Ratio of amplitude for two wave is 1:2 .Find the ratio of max:min Intensity?

Answer

To solve the problem step by step, we will follow the concepts of wave amplitudes and their relationship to intensity.

### Step 1: Understand the relationship between amplitude and intensity
The intensity (I) of a wave is proportional to the square of its amplitude (A). Mathematically, this can be expressed as:
\[ I \propto A^2 \]

### Step 2: Write down the given data
We are given the ratio of the amplitudes of two waves:
\[ \frac{A_1}{A_2} = \frac{1}{2} \]

### Step 3: Express the intensities in terms of amplitudes
Using the relationship between intensity and amplitude, we can express the intensities of the two waves:
\[ I_1 \propto A_1^2 \]
\[ I_2 \propto A_2^2 \]

From the ratio of amplitudes:
\[ A_1 = k \quad \text{and} \quad A_2 = 2k \]
for some constant \( k \).

Now, substituting these into the intensity expressions:
\[ I_1 = k^2 \]
\[ I_2 = (2k)^2 = 4k^2 \]

### Step 4: Find the ratio of intensities
Now we can find the ratio of the intensities:
\[ \frac{I_1}{I_2} = \frac{k^2}{4k^2} = \frac{1}{4} \]

### Step 5: Calculate maximum and minimum intensity
The maximum intensity \( I_{max} \) and minimum intensity \( I_{min} \) can be calculated using the following formulas:
- Maximum intensity:
\[ I_{max} = (\sqrt{I_1} + \sqrt{I_2})^2 \]
- Minimum intensity:
\[ I_{min} = (\sqrt{I_1} - \sqrt{I_2})^2 \]

Calculating \( I_{max} \):
\[ \sqrt{I_1} = \sqrt{k^2} = k \]
\[ \sqrt{I_2} = \sqrt{4k^2} = 2k \]
Thus,
\[ I_{max} = (k + 2k)^2 = (3k)^2 = 9k^2 \]

Calculating \( I_{min} \):
\[ I_{min} = (k - 2k)^2 = (-k)^2 = k^2 \]

### Step 6: Find the ratio of maximum to minimum intensity
Now we can find the ratio of maximum intensity to minimum intensity:
\[ \frac{I_{max}}{I_{min}} = \frac{9k^2}{k^2} = 9 \]

### Final Answer
The ratio of maximum intensity to minimum intensity is:
\[ \frac{I_{max}}{I_{min}} = 9:1 \]

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