Determine resultant amplitude after super position of given four waves with help of phasor diagram. `y_(1) = 15 sin omega t mm,y_(2) = 9 sin (omega t -pi//2) mm,y_(3)=7 sin (omega t +pi//2)mm` & `y_(4) = 13 sin omega t mm`.

Explanation

To determine the resultant amplitude after the superposition of the given four waves using a phasor diagram, we can follow these steps:

### Step 1: Identify the Waves
We have the following four waves:
1. \( y_1 = 15 \sin(\omega t) \) mm
2. \( y_2 = 9 \sin(\omega t - \frac{\pi}{2}) \) mm
3. \( y_3 = 7 \sin(\omega t + \frac{\pi}{2}) \) mm
4. \( y_4 = 13 \sin(\omega t) \) mm

### Step 2: Represent Each Wave as a Phasor
- **Wave 1 (\(y_1\))**: This wave has an amplitude of 15 mm and is along the positive y-axis.
- **Wave 2 (\(y_2\))**: This wave has an amplitude of 9 mm and is at a phase of \(-\frac{\pi}{2}\) (downward direction).
- **Wave 3 (\(y_3\))**: This wave has an amplitude of 7 mm and is at a phase of \(+\frac{\pi}{2}\) (upward direction).
- **Wave 4 (\(y_4\))**: This wave has an amplitude of 13 mm and is in phase with \(y_1\) (along the positive y-axis).

### Step 3: Draw the Phasor Diagram
1. Draw the phasor for \(y_1\) (15 mm) vertically upwards.
2. Draw the phasor for \(y_2\) (9 mm) vertically downwards.
3. Draw the phasor for \(y_3\) (7 mm) vertically upwards.
4. Draw the phasor for \(y_4\) (13 mm) vertically upwards.

### Step 4: Combine the Phasors
- Combine \(y_1\) and \(y_4\):
- \(y_1 + y_4 = 15 + 13 = 28\) mm (upward).
- Combine \(y_2\) and \(y_3\):
- \(y_2 + y_3 = -9 + 7 = -2\) mm (downward).

### Step 5: Resultant Amplitude Calculation
Now we have two resultant phasors:
- Upward phasor: 28 mm
- Downward phasor: 2 mm

Now, we can find the resultant amplitude:
- The net upward phasor is \(28 - 2 = 26\) mm.

### Step 6: Final Result
The resultant amplitude after the superposition of the given four waves is:
\[ A_{\text{resultant}} = 26 \text{ mm} \]