The equation of a wave is `y=2 sin pi (0.5x- 200 t)`. Where x and y are expressed in cm and t in sec. The wave velocity is

Options

A. 100 cm /sec
B. 200 cm / sec
C. 300 cm sec
D. 400 cm / sec

Answer

To find the wave velocity from the given wave equation \( y = 2 \sin(\pi(0.5x - 200t)) \), we will follow these steps:

### Step 1: Identify the wave equation format
The general form of a wave equation is given by:
\[ y = a \sin(kx - \omega t) \]
where:
- \( a \) is the amplitude,
- \( k \) is the wave number,
- \( \omega \) is the angular frequency.

### Step 2: Compare the given equation with the general form
From the given wave equation:
\[ y = 2 \sin(\pi(0.5x - 200t)) \]
we can rewrite it as:
\[ y = 2 \sin\left(\pi \cdot 0.5 x - 200 \pi t\right) \]
This allows us to identify:
- Amplitude \( a = 2 \)
- Wave number \( k = \pi \cdot 0.5 = 0.5\pi \)
- Angular frequency \( \omega = 200\pi \)

### Step 3: Calculate the wave velocity
The wave velocity \( v \) can be calculated using the relationship:
\[ v = \frac{\omega}{k} \]
Substituting the values of \( \omega \) and \( k \):
\[ v = \frac{200\pi}{0.5\pi} \]

### Step 4: Simplify the expression
Now we can simplify the expression:
\[ v = \frac{200\pi}{0.5\pi} = \frac{200}{0.5} = 200 \times 2 = 400 \text{ cm/s} \]

### Conclusion
Thus, the wave velocity is:
\[ \boxed{400 \text{ cm/s}} \]

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