The equation of a progressive wave is given by `y=a sin (628t-31.4x)`. If the distance are expressed in cms and time seconds, then the wave in this string wll be

Options

A. 314 cm
B. 628 cm
C. 5 cm
D. 400 cm

Explanation

To solve the problem, we need to analyze the given equation of the progressive wave and extract the relevant parameters to find the wavelength.

### Step-by-Step Solution

1. **Identify the wave equation**: The given equation of the wave is:
\[ y = a \sin(628t - 31.4x) \]
Here, \(a\) is the amplitude, \(t\) is time in seconds, and \(x\) is distance in centimeters.

2. **Compare with the standard wave equation**: The standard form of a progressive wave is:
\[ y = a \sin(\omega t - kx) \]
where \(\omega\) is the angular frequency and \(k\) is the wave number.

3. **Identify the parameters**: From the equation, we can identify:
- \(\omega = 628\) (angular frequency)
- \(k = 31.4\) (wave number)

4. **Relate wave number to wavelength**: The wave number \(k\) is related to the wavelength \(\lambda\) by the formula:
\[ k = \frac{2\pi}{\lambda} \]

5. **Rearranging for wavelength**: We can rearrange this equation to solve for \(\lambda\):
\[ \lambda = \frac{2\pi}{k} \]

6. **Substituting the value of \(k\)**: Now substitute \(k = 31.4\) into the equation:
\[ \lambda = \frac{2\pi}{31.4} \]

7. **Calculating the wavelength**: Using the approximate value of \(\pi \approx 3.14\):
\[ \lambda = \frac{2 \times 3.14}{31.4} = \frac{6.28}{31.4} \]

8. **Simplifying the calculation**: To simplify further:
\[ \lambda = \frac{628}{314} = 2 \text{ cm} \]

9. **Conclusion**: Therefore, the wavelength of the wave in the string is:
\[ \lambda = 2 \text{ cm} \]

### Final Answer
The wavelength of the wave in this string is **2 cm**.