Let the two waves `y_(1) = A sin ( kx - omega t)` and `y_(2) = A sin ( kx + omega t)` from a standing wave on a string . Now if an additional phase difference of `phi` is created between two waves , then

Options

A. the standing wave will have a different frequency
B. the standing wave will have a different amplitude for a given point
C. the spacing between two consecutives nodes will change
D. None of the above

Explanation

To solve the problem, we need to analyze the two waves and the effect of introducing an additional phase difference \( \phi \) between them. Let's break it down step by step.

### Step 1: Understand the Given Waves
We have two waves:
1. \( y_1 = A \sin(kx - \omega t) \)
2. \( y_2 = A \sin(kx + \omega t) \)

These waves represent two traveling waves moving in opposite directions, which can create a standing wave pattern.

### Step 2: Introduce the Phase Difference
When an additional phase difference \( \phi \) is introduced between the two waves, the equations of the waves become:
1. \( y_1 = A \sin(kx - \omega t) \)
2. \( y_2 = A \sin(kx + \omega t + \phi) \)

### Step 3: Resultant Wave
To find the resultant wave when these two waves interfere, we can use the principle of superposition. The resultant wave \( y \) can be expressed as:
\[ y = y_1 + y_2 \]
Using the sine addition formula, we can rewrite \( y_2 \):
\[ y_2 = A \sin(kx + \omega t + \phi) = A \left( \sin(kx + \omega t) \cos(\phi) + \cos(kx + \omega t) \sin(\phi) \right) \]

### Step 4: Combine the Waves
Now, let's combine \( y_1 \) and \( y_2 \):
\[ y = A \sin(kx - \omega t) + A \left( \sin(kx + \omega t) \cos(\phi) + \cos(kx + \omega t) \sin(\phi) \right) \]
This results in a new wave pattern that depends on the phase difference \( \phi \).

### Step 5: Analyze the Effects
1. **Frequency**: The frequency of the standing wave is determined by the source of the waves, and it remains unchanged despite the phase difference.
2. **Amplitude**: The amplitude of the resultant wave can change due to the phase difference, but it does not imply a change in amplitude for a given point.
3. **Spacing Between Nodes**: The introduction of a phase difference \( \phi \) alters the positions of the nodes and antinodes, which means the spacing between consecutive nodes will change.

### Conclusion
The correct conclusion is that the spacing between consecutive nodes will change due to the phase difference introduced between the two waves.

### Final Answer
The spacing between the consecutive nodes will change.

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