Question

A wave pulse in a string is described by the equation $y_1=\frac{5}{(3x-4t)^2+2}$ and another wave pulse in the same string is described by $y_2=\frac{-5}{(3x+4t-6)^2+2}$. The values of $y_1, y_2$ and $x$ are in meters and $t$ in seconds. Which of the following statement is correct?

• 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. both $y_1$ and $y_2$ travel along $+x$-direction
• D. at $x = 1$ m, $y_1$ and $y_2$ always cancel
• E. at time $t = 0.75$ s, $y_1$ and $y_2$ exactly cancel everywhere

Hint:

Two pulses traveling in opposite directions will perfectly cancel at the instant their arguments become identical if they have equal magnitudes but opposite signs.

Answer 1

Answer: (E)

Concept:
For wave pulses of the form $y = f(ax \pm bt)$: [itemsep=8pt]
• A pulse with $(ax - bt)$ travels in the positive x-direction.
• A pulse with $(ax + bt)$ travels in the negative x-direction.
• Complete cancellation (destructive interference) occurs everywhere if $y_1(x, t) = -y_2(x, t)$ at a specific time $t$.

Step 1: Analyze pulse directions.
$y_1$ contains $(3x - 4t)$, indicating it moves in the $+x$ direction.
$y_2$ contains $(3x + 4t - 6)$, indicating it moves in the $-x$ direction.

Step 2: Find the condition for exact cancellation.
Cancellation occurs when the denominators of $y_1$ and $y_2$ are identical: \[ (3x - 4t)^2 = (3x + 4t - 6)^2 \] This requires the spatial arguments to be equal: \[ 3x - 4t = 3x + 4t - 6 \] \[ -4t = 4t - 6 \] \[ 8t = 6 \implies t = \frac{6}{8} = 0.75 \text{ s} \]

Step 3: Verify at $t = 0.75$ s.
At $t = 0.75$ s, both arguments become $(3x - 3)$.
$y_1 = \frac{5}{(3x-3)^2 + 2}$ and $y_2 = \frac{-5}{(3x-3)^2 + 2}$.
Therefore, $y_1 + y_2 = 0$ for all values of $x$.