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### Wave and Sound

Wave motion is a type of motion in which the disturbance travels from one point of the medium to another but the particles of the medium do not travel from one point to another.For the propagation of wave, medium must have inertia and elasticity. These two properties of medium decide the speed of wave.

There are two types of waves

• Mechanical waves : These waves require material medium for their propagation. For example : sound waves, waves in stretched string etc.
• Non-mechanical waves or electromagnetic waves : These waves do not require any material medium for their propagation. For example : light waves, x-rays etc.

Types of wave motion

• Transverse waves : In the transverse wave, the particles of medium oscillate in a direction perpendicular to the direction of wave propagation.
• Longitudinal waves : In longitudinal waves particles of medium oscillate about their mean position along the direction of wave propagation.

EQUATION OF A HARMONIC WAVE

$y = Asin {2π({t \over T} - {x \over λ})}$

Speed of Sound

• Velocity of sound wave in a linear solid medium is given by
Young's modulus (Y) ${v = \sqrt{Y \over ρ}}$
Where Y = Young's Modulus of elasticity and
• Velocity of sound waves in a fluid medium (liquid or gas) is given by
Young's modulus (Y) ${v = \sqrt{B \over ρ}}$
Where B = Bulk modulus of the medium given by, and ρ is density of the medium
• Newtons Formula: Newton assumed propagation of sound through a gaseous medium to be an isothermal process
$v = {B \over ρ}$
• Laplace's Correction : $v = \sqrt{γP \over ρ }$
(b) Effect of moisture:-um/ud= √[ρd/ ρm] Since, ρm<ρd, then, um>ud This signifies sound travels faster in moist air. (c) Effect of pressure:- u=√γP/ρ=√γk = constant This signifies, change of pressure has no effect on the velocity of sound. (d) Effect of temperature:- ut/u0 =√ρ0/ρt= √T/T0 Thus, velocity of sound varies directly as the square root temperature on Kelvin’s scale. (e) Temperature coefficient of velocity of sound (α):- α = u0/546 = (ut-u0)/t

DOPPLER EFFECT

When a source of sound and an observer or both are in motion relative to each other there is an apparent change in frequency of sound as heard by the observer. This phenomenon is called the Doppler's effect.

Apparent change in frequency

When source is in motion and observer at rest

when source moving towards observer

$V_1 = v_o ({V \over {V - Vs}})$

When source is moving away from observer

$V_1 = v_o ({V \over {V + Vs}})$

Here V = velocity of sound
VS = velocity of source
ν0 = source frequency.

When source is at rest and observer in motion

when observer moving towards source

$V_2 = v_o ({{V + V_o}\over {V}})$

when observer moving away from source and
$V_o$ = velocity of observer.

$V_2 = v_o ({{V - V_o}\over {V}})$

When source and observer both are in motion

If source and observer both move away from each other.

$v_3 = ({{v - v_o} \over {V - V_s}}) v_o$

If source and observer both move towards from each other.

$v_3 = ({{v + v_o} \over {V - V_s}}) v_o$

Closed Organ Pipe

In a closed organ pipe, one end is closed and another end is open. In closed organ pipe, the closed end is always a node while the open end is always an antinode

For the first Mode λ1 = 4L, where L is the length of the pipe

For nth mode, $λ_{n} = {4L \over { 2n - 1}}$

Frequnecy, $f_n = { v \over λ_n } = {v(2n - 1) \over 4L}$

Open Organ Pipe

In an open organ pipe, both end are open. In an organ pipe, at both ends, there will be antinodes

For the first mode, $λ_1 = 2L$, where L is the length of the pipe.

In open organ pipe, all harmonics are present, whereas is a closed organ pipe only odd harmonics are present

For nth mode, $f_n = { v \over λ_n } = { nv \over 2L }$

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