two waves `y_1 = 2.5 sin100πt` and `y_2 = 2.5 sin102πt` ( where y is in meter and t is in second ) are traveling in same direction. the number of beat heard per second is

Options

A. one
B. two
C. three
D. four

Answer

To find the number of beats heard per second from the two waves given by the equations \( y_1 = 2.5 \sin(100\pi t) \) and \( y_2 = 2.5 \sin(102\pi t) \), we can follow these steps:

### Step 1: Identify the angular frequencies
The equations of the waves can be compared to the standard wave equation \( y = A \sin(\omega t) \), where \( \omega \) is the angular frequency.

For wave \( y_1 \):
\[ \omega_1 = 100\pi \, \text{rad/s} \]

For wave \( y_2 \):
\[ \omega_2 = 102\pi \, \text{rad/s} \]

### Step 2: Calculate the frequencies
The frequency \( f \) is related to the angular frequency \( \omega \) by the formula:
\[ f = \frac{\omega}{2\pi} \]

Calculating the frequency for \( y_1 \):
\[ f_1 = \frac{100\pi}{2\pi} = 50 \, \text{Hz} \]

Calculating the frequency for \( y_2 \):
\[ f_2 = \frac{102\pi}{2\pi} = 51 \, \text{Hz} \]

### Step 3: Calculate the beat frequency
The number of beats heard per second is given by the absolute difference between the two frequencies:
\[ \text{Beat frequency} = |f_2 - f_1| = |51 - 50| = 1 \, \text{Hz} \]

### Conclusion
The number of beats heard per second is \( 1 \, \text{beat per second} \).

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