The displacement wave in a string is `y=(3 cm)sin6.28(0.5x-50t)` where x is in centimetres and t in seconds. The velocity and wavelength of the wave is :-

Options

A. 2 cm, 100 `cms^(-1)`
B. 10 cm, 50 `cms^(-1)`
C. 20 cm, 2`ms^(-1)`
D. 2 m, 100`ms^(-1)`

Answer

To solve the problem, we start with the given wave equation:

\[ y = (3 \, \text{cm}) \sin(6.28(0.5x - 50t)) \]

### Step 1: Identify the wave parameters
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.

From the given equation, we can identify:
- Amplitude \( A = 3 \, \text{cm} \)
- Wave number \( k = 6.28 \)
- Angular frequency \( \omega = 50 \)

### Step 2: Calculate the wavelength (\( \lambda \))
The wave number \( k \) is related to the wavelength \( \lambda \) by the formula:

\[ k = \frac{2\pi}{\lambda} \]

Substituting the value of \( k \):

\[ 6.28 = \frac{2\pi}{\lambda} \]

Since \( 2\pi \approx 6.28 \), we can simplify:

\[ \lambda = \frac{2\pi}{6.28} \]

Calculating \( \lambda \):

\[ \lambda = 0.5 \, \text{cm}^{-1} \]

To find the wavelength in centimeters, we take the reciprocal:

\[ \lambda = 2 \, \text{cm} \]

### Step 3: Calculate the frequency (\( f \))
The angular frequency \( \omega \) is related to the frequency \( f \) by the formula:

\[ \omega = 2\pi f \]

Substituting the value of \( \omega \):

\[ 50 = 2\pi f \]

Solving for \( f \):

\[ f = \frac{50}{2\pi} \]

Calculating \( f \):

\[ f \approx \frac{50}{6.28} \approx 7.96 \, \text{Hz} \]

### Step 4: Calculate the velocity (\( v \))
The velocity \( v \) of the wave can be calculated using the formula:

\[ v = f \lambda \]

Substituting the values of \( f \) and \( \lambda \):

\[ v = 7.96 \, \text{Hz} \times 2 \, \text{cm} \]

Calculating \( v \):

\[ v \approx 15.92 \, \text{cm/s} \]

### Final Results
Thus, the wavelength \( \lambda \) is \( 2 \, \text{cm} \) and the velocity \( v \) is approximately \( 15.92 \, \text{cm/s} \).

### Summary of Results:
- Wavelength \( \lambda = 2 \, \text{cm} \)
- Velocity \( v \approx 15.92 \, \text{cm/s} \)

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