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Laplace's Correction to Newton's Formula

Assuming isothermal conditions to prevail when sound travels through air, Newton has applied Boyle's law to the changes in pressure and volume. In a region of compression, there is a slight increase in temperature and in a region of rarefaction, there is a slight decrease in temperature. These changes in pressure occur rapidly and air is a poor conductor of heat thus, equalization of temperature among the different regions was improbable, according to Laplace.. He was of the view that the changes in temperature occur under adiabatic conditions, i.e., no heat enters the gas from outside or leaves it from inside. The heat developed in the compressed layers remains fully confined to those layers and has no time to get dissipated into the entire body of the gas. Similarly, the cold caused in the rarefied layers cannot be compensated for, by flow of heat into it from other layers.

Thus Boyle's law does not apply in this case.

The relation between pressure and volume of a gas under adiabatic conditions is given by

where i.e., the ratio of the principle specific heats of the gas at constant pressure and constant volume respectively.

Let the pressure change by an amount DP, producing a change in volume by DV. Then

Taking out from the 2ⁿ^d factor in the above expression,

But from binomial approximation,

Canceling P on both sides and neglecting the term containing P.V because it is too small, we get

But the LHS in the above equation represents the bulk modulus, B.

Substituting in equation (1-23), we get

This is known as Newton-Laplace formula for the velocity of sound in a gas.

This is in close agreement with the experimental value.

Sub Topics

Effect of pressure, temperature, humidity and wind on the velocity of sound in a gas
Effect of pressure
Effect of temperature
Effect of humidity
Effect of wind

Effect of pressure, temperature, humidity and wind on the velocity of sound in a gas

Effect of pressure

Therefore, Newton-Laplace formula can be written as

If the temperature remains constant, then according to Boyle's law, PV remains constant. Thus, changes in pressure are accompanied by changes in volume, such that p/q is held a constant. Hence, the velocity of sound is independent of the pressure of the gas, provided the changes in pressure are sufficiently slow such that Boyle's law holds good. This has been verified experimentally and it has been found that the velocity of sound at high altitudes is the same as at sea-level, although the atmospheric pressure at the two places is different.

Effect of temperature

Therefore, the velocity is inversely proportional to the square root of the density

But density and volume are inversely proportional.

According to Charles' law of volume,

Where T and T₀ are the temperatures on the Kelvin scale corresponding to t^oC and 0^oc.

Thus, the velocity of sound in a gas varies directly as the square root of the absolute temperature.

Effect of humidity

The density of water vapor is less than that of dry air at ordinary temperatures. The density of water vapor at NTP is 0.8kg/m³ whereas the density of dry air at NTP is 1.293kg/m³. The presence of the water vapor in air lowers the density of air. Hence, the velocity of sound in moist air will be greater than that in dry air.

Effect of wind

Let S be a source emitting sound with a velocity v, towards a listener at L. let wind blow with a velocity w making an angle q with the direction of sound. Then component of wind velocity along the direction of sound = w cos q. Then the velocity of sound becomes v + w cos q. If q = 90^o, then wind will have no effect on the velocity of sound. If the wind is blowing opposite to the direction in which sound is traveling, then the velocity of sound = v - w.

The expression for the velocity of sound does not contain amplitude and wavelength of sound waves. Thus, the velocity of sound is independent of wavelength and amplitude. However, this is true for small amplitudes.