ध्वनि की गति: न्यूटन और लाप्लास के सूत्र

The calculation for the speed of sound ($v$) is a classic problem in physics. Sir Isaac Newton was the first to attempt a theoretical calculation, but his original formula yielded a value that was significantly lower than experimental observations.

Newton's Original Formula (Isothermal Assumption)

Newton derived his formula based on the assumption that the compression and rarefaction process of the air, as sound waves pass through it, occurs isothermally (at a constant temperature). His formula for the speed of sound was:

$$ v_{\\text{Newton}} = \\sqrt{\\frac{P}{\\rho}} $$

Where $P$ is the pressure and $\rho$ is the density. This formula predicted a speed that was too low by about 15%.

लाप्लास का सुधार (रुद्धोष्म धारणा)

The discrepancy was resolved by the French mathematician and astronomer पियरे-साइमन लाप्लास (Pierre-Simon Laplace) in 1816. Laplace correctly reasoned that the compression and rarefaction of air due to sound waves is too rapid for heat exchange to occur with the surroundings. Therefore, the process is not isothermal but adiabatic (no heat exchange).

रुद्धोष्म आयतन मापांक (Adiabatic Bulk Modulus): In an adiabatic process, the bulk modulus ($B$) is given by $B_{\\text{adiabatic}} = \\gamma P$, where $\gamma$ (gamma) is the adiabatic index (ratio of specific heat capacities).
Laplace's Corrected Formula: Laplace incorporated this into Newton's equation, resulting in the correct formula:

$$ v_{\\text{Laplace}} = \\sqrt{\\frac{\\gamma P}{\\rho}} $$

This modification, known as the Laplace correction, brought the theoretical speed of sound in air into excellent agreement with experimental values. Thus, पियरे-साइमन लाप्लास modified Newton's formula in 1816 to take into account adiabatic compression.

Physics Set 1

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