Ratio of amplitude for two wave is 6:7 .Find the ratio of intensity?

Answer

To solve the problem, we need to find the ratio of intensities of two waves given the ratio of their amplitudes. Here’s a step-by-step solution:

### Step 1: Write down the given data
We are given that the ratio of the amplitudes of two waves is:
\[ \frac{A_1}{A_2} = \frac{6}{7} \]

### Step 2: Understand the relationship between intensity and amplitude
The intensity \(I\) of a wave is directly proportional to the square of its amplitude. This can be expressed mathematically as:
\[ I \propto A^2 \]
This means that the ratio of the intensities of the two waves can be expressed in terms of the ratio of their amplitudes:
\[ \frac{I_1}{I_2} = \left(\frac{A_1}{A_2}\right)^2 \]

### Step 3: Substitute the given ratio of amplitudes into the intensity ratio formula
Using the ratio of amplitudes we have:
\[ \frac{I_1}{I_2} = \left(\frac{6}{7}\right)^2 \]

### Step 4: Calculate the square of the ratio
Now we will calculate the square of \(\frac{6}{7}\):
\[ \frac{I_1}{I_2} = \frac{6^2}{7^2} = \frac{36}{49} \]

### Step 5: State the final result
Thus, the ratio of the intensities of the two waves is:
\[ \frac{I_1}{I_2} = \frac{36}{49} \]