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 will analyze the given wave equation and extract the necessary information to find the frequency and the time taken for a wave pulse to reach a specific point.

### Step 1: Identify the wave equation
The given wave equation is:
\[ y = 4 \sin(30\pi t + 0.1x) \]

### Step 2: Determine the angular frequency (ω)
In the wave equation of the form \( y = A \sin(kx - \omega t) \), the term associated with time \( t \) is the angular frequency \( \omega \). From the equation, we can see that:
\[ \omega = 30\pi \, \text{rad/s} \]

### Step 3: Calculate the frequency (f)
The relationship between angular frequency \( \omega \) and frequency \( f \) is given by:
\[ \omega = 2\pi f \]
To find the frequency \( f \), we rearrange the equation:
\[ f = \frac{\omega}{2\pi} \]
Substituting the value of \( \omega \):
\[ f = \frac{30\pi}{2\pi} = 15 \, \text{Hz} \]

### Step 4: Determine the wave speed (v)
The wave number \( k \) is related to the spatial part of the wave equation. From the equation, we have:
\[ k = 0.1 \, \text{rad/cm} \]
The wave speed \( v \) can be calculated using the relationship:
\[ v = \frac{\omega}{k} \]
Substituting the values:
\[ v = \frac{30\pi}{0.1} = 300\pi \, \text{cm/s} \]

### Step 5: Calculate the time taken to reach a point 30 cm away
The time \( t \) taken for a wave to travel a distance \( d \) is given by:
\[ t = \frac{d}{v} \]
Here, \( d = 30 \, \text{cm} \). Thus:
\[ t = \frac{30}{300\pi} \]
Calculating this gives:
\[ t = \frac{1}{10\pi} \, \text{s} \]

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

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