SOUND

SOUND
Variation of velocity of sound in a gas with temperature.
The velocity of sound in a gas v = \(\sqrt {\frac{{\gamma p}}{\rho }} \)
 \(v = \sqrt {\frac{{\gamma p}}{\rho }} = \sqrt {\frac{{\gamma pv}}{m}} \)  (\(\because \,\,\rho = \frac{M}{V}\))
\(= \sqrt {\frac{{\gamma RT}}{M}} \)  \(\left( {\because PV = nRT} \right)\)
Where M is molecular weight of gas and T is absolute temperature of the gas
  \(V \propto \,\sqrt T \)  = \(\sqrt {273 + t} \)
\(\frac{{{V_1}}}{{{V_2}}} = \sqrt {\frac{{273 + {t_1}}}{{273 + {t_2}}}} \)
\(\frac{{{V_t}}}{{{V_0}}} = \sqrt {\frac{{273 + {t^0}C}}{{273}}} \) 
V_t = V₀ \(\left( {1 + \frac{{{t^0}C}}{{546}}} \right)\) v₀ = 332m/s
 V_t = V₀ +0.61t⁰C
Note: 
1.     The velocity of sound in air increases by 0.6m/s for every 10C rise of temperature.
2.     Velocity of sound in a gas is directly proportional to the square root of the absolute temperature
\(V = \sqrt {\frac{{\gamma RT}}{M}} \)    
3. But from kinetic theory of gases rms velocity of gaseous molecular is V_rms 
\(\sqrt {\frac{{3RT}}{m}} \)
\(\therefore \,\,\frac{V}{{{V_{rms}}}} = \sqrt {\frac{\gamma }{3}} \) \(\Rightarrow V = \sqrt {\frac{\gamma }{3}} {v_{rms}}\)
 velocity of sound in a gas is of the order of the velocity of its molecules
The velocity of sound in gas at absolute temperatures T and T¹ are V, nv then
  \(\frac{{nv}}{v} = \sqrt {\frac{{{T^1}}}{T}} \)      \({T^1} = {n^2}T\)
   \(v \propto \frac{1}{{\sqrt M }}\)       \(\frac{{{V_1}}}{{{V_2}}} = \sqrt {\frac{{{M_2}}}{{{M_1}}}} \)
EFFECT OF PRESSURE:
since \(\sqrt {\frac{{\gamma P}}{\rho }} = \sqrt {\frac{{\gamma RT}}{m}} \)
The velocity of sound in air is independent of pressure.
EFFECT OF DENSITY:
    At a constat pressure the velocity of sound depends on the density of the gas 
    \(v \propto \frac{1}{{\sqrt \rho }} \Rightarrow v\sqrt \rho = cons\tan t\)
    \(\frac{{{v_1}}}{{{v_2}}} = \sqrt {\frac{{{\rho _2}}}{{{\rho _1}}}} \)
Variation with humidity:-
    1. With increase in humidity, density of air decreases. So with increase of humidity velocity of sound increases.
    2. Sound travels faster in humid air (rainy season) than in dry air at the same temperature.
    P_{moist air}< P_{dry air}
     V  _{moist air} < V_{dry air}
Note:- Amplitude frequency, phase, loudness, pitch quality etc... Have practically no effect on velocity of sound.
Velocity of sound in a solid:-
     The velocity of sound in solids is v = \(\sqrt {\frac{y}{\rho }} \)
    Where y - is young's modulus of the material velocity of sound in steel is greater than in copper or brass (or) silver.
Velocity of sound in a liquid:-
    Velocity of sound in liquids V = \(\sqrt {\frac{k}{\rho }} \)
    Where k is the bulk modulus of the liquid.