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Question #42856

Difficulty: 8/10 | 🟢 Free

The Question

A transverse wave propagates along a long stretched string of linear mass density $\mu$ and tension $T$ . Due to a viscous surrounding medium, the wave amplitude attenuates exponentially. The transverse displacement of the string is given by: $y(x,t) = A_0 e^{-\gamma x} \sin(\omega t - kx)$ where $A_0$ , $\gamma$ , $\omega$ , and $k$ are positive constants. Which of the following expressions correctly represents the time-averaged total mechanical energy density (energy per unit length) $\langle u \rangle$ of the wave at a position $x$ ? Assume all quantities are expressed in coherent SI units.

Options

A.
$\frac{1}{4} A_0^2 e^{-2\gamma x} \left[ \mu \omega^2 + T(\gamma^2 + k^2) \right]$
B.
$\frac{1}{4} A_0^2 e^{-2\gamma x} \left[ \mu \omega^2 + T k^2 \right]$
C.
$\frac{1}{2} \mu \omega^2 A_0^2 e^{-2\gamma x}$
D.
$\frac{1}{2} A_0^2 e^{-2\gamma x} \left[ \mu \omega^2 + T(\gamma^2 + k^2) \right]$

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Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Variable_Constraint

Develop pure symbolic logic skills by mastering algebraic constraints without relying on numerical data.

Concepts Behind This Question

Energy Density in Transverse Waves|Damped Oscillation Equation|Mechanical Energy Conservation

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to derive wave energy from first principles rather than relying on memorized formulas for undamped waves.

The primary trap is assuming the potential energy density depends only on the wave number $k$ (i.e., $\frac{\partial y}{\partial x} \approx -k A \cos(\omega t - kx)$ ), which leads to the incorrect option B.

Students often forget that spatial attenuation introduces an additional term via the product rule when calculating the spatial derivative $\frac{\partial y}{\partial x}$ .

This cross-term significantly alters the potential energy density in highly dissipative media, making the standard $u = \frac{1}{2}\mu\omega^2 A^2$ approximation invalid.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Waves - Class 11 Part 2
Concept	Energy Density in Transverse Waves
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free