Mechanical Waves
The waves which needs material medium to propagate is called mechanical waves. Example: water waves, sound waves, etc.
Mechanical waves are of two types:
1. Longitudinal Waves
2. Transverse waves
(Explanation of types of mechanical waves is already in part 1: click here )
Speed of Sound Wave in any medium:
Experimentally it is found that speed of sound wave in any medium depends on two factors; elasticity of medium and density of medium.
Let 'V' be the speed of sound in any medium, 'E' be the elasticity and 'p' be the density of the medium. Then,
`V \alpha \rho^b` ---------(2)
Combining equation (1) and (2) we get,
`V \alpha E^a\rho^b`
`V = KE^a\rho^b`-----------(3)
Writing the dimension of each terms in eqn (3) we get,
Writing the dimension of each terms in eqn (3) we get,
`[M^0L^1T^-1]=[M^1L^-1T^-2]^a [M^1L^-3T^0]^b`
`[M^0L^1T^-1]=[M^(a+b)L^(-a-3b)T^(-2a)]`
Equating the power of M, L and T we get,
`a+b=0`, `-a-3b=1` and `-2a=-1`
Solving these equation we get,
`a=\frac{1}{2} and b=\frac{-1}{2}`
Putting the value of a and b in equation (3) we get,
Putting the value of a and b in equation (3) we get,
`V = K E^\frac{1}{2} \rho^\frac{-1}{2}`
`V = K \sqrt\frac{E}{\rho}`
`V = K \sqrt\frac{E}{\rho}`
Experimentally the value of K is found to be 1 so,
Velocity of sound in solid:
Elasticity(E)= Young's Modulus of Elasticity(Y)
`\therefore` velocity of sound in solid(`v_s`)= `\sqrt\frac{Y}{\rho}`
For steel, `Y = 2\times10^11 Nm^(-2)`
`\rho = 7800 kgm^(-3)`
`\therefore` velocity of sound in steel (`V_steel`)= `\sqrt\frac{2\times10^11}{7800}`
= 5064m/s
Velocity of sound in liquid
Elasticity(E) = Bulk modulus of Elasticity (`\beta`)
`\therefore` velocity of sound in liquid (`v_l`)= `\sqrt\frac{\beta}{\rho}`
For water, `\beta` = 2.04`\times10^9Nm^(-2)`
`\rho = 1000kgm^(-3)`
`\therefore` velocity of sound in water (`v_w`) = `\sqrt\frac{2.04\times10^9}{1000}`
= 1428 m/s
Velocity of sound in gaseous medium
i) Newton's formula for the velocity of sound in gas:
Elasticity(E) =Bulk modulus of elasticity(`\beta`) = Pressure exerted by gas (P)
`\therefore` velocity of sound in gas (`v_g`) = `\sqrt\frac{P}{\rho}`
At NTP, P = 1.013 `\times 10^5` N/`m^-2` and `\rho` = 1.29 kg/`m^3`
`\therefore` velocity of sound in gas (`v_g`) =`\sqrt\frac{1.013\times10^5}{1.29}`
= 280 m/s which does not match with the experimental result.
ii) Laplace correction for the velocity of sound in gas:
Elasticity(E)= Bulk modulus of Elasticity (`\beta`) = `\gammaP`
`\therefore` velocity of sound in gas (`v_g`) = `\sqrt\frac{\gammaP}{\rho}`
At NTP, P = 1.013 `\times 10^5` N/`m^-2` , `\rho` = 1.29 kg/`m^3` and `\gamma`=1.4
`\therefore` velocity of sound in gas (`v_g`) =`\sqrt\frac{1.4\times1.013\times10^5}{1.29}`
= 332 m/s which match with the experimental result.
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