Two identical travelling waves, moving in the same direction are out of phase by `pi//2` rad. What is the amplitude of the resultant wave in terms of the common amplitude `y_(m)` of the two combining waves?

Explanation

To find the amplitude of the resultant wave when two identical travelling waves are out of phase by \( \frac{\pi}{2} \) radians, we can follow these steps:

### Step 1: Define the Amplitudes
Let the amplitude of each wave be denoted as \( y_m \). Therefore, we have:
- Amplitude of wave 1, \( A_1 = y_m \)
- Amplitude of wave 2, \( A_2 = y_m \)

### Step 2: Identify the Phase Difference
The phase difference between the two waves is given as:
- \( \Delta \phi = \frac{\pi}{2} \) radians

### Step 3: Use the Formula for Resultant Amplitude
The formula for the resultant amplitude \( A_R \) when two waves combine is given by:
\[ A_R^2 = A_1^2 + A_2^2 + 2 A_1 A_2 \cos(\Delta \phi) \]

### Step 4: Substitute the Values
Substituting the values of \( A_1 \), \( A_2 \), and \( \Delta \phi \) into the formula:
\[ A_R^2 = (y_m)^2 + (y_m)^2 + 2 (y_m)(y_m) \cos\left(\frac{\pi}{2}\right) \]

### Step 5: Calculate the Cosine
We know that:
\[ \cos\left(\frac{\pi}{2}\right) = 0 \]
Thus, the equation simplifies to:
\[ A_R^2 = y_m^2 + y_m^2 + 2 (y_m)(y_m)(0) \]
\[ A_R^2 = y_m^2 + y_m^2 + 0 \]
\[ A_R^2 = 2y_m^2 \]

### Step 6: Take the Square Root
Now, we take the square root of both sides to find \( A_R \):
\[ A_R = \sqrt{2y_m^2} = y_m \sqrt{2} \]

### Step 7: Final Result
Thus, the amplitude of the resultant wave in terms of the common amplitude \( y_m \) is:
\[ A_R = y_m \sqrt{2} \approx 1.41 y_m \]