In terms of T, `mu` and l, find frequency of
(a) fourth overtone mode
(b) third harmonic .

Explanation

To solve the problem of finding the frequency in terms of T (tension), μ (linear mass density), and l (length of the string) for the fourth overtone mode and the third harmonic, we can follow these steps:

### Step-by-Step Solution:

#### Part (a): Frequency of the Fourth Overtone Mode

1. **Understand the Concept of Overtones**:
- The nth overtone corresponds to the (n+1)th harmonic. Therefore, the 4th overtone is the 5th harmonic.

2. **Identify the Value of n**:
- For the 4th overtone, we have:
\[ n = 5 \]

3. **Use the Frequency Formula**:
- The formula for the frequency \( f \) of a vibrating string is given by:
\[ f = \frac{n}{2l} \sqrt{\frac{T}{\mu}} \]

4. **Substitute the Value of n**:
- Substitute \( n = 5 \) into the frequency formula:
\[ f = \frac{5}{2l} \sqrt{\frac{T}{\mu}} \]

5. **Final Expression for the Fourth Overtone**:
- Thus, the frequency of the fourth overtone mode is:
\[ f = \frac{5}{2l} \sqrt{\frac{T}{\mu}} \]

#### Part (b): Frequency of the Third Harmonic

1. **Identify the Value of n**:
- For the third harmonic, we have:
\[ n = 3 \]

2. **Use the Frequency Formula**:
- Again, we use the same frequency formula:
\[ f = \frac{n}{2l} \sqrt{\frac{T}{\mu}} \]

3. **Substitute the Value of n**:
- Substitute \( n = 3 \) into the frequency formula:
\[ f = \frac{3}{2l} \sqrt{\frac{T}{\mu}} \]

4. **Final Expression for the Third Harmonic**:
- Thus, the frequency of the third harmonic is:
\[ f = \frac{3}{2l} \sqrt{\frac{T}{\mu}} \]

### Summary of Results:
- **Frequency of the Fourth Overtone Mode**:
\[ f = \frac{5}{2l} \sqrt{\frac{T}{\mu}} \]
- **Frequency of the Third Harmonic**:
\[ f = \frac{3}{2l} \sqrt{\frac{T}{\mu}} \]