Waves
On this page
Direct answer
Waves move energy, not matter: the air carrying a sound wave merely oscillates about its place while the disturbance travels onward as y(x,t) = A sin(kx - ωt + φ) at v = νλ. Speed belongs to the medium — sqrt(T/μ) on a string, sqrt(γP/ρ) in a gas after Laplace's correction — and superposition then produces standing waves, beats and the Doppler shifts that fill most of this chapter's NEET-UG numericals.
What you must remember
- Mechanical waves need a medium; transverse waves (string, light) oscillate perpendicular to travel, longitudinal waves (sound) along it.
- Speed on a stretched string v = sqrt(T/μ); in a gas v = sqrt(γP/ρ) — Laplace's adiabatic correction to Newton's isothermal guess, which fixed the predicted speed of sound in air to the measured 331.3 m/s at 0 °C.
- Sound speeds rise about 0.6 m/s per °C in air (about 343 m/s at 20 °C); in water about 1500 m/s, in steel several kilometres per second.
- Progressive wave: y = A sin(kx - ωt + φ) with k = 2π/λ; particle velocity (dy/dt) is a different animal from wave velocity.
- String fixed at both ends: harmonics f_n = nv/2L, all multiples present; the distance between adjacent nodes is λ/2.
- Organ pipes: closed at one end gives only odd harmonics (f1 = v/4L, then 3f1, 5f1 ...); open at both ends gives all harmonics (f1 = v/2L) — an open pipe of equal length sounds a higher fundamental.
- Beats: beat frequency = |f1 - f2|; instruments are tuned by driving the beat rate to zero.
- Doppler effect: f' = f(v ± v_o)/(v ∓ v_s) — motion toward each other raises the pitch; a source outrunning its own wave produces the shock cone of a sonic boom.
A typical exam case: beats and a closed pipe
A tuning fork of 256 Hz sounds with another fork and produces 4 beats per second; wax is loaded on the second fork and the beat rate falls, so loading lowered that fork's frequency — it must originally have been 260 Hz (the other candidate, 252 Hz, would move away from 256 on loading and the beats would worsen). Now let that 260 Hz fork resonate a closed pipe: the fundamental of a closed pipe is f1 = v/4L, so with v = 340 m/s the resonating length is L = 340/(4 × 260) ≈ 0.33 m. The closed pipe answers only at odd multiples of its fundamental, so the same fork also resonates when the pipe is three times as long — a favourite trap asking which length will not resonate. Both problems run on one skill: convert each statement into a frequency relation before touching algebra.
The Doppler finale: as an ambulance passes you, the heard pitch drops suddenly, not gradually — the radial velocity changes sign, switching the formula from f(v/(v - v_s)) to f(v/(v + v_s)).
Where students slip
Newton's speed-of-sound error is itself a question: he took the compressions as isothermal (v = sqrt(P/ρ)) and got about 280 m/s; Laplace showed the oscillations are too fast for heat exchange, so γ enters and the answer corrects itself. Second, the harmonic census of pipes: closed pipes are odd-harmonic-only, and an MCQ listing "second harmonic of a closed pipe" is testing whether you notice the animal does not exist. Third, Doppler sign discipline — the safe recipe is that approach, by source or observer, always raises frequency; if your algebra says otherwise the sign slipped. Fourth, wave speed is fixed by the medium while particle speed grows with amplitude and frequency; a louder shout does not arrive sooner. And when a question asks why thunder trails the flash, it is 3 × 10^8 m/s outrunning 343 m/s over the same kilometres.
Frequently asked questions
What was Laplace's correction to Newton's formula?
Sound compressions and rarefactions are too rapid for heat flow, so they are adiabatic, not isothermal; replacing P by γP gives v = sqrt(γP/ρ) and matches the measured 331.3 m/s in air at 0 °C.
Which harmonics does a closed organ pipe support?
Only odd multiples of the fundamental: f1 = v/4L, then 3f1, 5f1, and so on, because the closed end must be a node and the open end an antinode.
Two tuning forks of 256 Hz and 260 Hz are sounded together. What is heard?
Four beats per second. Loading one fork with wax lowers its frequency, so if the beat rate then decreases, the loaded fork was the higher one (260 Hz); if it increases, the loaded fork was the lower one.
Why does an open pipe sound richer than a closed pipe of the same length?
The open pipe contains all harmonics of a higher fundamental (v/2L against v/4L); the closed pipe suppresses every even harmonic.
What is a sonic boom?
The cone-shaped shock wave that forms when a source's speed exceeds the wave speed in the medium; inside the cone the disturbances pile up instead of outrunning each other.