Velocity of sound wave:
Sound waves are mechanical waves which need a material medium to propagate from one place to another. When sound waves travel in a medium, alternate compressions and rarefactions are formed. The faster these compressions or rarefactions travel in a medium, the more is the speed of sound in that medium. The speed of sound wave in a medium depends upon the two factors of the medium and they are elastic factor (ε) and density (ρ).
Determination of velocity of sound wave in a medium (Dimensional method):
The velocity of sound in a medium depends on elasticity and density. Let,
Velocity of sound ∝ elasticity x density
v ∝ Ex ρy
or, v = k Ex ρy …………. (i)
Writing dimensional formula on both sides, we have,
[LT-1] = [ML-1T-2]x [ML-3]y
Or, [MoLT-1] = [Mx+yL-x-3yT-2]
Comparing the powers on both sides, we get,
-1 = -2x
∴ x = ½
Or, x + y = 0
∴ ½ + y = 0
So, y = -1/2
Now, substituting the value of x and y in eqn. (i), we get,
v = k E1/2ρ-1/2
So, v = [latex]\frac{E^{1/2}}{\rho^{1/2}}[/latex]
= [latex]\frac{k\sqrt{E}}{\sqrt{\rho}}[/latex]
So, v = k[latex]\sqrt{\frac{E}{\rho}}[/latex]
Experimentally, the value of k is 1 [k = 1]
∴ Velocity of sound in a medium (v) = [latex]\sqrt{\frac{E}{\rho}}[/latex]
Velocity of sound in solid:
If the solid is in the form of rod, the elasticity is equal to Young’s modulus, ‘Y’. The velocity of sound in rod is:
v = [latex]\sqrt{\frac{Y}{\rho}}[/latex]
If the solid is in the form of bulk, then, bulk modulus of elasticity and modulus of rigidity is more effective. In this case, the velocity of sound in it is:
v = [latex]\sqrt{\frac{B+\frac{4}{3}\eta}{\rho}}[/latex], where, B is the bulk modulus of elasticity.
Velocity of sound in liquid:
In case of liquid, the bulk modulus of elasticity is effective, and hence the velocity of sound in liquid becomes,
v = [latex]\sqrt{\frac{B}{\rho}}[/latex]
Velocity of sound in gases: Newton’s formula:
The velocity of sound in a gas medium is given by:
v = [latex]\sqrt{\frac{B}{\rho}}[/latex] ………………. (i) where, B = bulk modulus of elasticity.
Newton assumed that, when sound wave travel through a gas medium, the compression and rarefaction are formed so slow that the temperature of the gas remains constant i.e. the process is an isothermal. For isothermal process, the equation is:
PV = constant
Differentiating both sides,
PdV + VdP = 0
Or, PdV = -VdP
Or, P = -V[latex]\frac{dP}{dV}[/latex]
∴ P = [latex]-\frac{dP}{\frac{dV}{V}}[/latex]
∴ P = B, where, B = [latex]-\frac{dP}{\frac{dV}{V}}[/latex].
Substituting the value of B in eqn. (i), we get,
v = [latex]\sqrt{\frac{P}{\rho}}[/latex] ………….. (ii)
This is called Newton’s formula for a sound in a gas medium.
At NTP (for air),
P = ρgh = 13600 x 9.8 x 0.76 N/m2
ρair = 1.293 Kg/m3
∴ v = [latex]\sqrt{\frac{P}{\rho}}[/latex] = [latex]\sqrt{\frac{13600\times 9.8\times 0.76}{}}[/latex] = 280 m/s
This value is about 16% less than the experimental value i.e. 332 m/s. This large difference cannot be attributed to only experimental error. Later, a French physicist Pierre Laplace explained the reason for this discrepancy.
Laplace correction:
Laplace pointed out that the propagation of sound wave in a gaseous medium is known as isothermal process but it is an adiabatic process. The compressions and rarefactions are formed so rapidly that no heat exchange takes place to the surrounding or from the surrounding i.e. the process is adiabatic. For adiabatic process, the eqn. is:
[latex]PV^{\gamma}[/latex] = constant …………. (i)
Where, [latex]\gamma = \frac{C_p}{C_v}[/latex] = ratio of molar heat capacities.
Differentiating eqn. (i) on both sides, we get,
[latex]P\gamma V^{\gamma – 1}dV + V^{\gamma}dP[/latex] = 0
Or, [latex]P\gamma \frac{V^\gamma}{V}dV + V^{\gamma}dP[/latex] = 0
Or, [latex]\gamma P\frac{dV}{V}+dP[/latex] = 0
So, [latex]\gamma P = -\frac{dP}{\frac{dV}{V}}[/latex] = B
Substituting the value of B in eqn. (i)
v = [latex]\sqrt{\frac{\gamma P}{\rho}}[/latex]
At NTP,
P = ρgh = 13600 x 9.8 x 0.76
[latex]\text{ } = 101292.8\ N/m²[/latex]
ρair = 1.293 Kg/m3
and, [latex]\gamma[/latex] = 1.4 (for air)
∴ v = [latex]\sqrt{\frac{\gamma P}{\rho}}[/latex] = [latex]\sqrt{\frac{1.4\times 101292.8}{1.293}}[/latex]
= 331.17 m/s, which is very close to experimental value.
Factors affecting the velocity of sound in air:
The velocity of sound in a gas medium is given by:
v = [latex]\sqrt{\frac{\gamma P}{\rho}}[/latex] …………….. (i)
a) Temperature:
Consider, 1 mole of a gas, then, the eqn. is:
PV = RT
So, P = [latex]\frac{RT}{V}[/latex]
Substituting the value of P in eqn. (i), we get,
v = [latex]\sqrt{\frac{\gamma RT}{V\rho}} = \sqrt{\frac{\gamma RT}{M}}[/latex]
where, M = Vρ = Molar mass of the gas.
Since, [latex]\gamma[/latex], M and R are constant, for the given gas, therefore, v ∝ [latex]\sqrt{T}[/latex]. Hence, the velocity of sound in a gas medium is directly proportional to the square root of absolute temperature.
If v1 and v2 are velocities of sound waves, in temperature T1 and T2 respectively, then,
[latex]\frac{v_1}{v_2} = \sqrt{\frac{T_1}{T_2}}[/latex]
b) Pressure:
Consider, 1 mole of a gas, then, eqn. of gas is:
PV = RT
So, P = [latex]\frac{RT}{V}[/latex]
Substituting the value of P in eqn. (i), we get,
v = [latex]\sqrt{\frac{\gamma RT}{V\rho}} = \sqrt{\frac{\gamma RT}{M}}[/latex]
where, M = Vρ = Molar mass of the gas.
If T is constant, [latex]\sqrt{\frac{\gamma RT}{M}}[/latex] is a constant.
Therefore, the velocity of sound is independent of pressure provided the temperature remains constant.
c) Density:
Let, v1 and v2 are the velocities of sound waves having densities ρ1 and ρ2 of the gases respectively, then,
[latex]\frac{v_1}{v_2} = \sqrt{\frac{\rho_2}{\rho_1}}[/latex]
d) Humidity:
The density of water vapour is less than that of dry air. Hence, the presence of water vapour in an air decreases its overall density. But, the velocity of sound is inversely proportional to the density. Therefore, the velocity of sound in humid air is greater than the velocity of sound in dry air.
Let, ρd and ρh are the densities of dry air and humid air respectively, then,
vd = [latex]\sqrt{\frac{\gamma P}{\rho_d}}[/latex] and vh = [latex]\sqrt{\frac{\gamma P}{\rho_h}}[/latex]
∴ [latex]\frac{v_d}{v_h} = \sqrt{\frac{\rho_h}{\rho_d}}[/latex].
Since, ρd is greater than ρh, vh > vd
e) Direction of wind:
Let and be the velocities of wind and sound in the medium and [latex]\theta[/latex] be the angle between them. Then, resultant velocity of sound in the given direction is given as
[latex]v = v_s + v_wcos\theta[/latex]
Hence, velocity of sound in the given direction increases when [latex]\theta[/latex] is acute and decreases when [latex]\theta[/latex] is obtuse and remains same when [latex]\theta = 90^o[/latex].
Note: Velocity of sound is independent of the frequency, wavelength and amplitude of the sound wave.