Question

Describe Newton’s formula for velocity of sound waves in air and also discuss Laplace’s correction.

Answer 1

Newton assumed that when sound propagates in air, the formation of compression and rarefaction takes place in a very slow manner so that the process is isothermal in nature. It is found that, the heat produced during compression (pressure increases, volume decreases), and heat lost during rarefaction (pressure decreases, volume increases) occur over a period of time such in a way that the temperature of the medium remains constant. Hence, by treating the air molecules to form an ideal gas, the changes in pressure and volume obey Boyle’s law, Mathematically

PV = Constant     …(1)

Differentiating equation (1), we get

PdV + Vdp = 0

(or) P = – V`"dp"/"dv" = "B"_"T"`   ....(2)

where, B_T is an isothermal bulk modulus of air.

v = `sqrt("B"/rho)`   .....(3)

Substituting equation (2) in equation (3), the speed of sound in air is

`"v"_"T" = sqrt("B"_"T"/rho) = sqrt("P"/rho)`

Since P is the pressure of air whose value at NTP (Normal Temperature and Pressure) is 76 cm of mercury, we have

P = (0.76 x 13.6 x 10³ x 9.8)Nm^-2

ρ = 1.293 kg m^-3.

Here p is density of air. Then the speed of sound in air at normal temperature and pressure (NTP) is

`"v"_"T" = sqrt((0.76 xx 13.6 xx 10^3 xx 9.8)/1.293)`

= 279.80 ms^-1

≈ 280 ms^-1 (theoretical value)

But the speed of sound in air at 0°C is experimentally observed as 332 ms^-1 that is close up to 16% more than the theoretical value.

Laplace correction:

Laplace satisfactorily corrected this discrepancy by assuming that when the sound propagates through a medium, the particles oscillate very rapidly such that the compression and rarefaction occur very fast. Hence the exchange of heat produced due to compression and cooling effect due to rarefaction do not take place, because, air (medium) is a poor conductor of heat. Since, temperature is no longer considered as a constant here, propagation of sound is an adiabatic process. By adiabatic considerations, the gas obeys Poisson’s law (not Boyle’s law as Newton assumed), that is

PV’ = Constant … (4)

where, γ = `("C"_"p")/("C"_"v")`, that is the ratio between specific heat at constant pressure and specific heat at constant volume.

Differentiating equation (4) on both the sides, we get

V^γ dP + P (γV^γ-1 dV) = 0

or, γP = – V`"dp"/"dv" = "B"_"A"`   ...(5)

where, B_A is the adiabatic bulk modulus of air.

v = `sqrt("B"/rho)`

Now, substituting equation (5) in equation (6), the speed of sound in air is

`"v"_"A" = sqrt("B"_"A"/rho) = sqrt((gamma"P")/rho)`

`= sqrt(gamma)"v"_"T"`    ...(7)

Since air contains mainly, nitrogen, oxygen, hydrogen etc, (diatomic gas), we take γ = 1.47. Hence, speed of sound in air is v_A = `(sqrt1.4)(280 "ms"^-1)` = 331.30 ms^-1, which is very much closer to experimental data.