A wave train has the equation `y = 4 sin (30pit + 0.1x)` where x is in cm is the frequency of the source? How much time does a wave-púlse take to reach a point 30 cm from it ?

Answer

To solve the problem, we need to extract the frequency of the wave and the time it takes for a wave pulse to reach a point 30 cm away.

### Step-by-Step Solution:

1. **Identify the wave equation**: The given wave equation is
\[ y = 4 \sin(30\pi t + 0.1x) \]
Here, \( \omega = 30\pi \) and \( k = 0.1 \).

2. **Calculate the frequency**: The frequency \( f \) can be calculated using the relationship between angular frequency \( \omega \) and frequency \( f \):
\[ f = \frac{\omega}{2\pi} \]
Substituting the value of \( \omega \):
\[ f = \frac{30\pi}{2\pi} = 15 \text{ Hz} \]

3. **Calculate the wave velocity**: The wave velocity \( v \) can be calculated using the formula:
\[ v = \frac{\omega}{k} \]
Substituting the values of \( \omega \) and \( k \):
\[ v = \frac{30\pi}{0.1} = 300\pi \text{ cm/s} \]

4. **Calculate the time taken to travel 30 cm**: We know that the distance \( d \) traveled by the wave pulse is 30 cm. The time \( t \) taken can be calculated using the formula:
\[ t = \frac{d}{v} \]
Substituting the values:
\[ t = \frac{30 \text{ cm}}{300\pi \text{ cm/s}} = \frac{1}{10\pi} \text{ seconds} \]

### Final Answers:
- The frequency of the source is \( 15 \text{ Hz} \).
- The time taken for the wave pulse to reach a point 30 cm away is \( \frac{1}{10\pi} \text{ seconds} \).