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 1: Understand the Power of Waves
The power of a wave is given by the formula:
\[ P = \frac{1}{2} \mu \omega^2 A^2 v \]
where:
- \( P \) is the power,
- \( \mu \) is the linear mass density,
- \( \omega \) is the angular frequency,
- \( A \) is the amplitude,
- \( v \) is the velocity of the wave.

### Step 2: Set Up the Power Equations
For the incident wave in the first medium:
\[ P_i = \frac{1}{2} \mu_1 \omega^2 A_i^2 v_1 \]
For the reflected wave in the first medium:
\[ P_r = \frac{1}{2} \mu_1 \omega^2 A_r^2 v_1 \]
For the transmitted wave in the second medium:
\[ P_t = \frac{1}{2} \mu_2 \omega^2 A_t^2 v_2 \]

### Step 3: Apply the Energy Conservation Principle
According to the principle of energy conservation, the power of the incident wave must equal the sum of the powers of the reflected and transmitted waves:
\[ P_i = P_r + P_t \]
Substituting the expressions for power:
\[ \frac{1}{2} \mu_1 \omega^2 A_i^2 v_1 = \frac{1}{2} \mu_1 \omega^2 A_r^2 v_1 + \frac{1}{2} \mu_2 \omega^2 A_t^2 v_2 \]

### Step 4: Simplify the Equation
We can cancel out the common terms \( \frac{1}{2} \omega^2 \) from both sides:
\[ \mu_1 A_i^2 v_1 = \mu_1 A_r^2 v_1 + \mu_2 A_t^2 v_2 \]

### Step 5: Relate Densities to Velocities
Using the relationship between the linear mass density and wave velocity:
\[ \mu_1 = \frac{T}{v_1^2} \quad \text{and} \quad \mu_2 = \frac{T}{v_2^2} \]
Substituting these into the equation gives:
\[ \frac{T}{v_1^2} A_i^2 v_1 = \frac{T}{v_1^2} A_r^2 v_1 + \frac{T}{v_2^2} A_t^2 v_2 \]
Cancelling \( T \) and \( v_1 \) from both sides:
\[ \frac{A_i^2}{v_1} = \frac{A_r^2}{v_1} + \frac{A_t^2 v_2}{v_2^2} \]

### Step 6: Rearranging the Equation
Rearranging gives:
\[ A_i^2 = A_r^2 + \frac{A_t^2 v_1}{v_2} \]

### Step 7: Use the Relationship Between Amplitudes
We know that:
\[ A_i = A_r + A_t \]
Substituting this into the equation:
\[ (A_r + A_t)^2 = A_r^2 + \frac{A_t^2 v_1}{v_2} \]

### Step 8: Solve for \( A_r \) and \( A_t \)
From the equations derived, we can isolate \( A_r \) and \( A_t \):
1. From the equation \( A_i^2 = A_r^2 + A_t^2 \) and substituting \( A_t = A_i - A_r \), we can derive the expressions for \( A_r \) and \( A_t \).

### Final Relations
After substituting and simplifying, we arrive at:
\[ A_r = \frac{v_2 - v_1}{v_1 + v_2} A_i \]
\[ A_t = \frac{2v_2}{v_1 + v_2} A_i \]