a) Does the propagation of sound wave cause change in thermodynamic condition of medium? Derive Laplace formula of velocity of sound in air. b) What is Doppler's effect? Derive an expression for the…

a) Propagation of Sound Waves and Laplace Formula

Does the propagation of sound wave cause change in thermodynamic condition of medium? No, the propagation of a sound wave in a medium does not cause any permanent change in the thermodynamic conditions (pressure, temperature, density) of the medium. Sound waves are adiabatic (isentropic) processes because they propagate too quickly for heat transfer to occur between different parts of the medium. The compression and rarefaction cycles are rapid, so no heat is exchanged with the surroundings, and the process is reversible.

Derivation of Laplace Formula for Velocity of Sound in Air The speed of sound in a gas is given by: $$v=\sqrt{\frac{\gamma P}{\rho }}$$ where:

• $\gamma =\frac{C_p}{C_v}$ (adiabatic index),
• $P$ = equilibrium pressure,
• $\rho$ = density of the medium.

For air at STP (0°C, 1 atm), using the ideal gas law $PV=nRT$, we can express $P$ and $\rho$ in terms of temperature $T$: $$P=\rho \frac{RT}{M}$$ where:

• $R$ = universal gas constant,
• $M$ = molar mass of air.

Substituting $P$ into the speed equation: $$v=\sqrt{\frac{\gamma RT}{M}}$$ For dry air at 0°C (273 K):

• $\gamma =1.4$,
• $R=8.314{\text{J mol}}^{-1}{\text{K}}^{-1}$,
• $M=0.029{\text{kg mol}}^{-1}$.

Thus, the Laplace formula for the speed of sound in air is: $$v=\sqrt{\frac{1.4\times 8.314\times 273}{0.029}}\approx 331\text{m/s}$$

---

b) Doppler’s Effect for a Moving Source

Definition Doppler’s effect describes the apparent change in frequency of a wave when there is relative motion between the source and the observer. For sound waves, this occurs due to the compression or rarefaction of wavefronts caused by the motion of the source or observer.

Expression for Apparent Frequency (Moving Source, Stationary Observer) When a source of sound moves away from a stationary observer, the wavelength increases, reducing the observed frequency. The relationship is given by: $$f^{\prime }=\frac{f}{1+\frac{v_s}{v}}$$ where:

• $f^{\prime }$ = apparent frequency (observed),
• $f$ = actual frequency of the source,
• $v_s$ = speed of the source (away from observer),
• $v$ = speed of sound in the medium.

Derivation

1. Let the source emit waves of frequency $f$ and wavelength $\lambda =\frac{v}{f}$.
2. If the source moves away at speed $v_s$, the effective wavelength becomes: $${\lambda }^{\prime }=\lambda +\frac{v_s}{f}=\frac{v}{f}+\frac{v_s}{f}=\frac{v+v_s}{f}$$
3. The apparent frequency $f^{\prime }$ is then: $$f^{\prime }=\frac{v}{{\lambda }^{\prime }}=\frac{v}{\frac{v+v_s}{f}}=\frac{vf}{v+v_s}=\frac{f}{1+\frac{v_s}{v}}$$

Key Observations

• If the source moves toward the observer, replace $+v_s$ with $-v_s$.
• If the observer moves (and the source is stationary), the formula becomes: $$f^{\prime }=f\left(1\pm \frac{v_o}{v}\right)$$ where $v_o$ is the observer’s speed.