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

Answer

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

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

### Step 2: Compare the given equation with the standard form
From the given equation \( y = 2 \sin(\pi(0.5x - 200t)) \):
- Amplitude \( A = 2 \)
- The term inside the sine function can be rewritten as:
\[ \pi(0.5x - 200t) = \pi \cdot 0.5x - \pi \cdot 200t \]
This means:
- \( k = \pi \cdot 0.5 \)
- \( \omega = \pi \cdot 200 \)

### Step 3: Calculate the wave number \( k \) and angular frequency \( \omega \)
From the above comparisons:
- \( k = \frac{2\pi}{\lambda} \) implies:
\[ \lambda = \frac{2\pi}{k} = \frac{2\pi}{\pi \cdot 0.5} = \frac{2}{0.5} = 4 \text{ cm} \]

- \( \omega = 2\pi f \) implies:
\[ f = \frac{\omega}{2\pi} = \frac{200\pi}{2\pi} = 100 \text{ Hz} \]

### Step 4: Calculate the wave velocity \( v \)
The wave velocity \( v \) is given by the formula:
\[ v = f \cdot \lambda \]
Substituting the values we found:
\[ v = 100 \text{ Hz} \cdot 4 \text{ cm} = 400 \text{ cm/s} \]

### Step 5: Convert the wave velocity to meters per second (if needed)
To convert from cm/s to m/s:
\[ v = 400 \text{ cm/s} = 4 \text{ m/s} \]

### Final Answer
The wave velocity is \( 400 \text{ cm/s} \) or \( 4 \text{ m/s} \).

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