A standing wave arises on a string when two waves of equal amplitude , frequency and wavelength travelling in opposite superimose. If the frequency of oscillation of the standing waves

Options

A. gets doubled
B. gets halved
C. remains unchanged
D. changes but not by a factor of `2 or 1//2`

Explanation

To solve the problem of finding the frequency of oscillation of the standing waves formed by the superposition of two waves traveling in opposite directions, we can follow these steps:

### Step 1: Understand the Components of the Waves
We have two waves traveling in opposite directions:
- Wave 1: \( y_1 = A \sin(\omega t - kx) \)
- Wave 2: \( y_2 = A \sin(\omega t + kx) \)

Here, \( A \) is the amplitude, \( \omega \) is the angular frequency, and \( k \) is the wave number.

**Hint:** Identify the parameters of the waves such as amplitude, frequency, and wavelength.

### Step 2: Superimpose the Two Waves
When these two waves superimpose, the resultant wave \( y \) can be expressed as:
\[ y = y_1 + y_2 = A \sin(\omega t - kx) + A \sin(\omega t + kx) \]

**Hint:** Write down the expression for the resultant wave by adding the two wave equations.

### Step 3: Use the Sine Addition Formula
Using the sine addition formula, we can simplify the resultant wave:
\[ y = 2A \sin\left(\omega t\right) \cos\left(kx\right) \]
This is derived from the identity \( \sin C + \sin D = 2 \sin\left(\frac{C+D}{2}\right) \cos\left(\frac{C-D}{2}\right) \).

**Hint:** Recall the sine addition formula to combine the two sine functions.

### Step 4: Identify the Frequency of the Standing Wave
From the expression \( y = 2A \sin(\omega t) \cos(kx) \), we can see that the standing wave oscillates in time with the term \( \sin(\omega t) \). The frequency \( f \) of the oscillation can be derived from the angular frequency \( \omega \) using the relationship:
\[ f = \frac{\omega}{2\pi} \]

**Hint:** Relate angular frequency to frequency using the formula \( f = \frac{\omega}{2\pi} \).

### Step 5: Compare with Individual Waves
The individual waves also have the same frequency \( f = \frac{\omega}{2\pi} \). Therefore, the frequency of oscillation of the standing wave is the same as that of either of the component waves.

**Hint:** Compare the frequency of the standing wave with the frequencies of the individual waves to conclude.

### Final Answer
Thus, the frequency of oscillation of the standing waves remains unchanged and is equal to the frequency of the individual component waves.

**Final Conclusion:** The frequency of oscillation of the standing wave is the same as that of either of the component waves.