A wave pulse in a string is described by the equation `y_1=5/((3x-4t)^2+2)` and another wave pulse in the same string is described by .. The values of `y_2=(-5)/((3x+4t-6)^2+2)` and x are in metres and t is in seconds. Which of the following statements is correct ?

Options

A. `y_1` travels along -x - direction and `y_2` along + x - direction
B. Both `y_1 and y_2` travel along +x - direction
C. At `x = 1m, y_1 and y_2` always cancel
D. At time t = 1s, `y_1 and y_2` exactly cancel everywhere

Answer

To solve the problem, we need to analyze the two wave pulse equations given:

1. \( y_1 = \frac{5}{(3x - 4t)^2 + 2} \)
2. \( y_2 = \frac{-5}{(3x + 4t - 6)^2 + 2} \)

We want to determine the relationship between \( y_1 \) and \( y_2 \) and find out which statement is correct regarding their behavior.

### Step 1: Analyze the equations

First, let's rewrite the equations for clarity:

- The first wave pulse \( y_1 \) is a positive function that depends on the term \( (3x - 4t) \).
- The second wave pulse \( y_2 \) is a negative function that depends on the term \( (3x + 4t - 6) \).

### Step 2: Substitute a specific value for \( x \)

Let's substitute \( x = 1 \) into both equations to see how they behave.

For \( y_1 \):
\[ y_1 = \frac{5}{(3(1) - 4t)^2 + 2} = \frac{5}{(3 - 4t)^2 + 2} \]

For \( y_2 \):
\[ y_2 = \frac{-5}{(3(1) + 4t - 6)^2 + 2} = \frac{-5}{(3 + 4t - 6)^2 + 2} = \frac{-5}{(4t - 3)^2 + 2} \]

### Step 3: Relate \( y_1 \) and \( y_2 \)

Now, we can express \( y_2 \) in terms of \( y_1 \):

Notice that:
\[ y_2 = \frac{-5}{(4t - 3)^2 + 2} \]
can be rewritten using the relationship between \( (3 - 4t) \) and \( (4t - 3) \):
\[ (4t - 3) = -(3 - 4t) \]

Thus, we can write:
\[ y_2 = \frac{-5}{(-(3 - 4t))^2 + 2} = -\frac{5}{(3 - 4t)^2 + 2} \]

This shows that:
\[ y_2 = -y_1 \]

### Step 4: Conclusion

From the relationship \( y_1 + y_2 = 0 \), we can conclude that \( y_1 \) and \( y_2 \) are equal in magnitude but opposite in sign. This means that at \( x = 1 \), the two wave pulses cancel each other out.

### Final Answer

The correct statement is that \( y_1 + y_2 = 0 \) at \( x = 1 \), indicating that the two wave pulses are opposite in direction.

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