Waves & Sound

Topics covered: Wave motion, standing waves, sound waves, organ pipes, beats and Doppler effect (JEE Main focus).

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1. Wave Motion

Wave is a process by which energy and momentum are transferred from one part of a medium to another without actual transfer of particles.

Wave Equation

Any function y(x, t) that satisfies:

d²y/dx² = (1/v²) · d²y/dt²

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2. Progressive (Travelling) Waves

General form:

y = f(ωt ± kx)

• Wave moves along +x direction if (ωt − kx)
• Wave moves along −x direction if (ωt + kx)

Wave speed:

v = ω / k

Particle velocity:

v_p = dy/dt

Slope of wave:

dy/dx

Relation:

v_p = −v × (slope)

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3. Plane Harmonic Progressive Wave

Along +x direction

y = A sin(ωt − kx)

y = A sin(ωt − kx + φ)

Along −x direction

y = A sin(ωt + kx)

y = A sin(ωt + kx + φ)

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4. Speed of Transverse Wave on Stretched String

v = √(T / μ)

• T = tension in string
• μ = mass per unit length

Also,

v = √(T / ρA)

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5. Standing (Stationary) Waves

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

If:

y₁ = a sin(ωt − kx)

y₂ = a sin(ωt + kx)

Resultant wave:

y = 2a cos(kx) sin(ωt)

Important Points

• Nodes: zero amplitude
• Antinodes: maximum amplitude
• Distance between successive nodes or antinodes = λ / 2
• Distance between node and nearest antinode = λ / 4
• Total energy of each loop remains constant

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6. Frequency of Vibrating String

Fundamental frequency:

f = (1 / 2l) √(T / μ)

p^th harmonic:

f_p = (p / 2l) √(T / μ)

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7. Sound Waves

Sound waves are longitudinal waves.

• Possible in solids, liquids and gases
• Consist of compressions and rarefactions

Velocity of longitudinal wave:

v = √(E / ρ)

• E = modulus of elasticity
• ρ = density of medium

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8. Speed of Sound

In solids

v = √(Y / ρ)

In fluids (liquid or gas)

v = √(B / ρ)

In air

v = √(γP / ρ)

v = √(γRT / M)

At 0°C:

v ≈ 331.3 m/s

Important Results

• Speed does not depend on pressure
• Increases by ≈ 0.61 m/s per 1°C rise in temperature
• Increases with humidity

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9. Displacement Wave and Pressure Wave

If displacement wave:

y = A sin(ωt − kx)

Pressure wave:

P = P₀ cos(ωt − kx)

Where:

P₀ = B A k

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10. Energy, Power and Intensity of Sound

Energy density:

(K.E.)_max = ½ ρ a² ω²

Power:

P = ½ ρ ω² a² v A

Intensity:

I = P / A = ½ ρ ω² a² v

Loudness level:

L = 10 log₁₀(I / I₀) dB

I₀ = 10⁻¹² W m⁻² (threshold of hearing)

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11. Organ Pipes

Closed End Organ Pipe

Fundamental frequency:

f₁ = v / 4l

• Only odd harmonics present
• Maximum wavelength = 4l

Open End Organ Pipe

Frequency:

f = n v / 2l

• All harmonics present
• Maximum wavelength = 2l

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12. Resonating Air Column

First resonance:

L₁ + e = λ / 4

Second resonance:

L₂ + e = 3λ / 4

Wavelength:

λ = 2(L₂ − L₁)

End correction:

e = (L₂ − 3L₁) / 2

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13. Beats

Beat frequency:

f_beat = |f₁ − f₂|

• Waxing tuning fork → frequency decreases
• Filing tuning fork → frequency increases

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14. Doppler Effect (Sound)

General expression:

n′ = n (V ± V_o) / (V ∓ V_s)

• Source towards observer: n′ increases
• Source away from observer: n′ decreases
• Observer moving towards source: n′ increases
• Observer moving away from source: n′ decreases
• Source at centre, observer in circular motion: n′ = n