in phase.
Stationary Waves in Stretched String
For a string of length L fixed at both ends, only those wavelengths fit which have nodes at both ends. Allowed harmonics:
fn = n·v/(2L), n = 1, 2, 3…
- n = 1: fundamental (first harmonic), λ = 2L.
- n = 2: second harmonic (first overtone).
- All harmonics (both odd and even) are present.
Speed of a transverse wave on a stretched string: v = √(T/μ), where T is the tension and μ is the mass per unit length.
Organ Pipes
An organ pipe sets up stationary waves in a column of air. End conditions determine which harmonics are allowed.
| Property | Closed pipe (one end closed) | Open pipe (both ends open) |
|---|---|---|
| End conditions | Closed end = node, open end = antinode | Both ends = antinode |
| Fundamental f1 | v / (4L) | v / (2L) |
| Harmonic formula | fn = (2n − 1) · v / (4L) | fn = n · v / (2L) |
| Harmonics produced | Odd only (1st, 3rd, 5th, …) | All (1st, 2nd, 3rd, …) |
| Tone quality | Hollow, fewer overtones | Brighter, richer overtones |
| Length for same f1 | L | 2L (twice as long) |
Speed of Sound (Newton's Formula)
Newton derived the speed of sound by assuming an isothermal compression-rarefaction process:
v = √(P/ρ)
For air at NTP this gives v ≈ 280 m/s — experimentally too low by ~16% (measured value ≈ 332 m/s).
Laplace's Correction
Laplace pointed out that sound vibrations are too rapid for heat to be exchanged with the surroundings. The compressions and rarefactions are therefore adiabatic, not isothermal. Replacing P by γP:
v = √(γP/ρ) = √(γRT/M)
For air, γ = 1.40 (diatomic), giving v ≈ 332 m/s — in excellent agreement with experiment.
Factors Affecting Speed of Sound
- Temperature: v ∝ √T (in kelvin). Sound travels faster in warm air. Approximate rule: v increases by ~0.6 m/s per °C rise.
- Density of medium: v ∝ 1/√ρ. Lower density media (e.g. helium) carry sound faster than denser ones.
- Humidity: moist air is less dense than dry air (water vapour mass is lower), so sound speed increases with humidity.
- Pressure: at constant temperature, P/ρ is constant for an ideal gas, so changes in pressure alone do not change the speed of sound.
- Wind: sound travels faster downwind than upwind.
Worked MCQs
Five MCQs that capture the high-yield testing patterns for this chapter.
Q1. A wave travels at 340 m/s. Its frequency is 1700 Hz. Its wavelength is:
v = fλ ⇒ λ = v/f = 340/1700 = 0.2 m.
Q2. A closed organ pipe of length L produces a fundamental frequency. The first overtone has frequency:
A closed organ pipe produces only odd harmonics (1, 3, 5…). The first overtone is the third harmonic, three times the fundamental.
Q3. Laplace corrected Newton's formula for the speed of sound by treating the compressions as:
Newton assumed the compressions and rarefactions were isothermal and underpredicted v. Laplace pointed out that sound oscillations are too fast for heat exchange — they are adiabatic.
Q4. The time period of a simple pendulum of length 1 m at a place where g = π² m/s² is:
T = 2π√(L/g) = 2π√(1/π²) = 2π/π = 2 s.
Q5. Two sound waves of frequencies 256 Hz and 260 Hz produce beats at frequency:
Beat frequency = |f1 − f2| = 260 − 256 = 4 Hz.
Quick Recap
- v = fλ; y = A sin(kx − ωt); k = 2π/λ, ω = 2πf.
- Transverse: oscillation ⊥ propagation; longitudinal: parallel.
- SHM: a = −ω²x; Tspring = 2π√(m/k); Tpend = 2π√(L/g).
- Newton: v = √(P/ρ); Laplace: v = √(γP/ρ).
- v in air ∝ √T; humidity raises v; pressure alone does not.
- Closed pipe: fn = (2n−1)v/(4L), odd harmonics only. Open: fn = nv/(2L), all harmonics.
- Beat frequency = |f1 − f2|; stretched string v = √(T/μ).