Two waves of equal frequency `f` and velocity `v` travel in opposite directions along the same path. The waves have amplitudes `A` and `3 A` . Then:

Options

A. the amplitude of the resulting wave varies with position between maxima of amplitude `4 A` and minima of zero amplitude.
B. the distance between a maxima and adjacent minima of amplitudes is `v//2f`
C. maximum amplitude is `4 A` and minimum amplitude is ` 2A`
D. The position of a maxima or minima of amplitude does not change with time

Explanation

To solve the problem, we need to analyze the two waves traveling in opposite directions and determine the characteristics of the resultant wave.

### Step 1: Write the equations of the two waves
Let’s denote the two waves as follows:
- Wave 1 (with amplitude A):
\[ y_1 = A \sin(\omega t - kx) \]
- Wave 2 (with amplitude 3A):
\[ y_2 = 3A \sin(\omega t + kx) \]

### Step 2: Write the equation for the resultant wave
The resultant wave \( y \) is the sum of the two waves:
\[ y = y_1 + y_2 = A \sin(\omega t - kx) + 3A \sin(\omega t + kx) \]

### Step 3: Use trigonometric identities to simplify
We can use the trigonometric identity for sine to combine the two waves:
\[ \sin A + \sin B = 2 \sin\left(\frac{A+B}{2}\right) \cos\left(\frac{A-B}{2}\right) \]
Applying this to our waves:
\[ y = A \sin(\omega t - kx) + 3A \sin(\omega t + kx) \]
This can be rewritten as:
\[ y = A \left( \sin(\omega t - kx) + 3 \sin(\omega t + kx) \right) \]

### Step 4: Combine the sine terms
Using the sine addition formula:
\[ \sin(\omega t + kx) = \sin(\omega t) \cos(kx) + \cos(\omega t) \sin(kx) \]
\[ \sin(\omega t - kx) = \sin(\omega t) \cos(kx) - \cos(\omega t) \sin(kx) \]
Substituting these into the equation gives:
\[ y = A \left( \sin(\omega t) \cos(kx) - \cos(\omega t) \sin(kx) + 3 \left( \sin(\omega t) \cos(kx) + \cos(\omega t) \sin(kx) \right) \right) \]
Combining like terms results in:
\[ y = 4A \sin(\omega t) \cos(kx) + 2A \cos(\omega t) \sin(kx) \]

### Step 5: Identify the maximum and minimum amplitudes
The maximum amplitude occurs when the sine and cosine functions reach their maximum values:
- Maximum amplitude = \( 4A \) (from the first term)
- Minimum amplitude = \( 0 \) (from the second term when \( \sin(kx) = 0 \))

### Step 6: Analyze the distance between maxima and minima
The distance between adjacent maxima and minima is given by:
- The distance between an antinode and the nearest node is \( \frac{\lambda}{4} \).
Using the wave relationship \( \lambda = \frac{v}{f} \):
\[ \text{Distance} = \frac{v}{4f} \]

### Step 7: Determine the position of maxima and minima
The positions of maxima and minima do not change with time; they are fixed points determined by the wave properties.

### Conclusion
From the analysis:
- The maximum amplitude is \( 4A \) and the minimum amplitude is \( 0 \).
- The distance between maxima and minima is \( \frac{v}{4f} \).
- The positions of maxima and minima do not change with time.

### Final Statements
- The amplitude of the resulting wave varies between a maximum of \( 4A \) and a minimum of \( 0 \).
- The distance between maxima and minima is \( \frac{v}{4f} \).
- The position of maxima or minima does not change with time.