The Laplace’s formula for sound waves is -

Options

A. `V = sqrt((P)/(rho))`
B. `V = sqrt((gammaP)/(rho))`
C. `V = sqrt((rho)/(gammaP))`
D. `V = sqrt((P)/(gammarho))`

Answer

To derive the Laplace's formula for sound waves, we will follow these steps:

### Step 1: Understand the Basic Formula
The speed of sound in a fluid (which can be air, water, or any liquid) is given by the formula:

\[ v = \sqrt{\frac{B}{\rho}} \]

where:
- \( v \) is the speed of sound,
- \( B \) is the bulk modulus of the fluid,
- \( \rho \) is the density of the fluid.

### Step 2: Define the Bulk Modulus
The bulk modulus \( B \) is defined as:

\[ B = -\frac{dP}{dV/V} \]

For small changes, this can be expressed as:

\[ B = -\frac{dP}{dB/B} \]

### Step 3: Consider the Adiabatic Process
Laplace assumed that the motion of sound in air is adiabatic. For an adiabatic process, we have:

\[ PV^\gamma = \text{constant} \]

where \( \gamma \) is the adiabatic index (ratio of specific heats).

### Step 4: Differentiate the Adiabatic Condition
We differentiate the adiabatic condition with respect to volume \( V \):

\[ \frac{dP}{dV} \cdot V^\gamma + P \cdot \gamma V^{\gamma-1} \cdot \frac{dV}{dV} = 0 \]

### Step 5: Rearranging the Equation
From the differentiation, we can rearrange to find \( \frac{dP}{dV} \):

\[ \frac{dP}{dV} = -\frac{P \cdot \gamma}{V} \]

### Step 6: Substitute into the Bulk Modulus Formula
Substituting this result into the bulk modulus formula, we have:

\[ B = -\frac{dP}{dV} = \frac{P \cdot \gamma}{V} \]

### Step 7: Substitute Bulk Modulus into the Speed of Sound Formula
Now, substituting \( B \) back into the speed of sound formula:

\[ v = \sqrt{\frac{B}{\rho}} = \sqrt{\frac{\gamma P}{\rho}} \]

### Final Result
Thus, the speed of sound in an adiabatic process can be expressed as:

\[ v = \sqrt{\frac{\gamma P}{\rho}} \]

This is the Laplace's formula for sound waves.

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