The ratio of the velocity of a wave in two media (1) and (2) is 2:1 and the ratio of their density is 3:1. When the wave enters the second medium the intensity is reduced to half of its value in the first medium. Find the ratio of the amplitude of the wave in the two media ?

Answer

To find the ratio of the amplitude of the wave in two media, we will use the relationship between intensity, amplitude, density, and velocity of the wave. Let's go through the solution step by step.

### Step 1: Understand the given ratios
We are given the following ratios:
- The ratio of the velocities of the wave in two media: \( V_1 : V_2 = 2 : 1 \)
- The ratio of their densities: \( \rho_1 : \rho_2 = 3 : 1 \)
- The intensity in the second medium is half of that in the first medium: \( I_1 : I_2 = 2 : 1 \)

### Step 2: Write the formula for intensity
The intensity \( I \) of a wave can be expressed as:
\[ I = 2 \pi^2 A^2 n^2 \rho V \]
where:
- \( A \) is the amplitude,
- \( n \) is the frequency,
- \( \rho \) is the density of the medium,
- \( V \) is the velocity of the wave.

### Step 3: Set up the intensity equations for both media
For medium 1:
\[ I_1 = 2 \pi^2 A_1^2 n^2 \rho_1 V_1 \]
For medium 2:
\[ I_2 = 2 \pi^2 A_2^2 n^2 \rho_2 V_2 \]

### Step 4: Use the ratio of intensities
From the problem, we know:
\[ \frac{I_1}{I_2} = \frac{2}{1} \]
Substituting the expressions for \( I_1 \) and \( I_2 \):
\[ \frac{2 \pi^2 A_1^2 n^2 \rho_1 V_1}{2 \pi^2 A_2^2 n^2 \rho_2 V_2} = 2 \]
This simplifies to:
\[ \frac{A_1^2 \rho_1 V_1}{A_2^2 \rho_2 V_2} = 2 \]

### Step 5: Substitute the known ratios
We know:
- \( \frac{V_1}{V_2} = 2 \) (implying \( V_1 = 2V_2 \))
- \( \frac{\rho_1}{\rho_2} = 3 \) (implying \( \rho_1 = 3\rho_2 \))

Substituting these into the intensity ratio equation gives:
\[ \frac{A_1^2 (3\rho_2) (2V_2)}{A_2^2 \rho_2 V_2} = 2 \]
This simplifies to:
\[ \frac{A_1^2 \cdot 6}{A_2^2} = 2 \]

### Step 6: Solve for the ratio of amplitudes
Rearranging gives:
\[ A_1^2 = \frac{2}{6} A_2^2 \]
\[ A_1^2 = \frac{1}{3} A_2^2 \]
Taking the square root of both sides:
\[ \frac{A_1}{A_2} = \frac{1}{\sqrt{3}} \]

### Final Answer
Thus, the ratio of the amplitudes of the wave in the two media is:
\[ A_1 : A_2 = 1 : \sqrt{3} \]