If the wave equation `y= 0.08 sin (2 pi)/(lambda) (200t-x)` then the velocity of the wave will be

Options

A. `400 sqrt(2)`
B. `200 sqrt(2)`
C. 400
D. 200

Answer

To find the velocity of the wave given by the equation \( y = 0.08 \sin\left(\frac{2\pi}{\lambda}(200t - x)\right) \), we can follow these steps:

### Step 1: Identify the wave equation format
The general 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 general form
From the given equation:
\[ y = 0.08 \sin\left(\frac{2\pi}{\lambda}(200t - x)\right) \]
we can identify:
- \( \omega = \frac{2\pi}{\lambda} \cdot 200 \)
- \( k = \frac{2\pi}{\lambda} \)

### Step 3: Calculate the angular frequency (\( \omega \))
From the equation, we can express \( \omega \):
\[ \omega = 200 \cdot \frac{2\pi}{\lambda} \]

### Step 4: Calculate the wave number (\( k \))
From the equation, we can express \( k \):
\[ k = \frac{2\pi}{\lambda} \]

### Step 5: Use the relationship between velocity, wave number, and angular frequency
The velocity \( v \) of the wave is given by the relationship:
\[ v = \frac{\omega}{k} \]

### Step 6: Substitute the values of \( \omega \) and \( k \)
Substituting the expressions for \( \omega \) and \( k \):
\[ v = \frac{200 \cdot \frac{2\pi}{\lambda}}{\frac{2\pi}{\lambda}} \]

### Step 7: Simplify the equation
The \( \frac{2\pi}{\lambda} \) terms cancel out:
\[ v = 200 \]

### Final Answer
The velocity of the wave is:
\[ v = 200 \, \text{m/s} \]

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