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

Options

A. `314 cm s^(-1)`
B. `628 cm s^(-1)`
C. `20 cm s^(-1)`
D. `400 cm s^(-1)`

Answer

To find the wave velocity from the given equation of the progressive wave, we can follow these steps:

### Step 1: Identify the wave equation
The wave equation is given as:
\[ y = a \sin(628t - 31.4x) \]

### Step 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,
- \( k \) is the wave number.

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

### Step 3: Calculate the wave velocity
The wave velocity \( v \) can be calculated using the formula:
\[ v = \frac{\omega}{k} \]

### Step 4: Substitute the values of \( \omega \) and \( k \)
Now, substituting the values we identified:
\[ v = \frac{628}{31.4} \]

### Step 5: Perform the division
Calculating the above expression:
\[ v = 20 \, \text{cm/s} \]

### Conclusion
Thus, the wave velocity is:
\[ v = 20 \, \text{cm/s} \]

### Final Answer
The correct option is C: 20 cm/s.

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