Ratio of amplitude for two wave is 1:3 .Find the ratio of maximum intensity to minimum intensity.

Explanation

To solve the problem, we need to find the ratio of maximum intensity to minimum intensity for two waves with a given amplitude ratio of 1:3. Here’s a step-by-step solution:

### Step 1: Understand the relationship between amplitude and intensity
The intensity (I) of a wave is proportional to the square of its amplitude (A). Therefore, if the amplitudes of two waves are A1 and A2, the ratio of their intensities can be expressed as:
\[ I_1 \propto A_1^2 \]
\[ I_2 \propto A_2^2 \]

### Step 2: Given data
We are given that the ratio of amplitudes is:
\[ \frac{A_1}{A_2} = \frac{1}{3} \]

### Step 3: Calculate the ratio of intensities
Using the relationship between intensity and amplitude, we can find the ratio of intensities:
\[ \frac{I_1}{I_2} = \left(\frac{A_1}{A_2}\right)^2 = \left(\frac{1}{3}\right)^2 = \frac{1}{9} \]

### Step 4: Identify I1 and I2
From the ratio we found:
Let \( I_1 = 1 \) and \( I_2 = 9 \).

### Step 5: Calculate maximum intensity (Imax)
The maximum intensity for two waves can be calculated using the formula:
\[ I_{max} = (\sqrt{I_1} + \sqrt{I_2})^2 \]
Substituting the values:
\[ I_{max} = (\sqrt{1} + \sqrt{9})^2 = (1 + 3)^2 = 4^2 = 16 \]

### Step 6: Calculate minimum intensity (Imin)
The minimum intensity can be calculated using the formula:
\[ I_{min} = (\sqrt{I_1} - \sqrt{I_2})^2 \]
Substituting the values:
\[ I_{min} = (\sqrt{1} - \sqrt{9})^2 = (1 - 3)^2 = (-2)^2 = 4 \]

### Step 7: Calculate the ratio of maximum intensity to minimum intensity
Now, we can find the ratio of maximum intensity to minimum intensity:
\[ \frac{I_{max}}{I_{min}} = \frac{16}{4} = 4 \]

### Final Answer
The ratio of maximum intensity to minimum intensity is:
\[ \frac{I_{max}}{I_{min}} = 4 \]

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