The speed of a sound wave depends on its wavelength `lamda`, and frequency v. Find an expression for the spped of sound.

Answer

To find the expression for the speed of sound in terms of its wavelength (λ) and frequency (μ), we can follow these steps:

### Step-by-Step Solution:

1. **Define the Variables**:
- Let the speed of sound be denoted as \( v \).
- Let the wavelength be denoted as \( \lambda \).
- Let the frequency be denoted as \( \mu \).

2. **Establish the Relationship**:
- According to the problem, the speed of sound depends on its wavelength and frequency. We can express this relationship as:
\[ v \propto \lambda^A \mu^B \]
- Here, \( A \) and \( B \) are constants that we need to determine.

3. **Introduce a Constant of Proportionality**:
- To eliminate the proportionality, we introduce a constant \( K \):
\[ v = K \lambda^A \mu^B \]

4. **Dimensional Analysis**:
- We need to analyze the dimensions of each term:
- The dimension of speed \( v \) is \( [L T^{-1}] \).
- The dimension of wavelength \( \lambda \) is \( [L] \).
- The dimension of frequency \( \mu \) is \( [T^{-1}] \).

5. **Write the Dimensional Equation**:
- Substituting the dimensions into the equation gives us:
\[ [L T^{-1}] = [L^A] [T^{-B}] \]
- This can be rewritten as:
\[ L^1 T^{-1} = L^A T^{-B} \]

6. **Equate the Exponents**:
- By equating the exponents of \( L \) and \( T \) from both sides, we get two equations:
- For \( L \): \( 1 = A \)
- For \( T \): \( -1 = -B \) which simplifies to \( B = 1 \)

7. **Substitute the Values of A and B**:
- Now that we have \( A = 1 \) and \( B = 1 \), we can substitute these values back into our equation:
\[ v = K \lambda^1 \mu^1 \]
- This simplifies to:
\[ v = K \lambda \mu \]

8. **Final Expression**:
- The expression for the speed of sound can be written as:
\[ v = K \lambda \mu \]
- If we consider \( K \) to be a constant that can be taken as 1 for simplicity in this context, we can express the speed of sound as:
\[ v = \lambda \mu \]

### Final Answer:
The expression for the speed of sound is:
\[ v = \lambda \mu \]