Sound waves are propagating in a medium. The moduli of isothermal and adiabatic elasticity of the medium are `E_(r) and E_(s)` respectively. The velocity of sound waves is proportional to -

Options

A. `(E_(s))/(E_(T))`
B. `sqrt(E_(r))`
C. `sqrt(E_(s))`
D. `E_(r)`

Answer

To determine the relationship between the velocity of sound waves and the moduli of isothermal and adiabatic elasticity in a medium, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Moduli of Elasticity**:
- The modulus of elasticity in the context of sound waves can be categorized into two types: isothermal elasticity (E_r) and adiabatic elasticity (E_s).
- For isothermal processes, the modulus of elasticity is represented as \( E_r = P \) (where P is the pressure).
- For adiabatic processes, the modulus of elasticity is represented as \( E_s = \gamma P \) (where \( \gamma \) is the adiabatic index).

2. **Velocity of Sound Formula**:
- The velocity of sound \( v \) in a medium can be expressed using the formula:
\[ v = \sqrt{\frac{E}{\rho}} \]
- Here, \( E \) is the modulus of elasticity (which can be either \( E_r \) or \( E_s \)), and \( \rho \) is the density of the medium.

3. **Applying the Adiabatic Modulus**:
- Since sound waves propagate through the medium as adiabatic processes, we will use the adiabatic modulus \( E_s \).
- Therefore, we can substitute \( E \) with \( E_s \) in the velocity formula:
\[ v = \sqrt{\frac{E_s}{\rho}} \]

4. **Proportionality of Velocity**:
- From the equation \( v = \sqrt{\frac{E_s}{\rho}} \), we can see that the velocity of sound \( v \) is proportional to the square root of the adiabatic modulus of elasticity:
\[ v \propto \sqrt{E_s} \]
- This implies that the velocity of sound is directly proportional to the square root of the adiabatic modulus of elasticity and inversely proportional to the square root of the density.

5. **Final Conclusion**:
- Therefore, we conclude that the velocity of sound waves is proportional to the square root of the adiabatic modulus of elasticity \( E_s \):
\[ v \propto \sqrt{E_s} \]