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| Laplace's Correction to Newton's Formula |
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| 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. |
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| Thus Boyle's law does not apply in this case. |
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| The relation between pressure and volume of a gas under adiabatic conditions is given by |
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where  i.e., the ratio of the
principle specific heats of the gas at constant pressure and constant volume respectively. |
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| Let the pressure change by an amount DP, producing a change in volume by DV. Then |
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Taking out  from the 2nd factor in the above expression, |
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| But from binomial approximation, |
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Canceling P on both sides and neglecting the term containing  P. V because it is too small, we get |
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| But the LHS in the above equation represents the bulk modulus, B. |
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| Substituting in equation (1-23), we get |
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| This is known as Newton-Laplace formula for the velocity of sound in a gas. |
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| This is in close agreement with the experimental value. |
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| Effect of pressure, temperature, humidity and wind on the velocity of sound in a gas |
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| Therefore, Newton-Laplace formula can be written as |
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| 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. |
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| Therefore, the velocity is inversely proportional to the square root of the density |
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| But density and volume are inversely proportional. |
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| According to Charles' law of volume, |
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| Where T and T0 are the temperatures on the Kelvin scale corresponding to toC and 0oc. |
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| Thus, the velocity of sound in a gas varies directly as the square root of the absolute temperature. |
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| 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. |
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| 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 = 90o, 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. |
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| 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. |
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