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
(a) `1:6`
(b) `1:8`
(c) `1:72`
(d) `1:3`

Options

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

Answer

To solve the problem, we need to find the amplitude ratio of two harmonic waves given their frequency and intensity ratios. Here’s a step-by-step solution:

### Step 1: Understand the given ratios
We are given:
- Frequency ratio: \( \frac{f_1}{f_2} = \frac{1}{2} \)
- Intensity ratio: \( \frac{I_1}{I_2} = \frac{1}{36} \)

### Step 2: Recall the relationship between intensity and amplitude
The intensity of a wave is related to the square of its amplitude. The relationship can be expressed as:
\[ I \propto A^2 \]
This means:
\[ \frac{I_1}{I_2} = \left(\frac{A_1}{A_2}\right)^2 \]

### Step 3: Substitute the intensity ratio into the equation
From the intensity ratio given:
\[ \frac{I_1}{I_2} = \frac{1}{36} \]
We can set this equal to the ratio of the squares of the amplitudes:
\[ \frac{1}{36} = \left(\frac{A_1}{A_2}\right)^2 \]

### Step 4: Solve for the amplitude ratio
Taking the square root of both sides gives:
\[ \frac{A_1}{A_2} = \sqrt{\frac{1}{36}} = \frac{1}{6} \]

### Step 5: Write the final answer
Thus, the amplitude ratio \( A_1 : A_2 \) is:
\[ A_1 : A_2 = 1 : 6 \]

### Conclusion
The correct answer is (a) \( 1:6 \).

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