Answer any one question. a) Describe Newton's formula for the velocity of sound in air. Explain why and how this formula is modified by Laplace. b) Describe an experiment with the necessary theory…

a) Newton’s Formula for Velocity of Sound in Air and Laplace’s Correction

Newton’s Formula

Newton derived the velocity of sound in air using the isothermal assumption (constant temperature). He considered air as an ideal gas and applied Boyle’s law (pressure ∝ density at constant temperature). The formula is:

$$v=\sqrt{\frac{P}{\rho }}$$

Where:

• $v$ = velocity of sound in air,
• $P$ = pressure of air,
• $\rho$ = density of air.

Derivation:

1. For an adiabatic process, pressure $P$ and density $\rho$ are related by $P=k{\rho }^{\gamma }$, where $\gamma =\frac{C_p}{C_v}$ (ratio of specific heats).
2. Newton assumed isothermal conditions ($\gamma =1$), so $P\propto \rho$.
3. The velocity of sound is derived from the restoring force due to pressure changes: $$v=\sqrt{\frac{dP}{d\rho }}=\sqrt{\frac{P}{\rho }}$$
4. Substituting $P=\rho RT$ (ideal gas law) gives: $$v=\sqrt{\frac{RT}{M}}$$ where $R$ = universal gas constant, $T$ = absolute temperature, and $M$ = molar mass of air.

Laplace’s Correction

Newton’s formula underestimates the speed of sound because it ignores the adiabatic nature of sound waves (rapid compression and rarefaction prevent heat exchange). Laplace corrected this by replacing $\gamma =1$ with the actual adiabatic index ($\gamma =1.4$ for air). The corrected formula is:

$$v=\sqrt{\frac{\gamma P}{\rho }}=\sqrt{\frac{\gamma RT}{M}}$$

Why the Correction?

• Sound waves propagate too quickly for heat transfer, so the process is adiabatic ($\Delta Q=0$).
• For air, $\gamma =\frac{C_p}{C_v}=1.4$ (diatomic gas).
• This increases the calculated speed by $\sqrt{\gamma }$, matching experimental values (~331 m/s at 0°C vs. Newton’s ~280 m/s).

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b) Resonance Tube Method to Determine Speed of Sound in Air

Theory

The resonance tube method uses standing waves in a gas column to measure the speed of sound. When a tuning fork vibrates at frequency $f$, it produces sound waves that travel down a tube partially filled with water. Resonance occurs when the length of the air column corresponds to an odd multiple of a quarter wavelength ($\frac{\lambda }{4},\frac{3\lambda }{4},\dots$).

The speed of sound is given by: $$v=2f\left(L_1-L_2\right)$$ where:

• $L_1$ = length of air column at first resonance,
• $L_2$ = length at second resonance,
• $f$ = frequency of the tuning fork.

Key Points:

1. End Correction ($e$): The open end of the tube does not behave as a perfect node; an end correction $e\approx 0.6r$ (where $r$ = tube radius) is added to the measured lengths.
2. Resonance condition: $$L_1+e=\frac{\lambda }{4},L_2+e=\frac{3\lambda }{4}$$ Subtracting gives: $$(L_1+e)-(L_2+e)=\frac{\lambda }{2}⟹\lambda =2(L_1-L_2)$$ Thus, $v=f\lambda =2f(L_1-L_2)$.

Experiment Setup

1. Apparatus: Resonance tube (glass tube with water reservoir), tuning fork (e.g., 512 Hz), ruler, soft board, water.
2. Procedure:
   • Fill the tube with water, leaving a small air column.
   • Strike the tuning fork and hold it near the tube’s open end.
   • Adjust the water level until loud resonance is heard (first resonance, $L_1$).
   • Lower the water further to find the second resonance ($L_2$).
   • Measure $L_1$ and $L_2$ using the ruler.
   • Calculate $v=2f(L_1-L_2)$.

Observations and Calculations

Assume:

• Frequency of tuning fork, $f=512$ Hz,
• Measured lengths: $L_1=16.5$ cm, $L_2=5.0$ cm,
• Tube radius $r=1.5$ cm → end correction $e=0.6\times 1.5=0.9$ cm.

Corrected lengths: $$L_1^{\prime }=L_1+e=16.5+0.9=17.4\text{ cm},$$ $$L_2^{\prime }=L_2+e=5.0+0.9=5.9\text{ cm}.$$

Wavelength: $$\lambda =2(L_1^{\prime }-L_2^{\prime })=2(17.4-5.9)=22.0\text{ cm}=0.22\text{ m}.$$

Speed of sound: $$v=f\lambda =512\times 0.22=112.64\text{ m/s}.$$ The slight discrepancy from the theoretical 343 m/s (at 20°C) is due to experimental errors (e.g., temperature, end correction approximation).