The displacement `y` of a particle executing periodic motion is given by `y = 4 cos^(2) ((1)/(2)t) sin(1000t)`
This expression may be considered to be a result of the superposition of

Options

A. (a) two
B. ( b ) three
C. ( c ) four
D. (d) five

Explanation

To solve the problem step by step, we will analyze the given displacement equation and express it in terms of sine waves.

### Step 1: Write down the given displacement equation
The displacement of the particle is given by:
\[ y = 4 \cos^2\left(\frac{1}{2}t\right) \sin(1000t) \]

### Step 2: Use a trigonometric identity
We can use the trigonometric identity for cosine squared:
\[ \cos^2(\theta) = \frac{1 + \cos(2\theta)}{2} \]
Applying this identity to our equation, we set \(\theta = \frac{1}{2}t\):
\[ \cos^2\left(\frac{1}{2}t\right) = \frac{1 + \cos(t)}{2} \]

### Step 3: Substitute the identity into the displacement equation
Substituting the identity into the displacement equation gives:
\[ y = 4 \left(\frac{1 + \cos(t)}{2}\right) \sin(1000t) \]
This simplifies to:
\[ y = 2(1 + \cos(t)) \sin(1000t) \]

### Step 4: Distribute the sine term
Now, we can distribute \(\sin(1000t)\):
\[ y = 2\sin(1000t) + 2\cos(t)\sin(1000t) \]

### Step 5: Use the product-to-sum identities
We can use the product-to-sum identities for the term \(2\cos(t)\sin(1000t)\):
\[ 2\cos(A)\sin(B) = \sin(A + B) - \sin(A - B) \]
Here, \(A = t\) and \(B = 1000t\):
\[ 2\cos(t)\sin(1000t) = \sin(1001t) - \sin(999t) \]

### Step 6: Combine all terms
Now we can combine all the terms:
\[ y = 2\sin(1000t) + \sin(1001t) - \sin(999t) \]

### Step 7: Identify the superposition of waves
From the final expression, we can identify the waves present:
1. \( \sin(1001t) \)
2. \( \sin(1000t) \)
3. \( \sin(999t) \)

Thus, we have a total of **three waves** in the superposition.

### Final Answer
The expression may be considered to be a result of the superposition of **three waves**: \(\sin(1001t)\), \(\sin(1000t)\), and \(\sin(999t)\).

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