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Question #79318

Difficulty: 5/10 | 🟢 Free

The Question

A transverse wave packet on a taut string is described by the wave function $y(x,t) = A \exp[-(ax^2 + bt^2 + cxt)]$ , where $a$ and $b$ are positive constants, and $c$ is a real constant. The wave propagates in the positive $x$ -direction without changing its shape. The string is rigidly fixed at $x = L$ . Which of the following represents the reflected wave $y_r(x,t)$ ?

Options

A.
$-A \exp\left[-\left(ax^2 + bt^2 + 2\sqrt{ab}xt - 4aLx - 4L\sqrt{ab}t + 4aL^2\right)\right]$
B.
$A \exp\left[-\left(ax^2 + bt^2 + 2\sqrt{ab}xt - 4aLx - 4L\sqrt{ab}t + 4aL^2\right)\right]$
C.
$-A \exp\left[-\left(ax^2 + bt^2 - 2\sqrt{ab}xt - 4aLx + 4L\sqrt{ab}t + 4aL^2\right)\right]$
D.
$-A \exp\left[-\left(ax^2 + bt^2 + 2\sqrt{ab}xt\right)\right]$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Multi-Concept_Bridge

Master problems that require linking physics and math concepts to solve complex competitive questions.

Concepts Behind This Question

Dimensional Consistency and Transcendental Functions|Phase Constant Determination|Boundary Condition Reflection

Why This Question Is Tricky

Common Mistakes

This question tests the synthesis of dimensional homogeneity, wave propagation kinematics, and boundary conditions.

Students often treat the exponent as a purely algebraic expression, failing to recognize that for a non-dispersive wave, the argument must form a perfect square $a(x \pm vt)^2$ .

A common trap is ignoring the propagation direction, leading to an incorrect sign for the cross-term $c$ (Option C).

Another trap is applying the free-boundary condition (Option B) or reflecting about the origin instead of $x=L$ (Option D).

The rigorous constraint of dimensional consistency forces the velocity to be $v = \sqrt{b/a}$ , linking the spatial and temporal coefficients.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Units and Measurements - Class 11 Part 1
Concept	Dimensional Consistency and Transcendental Functions
Difficulty	5
Cognitive Load	6
Target Solve Time	140 sec
Access	Free