Three one-dimensional mechanical waves in an elastic medium is given as
`y_1 = 3A sin (omegat - kx), y_2 = A sin (omegat - kx + pi) and y_3 = 2A sin (omegat + kx)`
are superimposed with each other. The maximum displacement amplitude of the medium particle would be

Options

A. 4A
B. 3A
C. 2A
D. A

Explanation

To find the maximum displacement amplitude of the medium particle when three one-dimensional mechanical waves are superimposed, we can follow these steps:

### Step 1: Identify the given waves
The three waves are:
1. \( y_1 = 3A \sin(\omega t - kx) \)
2. \( y_2 = A \sin(\omega t - kx + \pi) \)
3. \( y_3 = 2A \sin(\omega t + kx) \)

### Step 2: Simplify \( y_2 \)
The wave \( y_2 \) can be simplified using the identity \( \sin(x + \pi) = -\sin(x) \):
\[ y_2 = A \sin(\omega t - kx + \pi) = -A \sin(\omega t - kx) \]

### Step 3: Combine \( y_1 \) and \( y_2 \)
Now, we can find the resultant wave \( y_4 \) from \( y_1 \) and \( y_2 \):
\[ y_4 = y_1 + y_2 = 3A \sin(\omega t - kx) - A \sin(\omega t - kx) = (3A - A) \sin(\omega t - kx) = 2A \sin(\omega t - kx) \]

### Step 4: Combine \( y_4 \) and \( y_3 \)
Next, we need to find the resultant displacement by combining \( y_4 \) and \( y_3 \):
\[ y_3 = 2A \sin(\omega t + kx) \]
Thus, the total displacement \( y \) is:
\[ y = y_4 + y_3 = 2A \sin(\omega t - kx) + 2A \sin(\omega t + kx) \]

### Step 5: Use the principle of superposition
Using the principle of superposition, we can express the sum of two sine waves:
\[ y = 2A \left( \sin(\omega t - kx) + \sin(\omega t + kx) \right) \]
Using the identity for the sum of sine functions:
\[ \sin a + \sin b = 2 \sin\left(\frac{a+b}{2}\right) \cos\left(\frac{a-b}{2}\right) \]
we can rewrite:
\[ y = 2A \cdot 2 \sin\left(\omega t\right) \cos(kx) = 4A \sin(\omega t) \cos(kx) \]

### Step 6: Determine the maximum displacement amplitude
The maximum displacement amplitude of the resultant wave is given by the coefficient of the sine function:
\[ \text{Maximum Displacement Amplitude} = 4A \]

### Conclusion
Thus, the maximum displacement amplitude of the medium particle when the three waves are superimposed is \( 4A \).

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