The progressive waves produced by two sources of sound placed close to each other are given as
`y_(1) = 5 sin 300 pi t`
`y_(2) = 10 sin 290 pi t`
An observer standing near these two sources will observe.

Options

A. 3 beats/sec
B. 4 beats/sec
C. 5 beats/sec
D. 6 beats/sec

Explanation

To solve the problem, we need to analyze the two progressive waves given by the equations:

1. \( y_1 = 5 \sin(300 \pi t) \)
2. \( y_2 = 10 \sin(290 \pi t) \)

### Step 1: Identify the angular frequencies
The angular frequency \( \omega \) is given in the form \( \omega = 2 \pi f \), where \( f \) is the frequency.

For the first wave \( y_1 \):
- \( \omega_1 = 300 \pi \)
- Using the relation \( \omega = 2 \pi f \):
\[ 300 \pi = 2 \pi f_1 \implies f_1 = \frac{300 \pi}{2 \pi} = 150 \text{ Hz} \]

For the second wave \( y_2 \):
- \( \omega_2 = 290 \pi \)
- Similarly,
\[ 290 \pi = 2 \pi f_2 \implies f_2 = \frac{290 \pi}{2 \pi} = 145 \text{ Hz} \]

### Step 2: Calculate the beat frequency
The beat frequency \( f_b \) is given by the absolute difference between the two frequencies:
\[ f_b = |f_1 - f_2| = |150 \text{ Hz} - 145 \text{ Hz}| = 5 \text{ Hz} \]

### Step 3: Conclusion
An observer standing near these two sources will hear beats with a frequency of 5 beats per second.

### Final Answer
The observer will observe a beat frequency of 5 Hz.

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