Question

Beats are produced by waves $y_1 = a \sin 2000\pi t$ and $y_2 = a \sin 2008\pi t$. The number of beats heard per second is

• A. $4$
• B. $1$
• C. zero
• D. $8$

Hint:

To find the beat frequency directly from the $\omega$ values without calculating individual frequencies, use the formula $f_{\text{beat}} = \frac{|\omega_1 - \omega_2|}{2\pi}$. Here, $\frac{2008\pi - 2000\pi}{2\pi} = \frac{8\pi}{2\pi} = 4$.

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
We are given the equations of two sound waves and need to calculate the beat frequency, which is the number of beats heard per second when they superimpose.

Step 2: Key Formula or Approach:
The standard equation for a wave is $y = a \sin(\omega t)$, where $\omega = 2\pi n$ and $n$ is the frequency.
The beat frequency is the absolute difference between the frequencies of the two interfering waves: $n_{\text{beats}} = |n_1 - n_2|$.

Step 3: Detailed Explanation:
From the first wave equation, $y_1 = a \sin(2000\pi t)$:
$$\omega_1 = 2000\pi$$
$$2\pi n_1 = 2000\pi$$
$$n_1 = \frac{2000\pi}{2\pi} = 1000\ \text{Hz}$$
From the second wave equation, $y_2 = a \sin(2008\pi t)$:
$$\omega_2 = 2008\pi$$
$$2\pi n_2 = 2008\pi$$
$$n_2 = \frac{2008\pi}{2\pi} = 1004\ \text{Hz}$$
Now, calculate the beat frequency:
$$\text{Beat frequency} = |n_2 - n_1|$$
$$\text{Beat frequency} = |1004 - 1000| = 4\ \text{Hz}$$
This means 4 beats will be heard per second.

Step 4: Final Answer:
The number of beats heard per second is $4$, matching option (A).