Question

Discuss
Newton’s
formula
for
the
velocity
of
longitudinal
waves
in


air.
What
correction
was
applied
by
Laplace
and
why?​

Answer 1

Answer:

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Answer 2

Newton's formula for the velocity of sound in air. Newton was the first to show that velocity of a longitudinal wave (i.e., sound waves) in any medium is given by: v = √{p/ρ} ....(i) Newton assumed that during the compressions and rarefractions, the temperature of the medium does not change. Newton suggested that during compression, the temperature increases and the heat produced is immediately lost to the surrounding. During rarefractions, the heat is lost, which is compansated by gaining heat from the surrondings. Thus, when sound travels in air, conditions are "isothermal". i.e., pressure and volume of the air change. but the temperature remains constant. Suppose the pressure increases from p to (p + Δp) and the volume decreases from v to (v - Δv). As the temperature does not change, therefore, Boyle's law is applicable. i.e., pV = (p + Δp)(V - ΔV) = pV - pΔV + VΔp - Δp.ΔV Since, Δp and ΔV are very small, so the product Δp.ΔV can be neglected. Hence, pV = pV - p.ΔV + V.Δp or pΔV = V.Δp or p.{p}/{Δv/v} = k, Bulk modulus. Hence, eqn (i) can be written as v = √{p/ρ} ....(ii) which is the Newton's formula for the velocity of sound in air. Let us calculate the velocity of sound in air at normal temperature and pressure. At N.T.P we have p = 0.76 m of Hg. Now using p = hρg we get p = 0.76 m x 13,600 kg m x 9.8 ms-2 = (0.76 x 13600 x 9.8) Nm-2 and ρ (for air) = 1.293 kg m-3 Hence, v = √{0.76 x 13600 x 9.8 Nm-2}/{1.293 Kg m-3} = 280 ms-1 The experimental value of the velocity of the sound at N.T.P comes out to be 332 ms-1 which is much higher than the theoretical value (i.e., 280 ms-1). This means that there is some error in the Newton's formula. Newton could not give a satisfactory explanation for this error. In 1816, Laplace, a French Scientist explained the discrepency in the Newton's formula as follows: Laplace's Correction: Laplace pointed out that Newton was wrong in assuming that during compressions and rarefactions, the temperature remains constant. He on the other hand argued that compression and rarefaction follow each other so rapidly that they hardly get time to equalize their temperatures with the surroundings. During compressions there is an increase in temperature whereas during rarefactions the temperature falls. In otherwords, the process during the propagation of sound waves in the medium is adiabatic (where the temperature does not remain constant) and not isothermal as assumed by Newton. Thus Boyle's law is not applicable. For adiabatic process, we have. pvγ = Consatant ....(iii) [where γ = Cp/Cv] Differentiating equation (iii), we get pγpγ-1 dv + v2 dp = 0 or γppγ-1 dv = vγdp or, γp.{vγdp}/{pγ-1dv} = {dp}/{-dv/v = k} Hence, equation (i) becomes, v = √{γp/ρ} ....(iv) For air, the value of γ is 1.41. Therefore, the speed of sound in air at N.T.P. is given by v = [{1.41 x 76 x 13.6 x 980}/{1.293 x 10-3}]-1/2 = 332 ms-1 This result is in close agreement with the experimental value of the velocity of sound in air. Thus, equation (iv) gives the velocity of sound in the air. Read more on Sarthaks.com - https://www.sarthaks.com/570185/discuss-newtons-formula-for-the-velocity-of-sound-in-air-what-is-laplaces-correction