Waves
Waves and periodic motion are among the largest, highest-yield topics in MCAT physics. For the Chemical and Physical Foundations section, master simple harmonic motion, transverse vs longitudinal waves, v = fλ, sound (intensity, decibels, and the Doppler effect), beats, interference, and standing waves/resonance on strings and in pipes. This material is strictly algebra-based.
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
Sound is a longitudinal (pressure) wave. Its speed depends on the medium's stiffness and density — it travels fastest in solids, slower in liquids, slowest in gases (opposite to the intuition that denser = faster, because stiffness dominates). Approximate speeds: air ≈ 343 m/s (20°C), water ≈ 1500 m/s, steel ≈ 5000 m/s.
For an ideal gas the speed of sound is well modeled by:
v = √(γP/ρ) = √(γRT/M)
where γ = Cp/Cv (1.40 for air), P is pressure, ρ density, T absolute temperature, and M the molar mass. The compressions/rarefactions occur too fast for heat exchange, so the process is adiabatic (hence γ appears, rather than the simpler isothermal √(P/ρ), which underpredicts v).
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.
Sound Intensity & the Decibel Scale
Intensity (I) is power per unit area (W/m²). For a point source radiating in all directions, intensity falls off with the square of distance:
I ∝ 1/r² (so doubling the distance quarters the intensity). Intensity is proportional to amplitude squared: I ∝ A².
Loudness is measured on a logarithmic decibel (dB) scale relative to the threshold of hearing I0 = 10−12 W/m²:
β (dB) = 10 log10(I / I0)
- Every factor of 10 in intensity adds 10 dB; a factor of 100 adds 20 dB.
- An increase of 10 dB corresponds to a 10× increase in intensity (perceived as roughly twice as loud).
The Doppler Effect
The Doppler effect is the change in observed frequency when the source and observer move relative to each other. Approaching → higher observed frequency (pitch); receding → lower observed frequency. The general relation is:
fobserved = fsource · (v ± vobserver) / (v ∓ vsource)
where v is the speed of sound. Choose signs to make the shift go the right way: relative approach raises f, relative recession lowers f.
- Top signs (numerator +, denominator −) apply when the motion reduces the gap (approach).
- Medical context: Doppler ultrasound measures blood-flow velocity from the frequency shift of sound reflected off moving red blood cells; echocardiography uses the same principle.
- The same effect for light (redshift/blueshift) underlies the expansion of the universe and radar/speed guns.
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. An ambulance siren approaches a stationary observer at constant speed. Compared with the frequency emitted, the observer hears a frequency that is:
Doppler effect: an approaching source raises the observed frequency (pitch). At constant approach speed the shift is constant; the pitch drops only after the source passes and begins receding.
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); E = ½kA².
- Sound speed: v = √(γP/ρ) = √(γRT/M); fastest in solids; v ∝ √T in air; pressure alone does not change it.
- Intensity I ∝ 1/r² and I ∝ A²; loudness β = 10 log10(I/I0) dB (+10 dB = 10× intensity).
- Doppler: approach raises f, recession lowers f; basis of Doppler ultrasound.
- 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).