From energy conservation principle prove the relations,
`A_r = ((v_2-v_1)/(v_1 +v_2))A_i` and `A_t = ((2v_2)/(v_1 +v_2))A_i`
Here, symbols have their usual meanings.

Explanation

To prove the relations \( A_r = \frac{(v_2 - v_1)}{(v_1 + v_2)} A_i \) and \( A_t = \frac{2v_2}{(v_1 + v_2)} A_i \) using the principle of energy conservation, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Power Relationship**:
The average power of the incident wave \( P_i \) is equal to the sum of the average power of the reflected wave \( P_r \) and the average power of the transmitted wave \( P_t \). This can be expressed as:
\[ P_i = P_r + P_t \]

2. **Write the Power Formulas**:
The average power for a wave can be expressed as:
\[ P = \frac{1}{2} \mu \omega^2 A^2 v \]
where \( \mu \) is the linear mass density, \( \omega \) is the angular frequency, \( A \) is the amplitude, and \( v \) is the wave speed.

3. **Express Powers in Terms of Amplitudes**:
For the incident wave:
\[ P_i = \frac{1}{2} \mu_1 \omega^2 A_i^2 v_1 \]
For the reflected wave:
\[ P_r = \frac{1}{2} \mu_1 \omega^2 A_r^2 v_1 \]
For the transmitted wave:
\[ P_t = \frac{1}{2} \mu_2 \omega^2 A_t^2 v_2 \]

4. **Set Up the Energy Conservation Equation**:
From the conservation of energy, we can write:
\[ \frac{1}{2} \mu_1 \omega^2 A_i^2 v_1 = \frac{1}{2} \mu_1 \omega^2 A_r^2 v_1 + \frac{1}{2} \mu_2 \omega^2 A_t^2 v_2 \]
We can cancel out \( \frac{1}{2} \) and \( \omega^2 \) from all terms:
\[ \mu_1 A_i^2 v_1 = \mu_1 A_r^2 v_1 + \mu_2 A_t^2 v_2 \]

5. **Substitute Mass Densities**:
The mass densities can be related to wave speeds:
\[ \mu_1 = \frac{T}{v_1}, \quad \mu_2 = \frac{T}{v_2} \]
Substituting these into the equation gives:
\[ \frac{T}{v_1} A_i^2 v_1 = \frac{T}{v_1} A_r^2 v_1 + \frac{T}{v_2} A_t^2 v_2 \]
Cancelling \( T \) and \( v_1 \):
\[ A_i^2 = A_r^2 + \frac{v_1}{v_2} A_t^2 \]

6. **Relate Amplitudes**:
Using the relationship \( A_r = A_t - A_i \):
Substitute \( A_r \) into the equation:
\[ A_i^2 = (A_t - A_i)^2 + \frac{v_1}{v_2} A_t^2 \]
Expanding and simplifying gives:
\[ A_i^2 = A_t^2 - 2A_t A_i + A_i^2 + \frac{v_1}{v_2} A_t^2 \]
Rearranging leads to:
\[ 2A_t A_i = A_t^2\left(1 + \frac{v_1}{v_2}\right) \]

7. **Solve for \( A_t \)**:
Rearranging gives:
\[ A_t = \frac{2A_i v_2}{v_1 + v_2} \]
Thus, we have proved:
\[ A_t = \frac{2v_2}{v_1 + v_2} A_i \]

8. **Solve for \( A_r \)**:
Now substituting \( A_t \) back into the equation for \( A_r \):
\[ A_r = A_t - A_i = \frac{2v_2}{v_1 + v_2} A_i - A_i \]
This simplifies to:
\[ A_r = \left(\frac{2v_2 - v_1 - v_2}{v_1 + v_2}\right) A_i = \frac{(v_2 - v_1)}{(v_1 + v_2)} A_i \]

### Conclusion:
Thus, we have proved both relations:
\[ A_r = \frac{(v_2 - v_1)}{(v_1 + v_2)} A_i \quad \text{and} \quad A_t = \frac{2v_2}{(v_1 + v_2)} A_i \]