Question

Two waves $Y_1 = 0.25 \sin(316t)$ and $Y_2 = 0.25 \sin(310t)$ are propagating in the same direction. The number of beats produced per second are

• A. $\frac{3}{\pi}$
• B. $\frac{\pi}{3}$
• C. $\frac{\pi}{2}$
• D. $\frac{2}{\pi}$

Hint:

You can find the beat frequency directly using angular frequencies via $f_{\text{beat}} = \frac{\Delta \omega}{2\pi}$. Here, $\Delta \omega = 316 - 310 = 6$, so the beat frequency is immediately evaluated as $\frac{6}{2\pi} = \frac{3}{\pi}$.

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
We are given the mathematical wave equations for two interfering waves traveling along the same line. We need to determine the beat frequency, which corresponds directly to the number of beats heard per second.

Step 2: Key Formula or Approach:
The standard equation for a harmonic wave as a function of time is:
$$Y = A \sin(\omega t)$$ where $\omega$ is the angular frequency related to the linear frequency $f$ by the formula $\omega = 2\pi f$. The beat frequency $f_{\text{beat}}$ is defined as the absolute difference between the linear frequencies of the two interacting waves:
$$f_{\text{beat}} = |f_1 - f_2|$$

Step 3: Detailed Explanation:
From the given equations, let's extract the angular frequencies:
For the first wave, $\omega_1 = 316\ \text{rad/s}$. For the second wave, $\omega_2 = 310\ \text{rad/s}$. Now, map these values to convert them into linear frequencies ($f = \frac{\omega}{2\pi}$):
$$f_1 = \frac{316}{2\pi}$$ $$f_2 = \frac{310}{2\pi}$$ Substitute these expressions directly into the beat frequency relation:
$$f_{\text{beat}} = \left| \frac{316}{2\pi} - \frac{310}{2\pi} \right|$$ Since the denominators are identical, combine the numerators directly:
$$f_{\text{beat}} = \frac{316 - 310}{2\pi} = \frac{6}{2\pi} = \frac{3}{\pi}$$ The number of beats produced per second matches the beat frequency value.

Step 4: Final Answer:
The number of beats produced per second is $\frac{3}{\pi}$, which corresponds perfectly to option (A).