Two harmonic waves travelling in the same medium have frequency in the ratio 1:2 and intensity in the ratio 1:36. Their amplitude ratio is:

Options

A. `1:6`
B. `1:8`
C. `1:72`
D. 1:3`

Explanation

To find the amplitude ratio of the two harmonic waves given their frequency and intensity ratios, we can follow these steps:

### Step 1: Understand the relationship between intensity, amplitude, and frequency
The intensity \( I \) of a wave is related to its amplitude \( A \) and frequency \( f \) by the formula:
\[ I \propto A^2 f^2 \]
This means that intensity is proportional to the square of the amplitude and the square of the frequency.

### Step 2: Set up the ratios
Given:
- Frequency ratio \( \frac{f_1}{f_2} = \frac{1}{2} \)
- Intensity ratio \( \frac{I_1}{I_2} = \frac{1}{36} \)

### Step 3: Express intensity in terms of amplitude and frequency
From the relationship stated above, we can express the intensities as:
\[ I_1 = k A_1^2 f_1^2 \]
\[ I_2 = k A_2^2 f_2^2 \]
where \( k \) is a constant that depends on the medium.

### Step 4: Substitute the frequency ratio into the intensity ratio
Using the frequency ratio \( \frac{f_1}{f_2} = \frac{1}{2} \), we can express \( f_2 \) as:
\[ f_2 = 2f_1 \]
Now substitute this into the intensity ratio:
\[ \frac{I_1}{I_2} = \frac{A_1^2 f_1^2}{A_2^2 (2f_1)^2} = \frac{A_1^2 f_1^2}{A_2^2 \cdot 4f_1^2} = \frac{A_1^2}{4A_2^2} \]

### Step 5: Set up the equation using the given intensity ratio
We know that:
\[ \frac{I_1}{I_2} = \frac{1}{36} \]
Thus, we can equate:
\[ \frac{A_1^2}{4A_2^2} = \frac{1}{36} \]

### Step 6: Solve for the amplitude ratio
Cross-multiplying gives:
\[ A_1^2 \cdot 36 = 4A_2^2 \]
Rearranging this gives:
\[ \frac{A_1^2}{A_2^2} = \frac{4}{36} = \frac{1}{9} \]
Taking the square root of both sides gives:
\[ \frac{A_1}{A_2} = \frac{1}{3} \]

### Final Answer
The amplitude ratio \( \frac{A_1}{A_2} \) is \( \frac{1}{3} \).

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