Speed of sound waves in a fluid depends

Options

A. directly on density of the medium
B. square of Bulk modulus of the medium
C. inversly on the square root of density
D. directly on the square root of bulk modulus of the medium

Explanation

To determine how the speed of sound waves in a fluid depends on various factors, we can analyze the relationship using the formula for the speed of sound in a medium.

### Step-by-Step Solution:

1. **Understanding the Speed of Sound**:
The speed of sound in a fluid is influenced by the properties of the medium through which it travels. The primary factors are the elasticity (bulk modulus) and the density of the fluid.

2. **Formula for Speed of Sound**:
The speed of sound \( v \) in a fluid can be expressed as:
\[ v = \sqrt{\frac{E}{\rho}} \]
where \( E \) is the elasticity (bulk modulus) of the fluid, and \( \rho \) is the density of the fluid.

3. **Elasticity (Bulk Modulus)**:
For fluids, the elasticity is represented by the bulk modulus \( B \), which is defined as:
\[ B = \frac{\text{Pressure}}{\text{Volumetric Strain}} = \frac{P}{\Delta V/V} \]
This indicates how incompressible a fluid is; higher bulk modulus means the fluid is less compressible, which allows sound to travel faster.

4. **Density**:
The density \( \rho \) of the fluid is the mass per unit volume. The relationship indicates that as the density increases, the speed of sound decreases.

5. **Proportional Relationships**:
- The speed of sound is **directly proportional** to the square root of the bulk modulus:
\[ v \propto \sqrt{B} \]
- The speed of sound is **inversely proportional** to the square root of the density:
\[ v \propto \frac{1}{\sqrt{\rho}} \]

6. **Conclusion**:
From the above relationships, we can conclude:
- The speed of sound increases with an increase in the bulk modulus.
- The speed of sound decreases with an increase in density.

### Final Answer:
The speed of sound waves in a fluid depends on:
- Directly on the square root of the bulk modulus of the medium.
- Inversely on the square root of the density of the medium.