A harmonic wave is travelling on a stretched string . At any particular instant , the smallest distance between two particles having same displacement , equal to half of amplitude is `8 cm`. Find the smallest separation between two particles which have same values of displacement (magnitude only) equal to half of amplitude.

Options

A. (a)`8 cm`
B. (b)`24 cm`
C. (c)`12 cm`
D. (d)`4 cm`

Explanation

To solve the problem step by step, we will analyze the given information and apply the concepts of wave motion and harmonic waves.

### Step 1: Understand the given information
We know that the smallest distance between two particles having the same displacement equal to half of the amplitude is given as 8 cm. This means that the distance between two points on the wave where the displacement is \( \frac{A}{2} \) is 8 cm.

### Step 2: Relate the distance to the wavelength
In a harmonic wave, points that have the same displacement repeat at regular intervals, specifically at intervals of the wavelength \( \lambda \). The displacement of \( \frac{A}{2} \) occurs at specific positions along the wave. The distance between two consecutive points with the same displacement of \( \frac{A}{2} \) is \( \frac{\lambda}{3} \) because the sine function has the same value at multiple angles within one wavelength.

### Step 3: Set up the equation
From the problem, we have:
\[ \text{Distance} = \frac{\lambda}{3} = 8 \text{ cm} \]
From this, we can find the wavelength \( \lambda \):
\[ \lambda = 8 \times 3 = 24 \text{ cm} \]

### Step 4: Find the smallest separation for the same displacement
Now, we need to find the smallest separation between two particles that have the same displacement equal to \( \frac{A}{2} \). The smallest separation for the same displacement occurs at \( \frac{\lambda}{6} \) because the sine function will repeat its values every \( \frac{\lambda}{6} \) for the same amplitude.

### Step 5: Calculate the smallest separation
Using the wavelength we found:
\[ \text{Smallest separation} = \frac{\lambda}{6} = \frac{24 \text{ cm}}{6} = 4 \text{ cm} \]

### Final Answer
The smallest separation between two particles which have the same value of displacement equal to half of the amplitude is **4 cm**.

---