Question

Newton formula and laplace correction derivation

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

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

thus,substituting B=P we get

v=√P/p

where P= pressure

p= density....

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hope its help you...