A progressive wave is represented by the equation `y= 0.5 sin ( 314t- 12.56x) ` where y and x are in metre and t is in second .Its wavelength is

Options

A. 0.5 m
B. 0.2 m
C. 1 m
D. 2m

Answer

To find the wavelength of the progressive wave represented by the equation \( y = 0.5 \sin(314t - 12.56x) \), we can follow these steps:

### Step 1: Identify the wave equation
The given wave equation is:
\[ y = 0.5 \sin(314t - 12.56x) \]
This is in the standard form of a progressive wave, which can be expressed as:
\[ y = A \sin(\omega t - kx) \]
where:
- \( A \) is the amplitude,
- \( \omega \) is the angular frequency,
- \( k \) is the wave number.

### Step 2: Identify the wave number \( k \)
From the equation, we can see that the coefficient of \( x \) is \( k \):
\[ k = 12.56 \]

### Step 3: Relate wave number \( k \) to wavelength \( \lambda \)
The relationship between the wave number \( k \) and the wavelength \( \lambda \) is given by the formula:
\[ k = \frac{2\pi}{\lambda} \]

### Step 4: Rearrange the formula to find wavelength \( \lambda \)
We can rearrange this formula to solve for \( \lambda \):
\[ \lambda = \frac{2\pi}{k} \]

### Step 5: Substitute the value of \( k \)
Now, substituting the value of \( k \) into the equation:
\[ \lambda = \frac{2\pi}{12.56} \]

### Step 6: Calculate \( \lambda \)
Using the value of \( \pi \approx 3.14 \):
\[ 2\pi \approx 2 \times 3.14 = 6.28 \]
Now, substituting this value:
\[ \lambda = \frac{6.28}{12.56} = 0.5 \text{ meters} \]

### Final Answer
The wavelength \( \lambda \) is:
\[ \lambda = 0.5 \text{ meters} \]

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