Two waves of equal frequencies have their amplitudes in the ratio of 3:5. They are superimposed on each other. Calculate the ratio of maximum and minimum intensities of the resultant wave.

Explanation

To solve the problem step by step, we will calculate the ratio of maximum and minimum intensities of the resultant wave formed by the superposition of two waves with given amplitudes.

### Step 1: Understand the relationship between amplitude and intensity
The intensity \( I \) of a wave is proportional to the square of its amplitude \( A \):
\[ I \propto A^2 \]

### Step 2: Define the amplitudes of the two waves
Let the amplitudes of the two waves be \( A_1 \) and \( A_2 \). According to the problem, the ratio of their amplitudes is given as:
\[ \frac{A_1}{A_2} = \frac{3}{5} \]
We can express this as:
\[ A_1 = 3k \quad \text{and} \quad A_2 = 5k \]
for some constant \( k \).

### Step 3: Calculate the intensities of the two waves
Using the relationship between intensity and amplitude, we can express the intensities \( I_1 \) and \( I_2 \) as:
\[ I_1 = A_1^2 = (3k)^2 = 9k^2 \]
\[ I_2 = A_2^2 = (5k)^2 = 25k^2 \]

### Step 4: Find the maximum intensity
The maximum intensity \( I_{\text{max}} \) occurs when the two waves are in phase (i.e., when the phase difference \( \phi = 0 \)):
\[ I_{\text{max}} = I_1 + I_2 + 2\sqrt{I_1 I_2} \]
Substituting the values of \( I_1 \) and \( I_2 \):
\[ I_{\text{max}} = 9k^2 + 25k^2 + 2\sqrt{9k^2 \cdot 25k^2} \]
Calculating the square root:
\[ \sqrt{9k^2 \cdot 25k^2} = \sqrt{225k^4} = 15k^2 \]
Thus, we have:
\[ I_{\text{max}} = 9k^2 + 25k^2 + 30k^2 = 64k^2 \]

### Step 5: Find the minimum intensity
The minimum intensity \( I_{\text{min}} \) occurs when the two waves are out of phase (i.e., when the phase difference \( \phi = \pi \)):
\[ I_{\text{min}} = I_1 + I_2 - 2\sqrt{I_1 I_2} \]
Substituting the values of \( I_1 \) and \( I_2 \):
\[ I_{\text{min}} = 9k^2 + 25k^2 - 2\sqrt{9k^2 \cdot 25k^2} \]
Using the previously calculated square root:
\[ I_{\text{min}} = 9k^2 + 25k^2 - 30k^2 = 4k^2 \]

### Step 6: Calculate the ratio of maximum to minimum intensity
Now, we can find the ratio of maximum intensity to minimum intensity:
\[ \frac{I_{\text{max}}}{I_{\text{min}}} = \frac{64k^2}{4k^2} = 16 \]

### Final Answer
The ratio of maximum and minimum intensities of the resultant wave is:
\[ \boxed{16:1} \]