A clamped string is oscillating in `n^(th)` harmonic, then :-

A. Total energy of oscilations will be `n^(2)` times that of fundamental frequency
B. Total energy of oscillations will be `(n-1)^(2)` times that of fundamental frequency
C. Average kinetic energy of the string over a complete oscillations is half of that the total energy of the string
D. None of these

Options

A. Total energy of oscilations will be `n^(2)` times that of fundamental frequency
B. Total energy of oscillations will be `(n-1)^(2)` times that of fundamental frequency
C. Average kinetic energy of the string over a complete oscillations is half of that the total energy of the string
D. None of these

Explanation

To solve the problem, we need to analyze the oscillation of a clamped string in the nth harmonic and determine the relationships between the total energy and the fundamental frequency.

### Step-by-Step Solution:

1. **Understanding the Total Energy of the String**:
The total energy \( W_n \) of a wave on a clamped string is given by the formula:
\[ W_n = \frac{1}{2} \rho \omega^2 A^2 \]
where \( \rho \) is the linear mass density, \( \omega \) is the angular frequency, and \( A \) is the amplitude of the wave.

2. **Relating Angular Frequency to Frequency**:
The angular frequency \( \omega_n \) for the nth harmonic is related to the frequency \( f_n \) by:
\[ \omega_n = 2 \pi f_n \]

3. **Finding the Frequency of the nth Harmonic**:
For a clamped string, the frequency of the nth harmonic is given by:
\[ f_n = n f_1 \]
where \( f_1 \) is the fundamental frequency.

4. **Substituting Frequency into the Energy Formula**:
Substituting \( f_n \) into the energy formula gives:
\[ W_n = \frac{1}{2} \rho (2 \pi f_n)^2 A^2 = \frac{1}{2} \rho (2 \pi n f_1)^2 A^2 \]
Simplifying this, we get:
\[ W_n = \frac{1}{2} \rho (4 \pi^2 n^2 f_1^2) A^2 \]

5. **Finding the Total Energy for the Fundamental Frequency**:
The total energy for the fundamental frequency (1st harmonic) is:
\[ W_1 = \frac{1}{2} \rho (2 \pi f_1)^2 A^2 = \frac{1}{2} \rho (4 \pi^2 f_1^2) A^2 \]

6. **Comparing Energies**:
Now, we can express the total energy of the nth harmonic in terms of the fundamental frequency:
\[ W_n = n^2 W_1 \]
This shows that the total energy in the nth harmonic is \( n^2 \) times that of the fundamental frequency.

7. **Average Kinetic Energy**:
The total energy of the string is equally divided between kinetic and potential energy in a complete oscillation. Thus, the average kinetic energy \( K \) is:
\[ K = \frac{1}{2} W_n \]

8. **Conclusion**:
From the analysis, we find:
- Option A is correct: Total energy of oscillations will be \( n^2 \) times that of the fundamental frequency.
- Option C is also correct: Average kinetic energy of the string over a complete oscillation is half of the total energy of the string.
- Options B and D are incorrect.

### Final Answer:
- A. Total energy of oscillations will be \( n^2 \) times that of fundamental frequency.
- C. Average kinetic energy of the string over a complete oscillation is half of that of the total energy of the string.