Four waves are to be sent along the same string, in the same direction :
`y_(1) (x,t)=(5.00 mm) sin (4pi x-400pit)`
`y_(2) (x,t) =(5.00 mm) sin (4pi x-400 pit+0.8pi)`
`y_(3) (x,t) =(5.00 mm) sin (4pi x-400pit+pi)`
`y_(4) (x,t)=(5.00 mm) sin (4pi x-400 pit+1.8pi)`
What is the amplitude of the resultant wave?

Answer

To find the amplitude of the resultant wave formed by the four given waves, we can follow these steps:

### Step 1: Identify the given waves
The four waves are:
1. \( y_1(x,t) = (5.00 \, \text{mm}) \sin(4\pi x - 400\pi t) \)
2. \( y_2(x,t) = (5.00 \, \text{mm}) \sin(4\pi x - 400\pi t + 0.8\pi) \)
3. \( y_3(x,t) = (5.00 \, \text{mm}) \sin(4\pi x - 400\pi t + \pi) \)
4. \( y_4(x,t) = (5.00 \, \text{mm}) \sin(4\pi x - 400\pi t + 1.8\pi) \)

### Step 2: Analyze the phase shifts
The phase shifts for each wave are:
- Wave 1: \( 0 \)
- Wave 2: \( 0.8\pi \)
- Wave 3: \( \pi \)
- Wave 4: \( 1.8\pi \)

### Step 3: Convert phase shifts to sine values
Using the sine function:
- \( \sin(0) = 0 \)
- \( \sin(0.8\pi) = \sin(144^\circ) \approx 0.5878 \)
- \( \sin(\pi) = 0 \)
- \( \sin(1.8\pi) = \sin(324^\circ) \approx -0.5878 \)

### Step 4: Calculate the resultant amplitude
The resultant amplitude \( A \) can be calculated using the formula for the sum of waves with the same amplitude but different phases:
\[ A = A_0 \left( \sin(0) + \sin(0.8\pi) + \sin(\pi) + \sin(1.8\pi) \right) \]
Substituting the sine values:
\[ A = 5.00 \, \text{mm} \left( 0 + 0.5878 + 0 - 0.5878 \right) \]
\[ A = 5.00 \, \text{mm} \cdot 0 = 0 \, \text{mm} \]

### Step 5: Conclusion
The amplitude of the resultant wave is \( 0 \, \text{mm} \).

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