Question

Establish Newton's formula for velocity of sound in an elastic medium. What is the Laplace's correction?

Answer 1

Newton's theory about velocity of sound :- according to Newton, propagation of sound in air is an isothermal process. Because when sound is propagated in air, compression and rarefraction is formed . formation of these are very slow process. And heat produced due to compression is given to give surrounding and heat losed due to rarefraction is taken from surrounding. The whole process temperature remains the same. So, he assume propagation of sound is an isothermal process.

Now, according to Boyle's law
PV = constant [ at constant temperature ]
differentiate both sides,
VdP + PdV = 0
P = - dP/{dV/V} = Β [ Β is bulk modulus ] --------(1)

We know, velocity of sound is given by
v = √{Β/ρ} , here ρ is density and v is velocity of sound
Now, v = √{P/ρ} [ from equation (1)]

Hence, according to Newton's theory formula of sound in air is given by
v = √{P/ρ} , here P denotes the pressure and ρ denotes the density .


Laplace correction :- We learnt above Newton's theory , according to him propagation of sound is an isothermal process. But Laplace absorbed that propagation of sound in air isn't an isothermal process , it is an adiabatic process. Means when sound propagates in air , heat remains constant.

So, PV ≠ constant , It's $\bold{PV^{\gamma}=Constant}$
differentiate both sides,
V^γ.dP + γV^{γ-1}PdV = 0
P = -dP/{γdV/V} = Β/γ
B = γP -----(1)
∴ velocity of sound , v = √{γP/ρ}
Hence, according to Laplace correction , velocity of sound , v = √{γP/ρ}

Answer 2

Newton's Formula for velocity of Sound  :

V=√E/dwhere E is elasticity of medium and d is density of medium.

Newton assumed that , when sound travels through air producing compression and rare fractions, no change in temperature takes place.
Such a process taking place at constant temperature is called Isothermal process.
 In this case Boyle's law is obeyed

.PressureXVolume =Constant. 
PV =Constant

under Isothermal conditions the elasticity E of gas is numerically to pressure P of the gas .
 For air elasticity constant is bulk modulus.Elasticity =E=Bulk modulus K=pressure P
Hence Velocity V=√p/d
Newton's Formula gave us the velocity of sound at N.T.P to be 279.89m/s. This value is lower than experimental value.

Laplace's Correction to Newton's Formula :
French Mathematician Laplace suggested a correction in Newton's Formula  in the year 1816.He explained air is a bad conductor of heat and compression and rare fractions are produced rapidly in air.There is no time for equalization of temperature .
Hence when  sound travels in air, a change in temperature is produced .The changes are adiabatic and not isothermal.

Under adiabatic conditions the elasticity of the gas is equal to γp where γ is the ratio of two specific heats .
P is pressure.In case of Adiabatic conditions,
the elasticity or bulk modulus of gas is numerically equal to γ times pressure.
K=γP
Hence Velocity of Sound waves in air :
 V=√γp/d
From this formula the velocity of sound in gas =332m/s which is in close agreement with experimental value.