Equation of a progressive wave is given by `y= a sin pi [(t)/(2)-(x)/(4)]`, where t is in seconds and x is in meters. The distance through which the wave moves in 8 sec is (in meter)

Options

A. 8
B. 16
C. 2
D. 4

Answer

To solve the problem, we need to find the distance through which the wave moves in 8 seconds given the equation of the wave. The equation of the wave is:

\[ y = a \sin\left(\pi \left(\frac{t}{2} - \frac{x}{4}\right)\right) \]

### Step-by-Step Solution:

**Step 1: Identify the coefficients in the wave equation.**

The wave equation can be rewritten as:

\[ y = a \sin\left(\frac{\pi}{2} t - \frac{\pi}{4} x\right) \]

From this, we can identify:
- Coefficient of \( t \) (angular frequency) = \( \frac{\pi}{2} \)
- Coefficient of \( x \) (wave number) = \( \frac{\pi}{4} \)

**Step 2: Calculate the wave velocity.**

The wave velocity \( v \) can be calculated using the formula:

\[ v = \frac{\text{Coefficient of } t}{\text{Coefficient of } x} \]

Substituting the coefficients we identified:

\[ v = \frac{\frac{\pi}{2}}{\frac{\pi}{4}} \]

**Step 3: Simplify the expression for velocity.**

Now, simplifying the expression:

\[ v = \frac{\frac{\pi}{2}}{\frac{\pi}{4}} = \frac{\pi}{2} \times \frac{4}{\pi} = 2 \times 4 = 4 \text{ m/s} \]

**Step 4: Calculate the distance traveled in 8 seconds.**

Using the formula for distance \( d \):

\[ d = v \times t \]

Substituting the values we have:

\[ d = 4 \text{ m/s} \times 8 \text{ s} = 32 \text{ m} \]

### Final Answer:

The distance through which the wave moves in 8 seconds is **32 meters**.

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