write expression for speed of a longitudinal wave in a liquid or a gas & long solid rod. Obtain Newton's formula for the speed of sound in gas . Why and What correction was applied by Laplace in this formula
Answers
Newton assumed that when sound propagates through air, temperature remains constant (i.e. the process is isothermal).
So, bulk modulus of elasticity B = BT = p
(isothermal bulk modulus BT of a gas is equal to its pressure).
Therefore at NTP
p = 1.01 × 105 N/m2 and ρ = 1.3 kg/m3
= 279 m/s
The experimental value of v in air is 332 m/s at NTP. This discrepancy was removed by Laplace.
LAPLACE’S CORRECTION :
Laplace assumed that the propagation of sound in air is an adiabatic process not the isothermal.
B = Bs = γP [Adiabatic bulk modulus Bs of a gas = γP]
Where γ = Cp/Cv = 1.41 for air
Which is in agreement with the experimental value (332 m/s) thus,
We can conclude that sound waves propagate through gases adiabatically
a) Effect of density
Clearly , velocity of sound in gas is inversely proportional to the square root of density of the gas .
(b) Effect of temperature:- In a gas
v ∝√T
i.e. with increase in temperature velocity of sound in a gas increases.
Let us find velocity of sound in air at t°C.
At NTP vair = vo°C = 332 m/s
When t is small-
Putting vo = 332 m/s
we have, vt = (332 + 0.61 t) m/s
i.e. for small temperature variations at 0° C, velocity of sound changes by 0.61 m/s when temperature changes by 1°C.
(c) Effect of pressure: In a gas ;
Change in pressure has no effect on velocity of sound in a gas, so long as temperature remains constant; because ;
P/ρ = constant; as long as temperature is constant.
(d) Effect of relative humidity: When humidity increases, there is an increase in the relative number of water molecules and hence a decrease in the molar mass (avg. molecular wt.), and the speed of sound increases.
Newton’s Formula & Laplace’s correction
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Newton assumed that when sound propagates through air, temperature remains constant (i.e. the process is isothermal).
Newton assumed that when sound propagates through air, temperature remains constant (i.e. the process is isothermal).So, bulk modulus of elasticity B = BT = p
Newton assumed that when sound propagates through air, temperature remains constant (i.e. the process is isothermal).So, bulk modulus of elasticity B = BT = p(isothermal bulk modulus BT of a gas is equal to its pressure).
Newton assumed that when sound propagates through air, temperature remains constant (i.e. the process is isothermal).So, bulk modulus of elasticity B = BT = p(isothermal bulk modulus BT of a gas is equal to its pressure).Therefore at NTP
Newton assumed that when sound propagates through air, temperature remains constant (i.e. the process is isothermal).So, bulk modulus of elasticity B = BT = p(isothermal bulk modulus BT of a gas is equal to its pressure).Therefore at NTPp = 1.01 × 105 N/m2 and ρ = 1.3 kg/m3
Newton assumed that when sound propagates through air, temperature remains constant (i.e. the process is isothermal).So, bulk modulus of elasticity B = BT = p(isothermal bulk modulus BT of a gas is equal to its pressure).Therefore at NTPp = 1.01 × 105 N/m2 and ρ = 1.3 kg/m3= 279 m/s
Newton assumed that when sound propagates through air, temperature remains constant (i.e. the process is isothermal).So, bulk modulus of elasticity B = BT = p(isothermal bulk modulus BT of a gas is equal to its pressure).Therefore at NTPp = 1.01 × 105 N/m2 and ρ = 1.3 kg/m3= 279 m/sThe experimental value of v in air is 332 m/s at NTP. This discrepancy was removed by Laplace.
LAPLACE’S CORRECTION :
Laplace assumed that the propagation of sound in air is an adiabatic process not the isothermal.
Laplace assumed that the propagation of sound in air is an adiabatic process not the isothermal.B = Bs = γP [Adiabatic bulk modulus Bs of a gas = γP]
Laplace assumed that the propagation of sound in air is an adiabatic process not the isothermal.B = Bs = γP [Adiabatic bulk modulus Bs of a gas = γP]Where γ = Cp/Cv = 1.41 for air
Laplace assumed that the propagation of sound in air is an adiabatic process not the isothermal.B = Bs = γP [Adiabatic bulk modulus Bs of a gas = γP]Where γ = Cp/Cv = 1.41 for air
Laplace assumed that the propagation of sound in air is an adiabatic process not the isothermal.B = Bs = γP [Adiabatic bulk modulus Bs of a gas = γP]Where γ = Cp/Cv = 1.41 for air vair =√yP/P
Laplace assumed that the propagation of sound in air is an adiabatic process not the isothermal.B = Bs = γP [Adiabatic bulk modulus Bs of a gas = γP]Where γ = Cp/Cv = 1.41 for air vair =√yP/Pvair = √(1.0×10*5)(1.41)/1.3
Laplace assumed that the propagation of sound in air is an adiabatic process not the isothermal.B = Bs = γP [Adiabatic bulk modulus Bs of a gas = γP]Where γ = Cp/Cv = 1.41 for air vair =√yP/Pvair = √(1.0×10*5)(1.41)/1.3= 331.3 m/s
Which is in agreement with the experimental value (332 m/s) thus,
Which is in agreement with the experimental value (332 m/s) thus,We can conclude that sound waves propagate through gases adiabatically