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Question #41997

Difficulty: 8/10 | 🟢 Free

The Question

A transverse wave packet on a taut string is described by the incident wave equation $Y_i(x,t) = A \exp[-(ax^2 + bt^2 + cxt)]$ , where $A, a, b$ are positive constants and $c$ is a negative constant. The wave is incident on a fixed rigid boundary located at $x = 0$ . By applying the principle of dimensional homogeneity and boundary conditions, determine the wave propagation speed $v$ and the equation of the reflected wave $Y_r(x,t)$ .

Options

A.
$v = \sqrt{\frac{b}{a}}, \quad Y_r(x,t) = -A \exp[-(ax^2 + bt^2 - cxt)]$
B.
$v = \frac{|c|}{a}, \quad Y_r(x,t) = -A \exp[-(ax^2 + bt^2 + cxt)]$
C.
$v = \sqrt{\frac{b}{a}}, \quad Y_r(x,t) = A \exp[-(ax^2 + bt^2 - cxt)]$
D.
$v = \frac{|c|}{2b}, \quad Y_r(x,t) = A \exp[-(ax^2 + bt^2 + cxt)]$

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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 with Galilean wave transformations and boundary conditions.

Students often fall into the trap of assuming velocity is simply $v = |c|/a$ because it is dimensionally consistent ( $[L^{-1}T^{-1}]/[L^{-2}] = [LT^{-1}]$ ), ignoring that a valid traveling wave packet requires the exponent to be a perfect square, yielding $v = \sqrt{b/a} = |c|/(2a)$ .

Furthermore, the reflection at a rigid boundary requires both a spatial inversion (changing the sign of the cross-term $cxt$ to represent propagation in the $-x$ direction) and a $\pi$ phase shift (amplitude inversion to $-A$ ), filtering out candidates who treat the exponent as a disconnected algebraic sum rather than a coherent physical phase argument.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Units and Measurements - Class 11 Part 1
Concept	Dimensional Consistency and Transcendental Functions
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free