Class 11 · Chapter 15

Waves

Overview, notes, short notes, formula sheet, daily practice problems, previous year questions, and videos for this chapter — all in one place.

Waves Short Notes

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Condensed revision points for Waves — for quick recall before exams, not a substitute for the full notes.

Wave = disturbance carrying energy/momentum, no net matter transport.
Mechanical waves need a medium (sound, water ripples); EM waves don't (light, radio).
Transverse: particle motion ⊥ propagation — only in solids & liquid surfaces (needs shear).
Longitudinal: particle motion ∥ propagation — only possible kind in gases/liquids.
In solids: longitudinal waves travel faster than transverse waves.
Progressive wave: energy travels continuously. Stationary wave: confined, no net energy transfer.

v_p = ∂y/∂t = Aω cos(ωt − kx); max |v_p| = Aω at mean position, zero at extremes.
a_p = −ω²y (same as SHM).
∂y/∂x = −(1/V)(∂y/∂t) → particle velocity = −(wave velocity)×(slope).
Differential wave equation: ∂²y/∂x² = (1/V²)∂²y/∂t².
Wave velocity: constant for the medium. Particle velocity: oscillates between ±Aω.

String: V = √(T/μ); μ = mass/length = πD²ρ/4.
Solid (longitudinal): V = √(Y/ρ). Liquid: V = √(B/ρ). Gas: V = √(γP/ρ) = √(γRT/M).
Newton (isothermal) gives 279 m/s for air; Laplace (adiabatic, γ=1.41) gives 331.3 m/s ≈ experimental 332 m/s.
v_solid > v_liquid > v_gas always.
Temperature: V ∝ √T; for air, V_t = 332 + 0.61t.
Pressure: no effect (P/ρ constant). Humidity: speed increases (density decreases). Frequency/amplitude/phase: no effect.
Wind adds vectorially: V + w along wind, V − w against it.

Displacement wave: y = A sin(ωt − kx). Pressure wave: ΔP = P₀cos(ωt − kx), P₀ = ABk.
Pressure & displacement waves are 90° out of phase; max displacement ↔ min pressure change.
Pressure wave: no phase change at rigid end, π change at free end (opposite of displacement wave).
Loudness L = 10log₁₀(I/I₀), unit dB; depends on intensity + source size/shape.
Pitch ∝ frequency only. Quality (timbre) depends on waveform/overtones present.

I = I₁ + I₂ + 2√(I₁I₂)cosΔφ.
Maxima: Δφ = 2Nπ, Δx = Nλ → I_max = (√I₁+√I₂)².
Minima: Δφ = (2N+1)π, Δx = (2N+1)λ/2 → I_min = (√I₁−√I₂)².
Coherent sources: phase difference constant in time — required for interference.
Equal amplitudes (a₁=a₂): I_max = 4I, I_min = 0, degree of interference = 100%.
Interference: I = f(position), permanent. Beats: I = f(time), temporary.
Beat frequency = |n₁ − n₂|; resultant frequency = (n₁+n₂)/2.
Ear resolves beats only up to ~10/sec.
Filing a fork's prongs → frequency increases. Loading (wax) → frequency decreases.

Rigid end: phase change of π on reflection (inverted). Free end: no phase change (not inverted).
Rigid-end stationary wave: y = −2a sinkx·cosωt (node at boundary).
Free-end stationary wave: y = 2a coskx·sinωt (antinode at boundary).
Node: zero displacement & velocity, max strain. Antinode: max displacement & velocity, zero strain.
Distance between consecutive nodes (or antinodes) = λ/2.
Particles between adjacent nodes vibrate in phase; across a node, vibration is opposite phase.
Stationary wave: zero net energy transfer, zero wave velocity.
Longitudinal stationary wave: pressure/density variation max at node, min (constant) at antinode.

String fixed both ends, p loops: λ = 2L/p, n_p = (p/2L)√(T/m). All harmonics allowed (1:2:3:4...).
Closed pipe: node at closed end, antinode at open end. n = V/4L. Only odd harmonics (1:3:5...).
Closed pipe overtones: 1st overtone = 3rd harmonic, 2nd overtone = 5th harmonic.
Open pipe: antinodes at both ends, node in middle. n = V/2L. All harmonics (1:2:3...).
Open pipe overtones: 1st overtone = 2nd harmonic, 2nd overtone = 3rd harmonic.
Same length: open pipe fundamental = 2 × closed pipe fundamental.
End correction e ≈ 0.6r. Closed: L_eff = L + 0.6r. Open: L_eff = L + 1.2r.
Open pipe half-submerged in water = closed pipe of half length, same fundamental frequency.
Resonance tube: λ = 2(l₂ − l₁); v = nλ — end correction cancels automatically.

Source → observer (approaching): n' = nV/(V − v_s) [n' > n].
Source ← observer (receding): n' = nV/(V + v_s) [n' < n].
Observer → source (approaching): n' = n(V + v₀)/V [n' > n].
Observer ← source (receding): n' = n(V − v₀)/V [n' < n].
General (wind v_m): n' = n(V ± v_m ± v₀)/(V ± v_m ∓ v_s).
No Doppler shift if: both at rest; same velocity & direction; perpendicular relative motion; only medium moves; constant separation.
Light Doppler is symmetric (depends only on relative speed); sound is not.
Light (v<<c): approach n'≈n(1+v/c); recession n'≈n(1−v/c); Δλ/λ = ±v/c.
Echo/SONAR/RADAR: reflected-wave beat frequency ≈ 2v_target·n/V (sound) — used to find target speed.
Shock wave (supersonic source): sinθ = V/v_s = 1/(Mach number); Mach number = speed/speed of sound.

Wave velocity is NEVER the same as particle velocity — don't confuse Aω with V = nλ.
Pressure-wave and displacement-wave reflection rules are OPPOSITE at the same boundary — a classic NEET trap.
Closed pipe gives ONLY odd harmonics — never assume the 2nd harmonic exists there.
End correction direction: forgetting +0.6r (closed) vs +1.2r (open, both ends) is the most common numerical slip.
In Doppler problems, only the velocity component ALONG the source–observer line matters — perpendicular motion gives zero shift.
Pressure has NO effect on speed of sound at constant temperature — a frequently-tested 'no effect' fact.
Beats need frequencies to be close (not equal, not very different) — equal frequencies give pure interference, not beats.
Don't confuse 'degree of interference' (contrast, %) with the intensity ratio I_max/I_min — they're related but not identical.

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