Waves - Class 11 Physics

Waves

Overview: Propagation of disturbance, Types of waves, Sound waves, Superposition, Beats, and Doppler Effect.

1. Types of Waves

Mechanical Waves: Require medium (Sound, Water).
Electromagnetic Waves: No medium required (Light).
Matter Waves: Associated with particles.

Transverse vs Longitudinal

Transverse: Particle motion perpendicular to wave motion (Light, String).
Longitudinal: Particle motion parallel to wave motion (Sound). Compression and Rarefaction.

2. Progressive Wave Equation

y(x, t) = a sin(kx - ωt + φ)

Where k = Wave Number = 2π/λ, ω = Angular Frequency = 2πf.

Wave Velocity v = ω/k = fλ.

3. Speed of Waves

On a Stretched String

v = √(T/μ)

Where T = Tension, μ = Linear mass density (m/L).

Sound in Gas (Newton-Laplace Formula)

v = √(γP/ρ) = √(γRT/M)

4. Superposition & Standing Waves

Principle of Superposition: y = y1 + y2.

Standing Waves

Formed by superposition of two identical waves traveling in opposite directions.

y = 2a sin(kx) cos(ωt)

Nodes: Zero displacement. Antinodes: Max displacement.

Organ Pipes

Closed Pipe: Only odd harmonics (f, 3f, 5f...). Fundamental f = v/4L.
Open Pipe: All harmonics (f, 2f, 3f...). Fundamental f = v/2L.

5. Beats & Doppler Effect

Beats

Periodic variation in intensity due to superposition of waves with slightly different frequencies.

f_beat = |f1 - f2|

Doppler Effect

Apparent change in frequency due to relative motion.

f' = f [(v ± v_o) / (v ∓ v_s)]

Numericals - Waves

Numericals

Wave Speed

Q1. A steel wire 0.72 m long has mass 5×10^-3 kg. T = 60N. Speed of wave?

μ = m/L = 5×10^-3 / 0.72 = 0.0069 kg/m

v = √(T/μ)

v = √(60 / 0.0069)

v = √(8695) ≈ 93 m/s

Wave Equation

Q2. y = 0.05 sin(80πt - 4πx). Find wavelength and velocity.

Compare with y = A sin(ωt - kx)

ω = 80π, k = 4π

λ = 2π/k = 2π/4π = 0.5 m

v = ω/k = 80π/4π = 20 m/s

Beats

Q3. Two tuning forks 256 Hz and 260 Hz sounded together. Beats per second?

f_beat = |f1 - f2|

f_beat = |256 - 260| = 4

4 beats per second.

Open Organ Pipe

Q4. Length of open pipe is 30cm. Speed of sound 330 m/s. Fundamental freq?

f = v / 2L

L = 0.3 m

f = 330 / (2 × 0.3) = 330 / 0.6

f = 550 Hz

Closed Organ Pipe

Q5. For same pipe (30cm) if closed at one end?

f_closed = v / 4L

f = 330 / (4 × 0.3) = 330 / 1.2

f = 275 Hz

Doppler Effect

Q6. Train moves at 30 m/s towards stationary observer. Whistle freq 400Hz. v_sound=340 m/s. Apparent freq?

Source moving towards observer. f increases.

f' = f [v / (v - vs)]

f' = 400 [340 / (340 - 30)]

f' = 400 [340 / 310] = 400 × 1.096

f' ≈ 439 Hz

Echo Depth

Q7. Sonar pulse returns in 2s. v_water = 1450 m/s. Depth?

2d = v × t

d = (v × t) / 2

d = (1450 × 2) / 2 = 1450 m

Phase Difference

Q8. Path difference = 1 m. λ = 4 m. Phase Diff?

Δφ = (2π/λ) Δx

Δφ = (2π/4) × 1

Δφ = π/2 rad

String Resonance

Q9. String vibrates in 4 loops. L=1m. λ?

Number of loops n = 4.

L = n (λ/2)

1 = 4 λ/2 = 2λ

λ = 0.5 m

Newton's Formula Error

Q10. Calculate % error if Newton's formula used for sound in air. (Actual=332, Newton=280).

Error = 332 - 280 = 52

% Error = (52/332) × 100

≈ 15.6%

Formulas & Facts - Waves

Equations & Formulas

Concept	Formula
Wave Eq	y = A sin(kx - ωt + φ)
Wave Speed	v = λf = ω/k
Wave Number	k = 2π/λ
Speed on String	v = √(T/μ)
Speed in Gas	v = √(γRT/M)
Open Pipe	f_n = n v/2L (n=1,2,3..)
Closed Pipe	f_n = (2n-1) v/4L (n=1,2,3..)
Beat Freq	f_b = \|f1 - f2\|
Doppler	f' = f (v ± v_o)/(v ∓ v_s)
Intensity	I = 2π² f² A² ρ v

50 NEET Facts

Key points for Waves.

1. Mechanical Wave Requires material medium. Cannot travel in vacuum.

2. EM Wave Does not require medium. Can travel in vacuum.

3. Transverse Wave Particle vibration perpendicular to propagation. (Light, String).

4. Longitudinal Wave Particle vibration parallel to propagation. (Sound).

5. Polarization Only Transverse waves can be polarized. Sound cannot be polarized.

6. Speed in Solid v = √(η/ρ).

7. Speed in Medium Elasticity/Inertia ratio.

8. Newton's Assumption Sound propagation is Isothermal. (Wrong value 280 m/s).

9. Laplace Correction Sound propagation is Adiabatic. (Correct value ~332 m/s).

10. Effect of Pressure No effect on sound speed (if T constant).

11. Effect of Density v ∝ 1/√ρ. Hydrogen has higher speed than Oxygen.

12. Effect of Temp v ∝ √T. Speed increases by 0.61 m/s for every 1°C rise.

13. Effect of Humidity Humid air is less dense than dry air. Sound travels faster in humid air.

14. Constructive Interference Path diff = nλ. Max Intensity (4I).

15. Destructive Interference Path diff = (2n-1)λ/2. Min Intensity (0).

16. Stationary Waves Energy is not transferred. It is confined.

17. Nodes Points of zero displacement. Pressure change is maximum.

18. Antinodes Points of max displacement. Pressure change is zero.

19. Distance Node-Node λ/2.

20. Distance Node-Antinode λ/4.

21. Fundamental Mode Simplest mode of vibration. Lowest frequency. (First Harmonic).

22. Open Organ Pipe Produces All Harmonics (Odd and Even). Richer sound.

23. Closed Organ Pipe Produces Only Odd Harmonics.

24. End Correction (e) Antinode forms slightly outside the pipe. e = 0.6r.

25. Corrected Length Closed: L+e. Open: L+2e.

26. Beats Condition Difference in freq must be small (<10 Hz).

27. Beat Period Reciprocal of beat frequency.

28. Tuning Fork Wax Loading with wax decreases frequency.

29. Filing Tuning Fork Filing increases frequency.

30. Doppler Limit Not applicable if v_source > v_sound (Supersonic).

31. Shock Waves Cone shaped wave produced by supersonic object.

32. Mach Number Ratio of object speed to sound speed.

33. Red Shift Star moving away. Freq reduces. Wavelength increases towards red.

34. Blue Shift Star moving towards. Freq increases.

35. Radar Speed Gun Uses Doppler effect of radio waves.

36. Sonar Uses reflection of ultrasound.

37. Wave Intensity Proportional to square of Amplitude.

38. Loudness Depends on Intensity. Measured in Decibels (dB). Logarithmic scale.

39. Pitch Depends on Frequency.

40. Quality (Timbre) Depends on waveform/number of harmonics.

41. Echo Reflection of sound. Min distance 17m to hear distinct echo.

42. Reverberation Persistence of sound due to multiple reflections.

43. Resonance Column Exp to find speed of sound using water level.

44. Melde's Experiment Demonstrates transverse and longitudinal modes on string.

45. Quincke's Tube Demonstrates interference of sound.

46. Sonometer Verifies laws of vibrating strings.

47. Law of Length f ∝ 1/L.

48. Law of Tension f ∝ √T.

49. Law of Mass f ∝ 1/√μ.

50. Human Hearing Range 20 Hz to 20,000 Hz.

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