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Question #42648

Difficulty: 8/10 | 🟢 Free

The Question

A long taut string of linear mass density $\mu$ and tension $T$ is immersed in a viscous fluid. The fluid exerts a damping force per unit length equal to $-b \frac{\partial y}{\partial t}$ , where $y(x,t)$ is the transverse displacement and $b$ is a positive constant. A harmonic wave of angular frequency $\omega$ is generated at $x=0$ and propagates in the $+x$ direction. Assuming the damping is small ( $b \ll \mu \omega$ ), the wave amplitude decays exponentially as $e^{-\alpha x}$ . Which of the following expressions correctly represents the spatial attenuation constant $\alpha$ and the phase velocity $v_p$ of the wave, up to the second order in the dimensionless parameter $(b/\mu\omega)$ ?

Options

A.
$\alpha = \frac{b}{2\sqrt{\mu T}}, \quad v_p = \sqrt{\frac{T}{\mu}}\left(1 - \frac{b^2}{8\mu^2\omega^2}\right)$
B.
$\alpha = \frac{b}{2\sqrt{\mu T}}, \quad v_p = \sqrt{\frac{T}{\mu}}\left(1 + \frac{b^2}{8\mu^2\omega^2}\right)$
C.
$\alpha = \frac{b}{2\sqrt{\mu T}}, \quad v_p = \sqrt{\frac{T}{\mu}}\left(1 - \frac{b^2}{4\mu^2\omega^2}\right)$
D.
$\alpha = \frac{b}{\sqrt{\mu T}}, \quad v_p = \sqrt{\frac{T}{\mu}}\left(1 + \frac{b^2}{4\mu^2\omega^2}\right)$

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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

Wave Equation Sign Convention|Damped Oscillation Equation|Approximation (Taylor/Binomial)

Why This Question Is Tricky

Common Mistakes

The question tests the student's ability to handle wave propagation in a dissipative medium by introducing a complex wavenumber.

A common trap is assuming that phase velocity is given by $v_p = \omega / |k|$ instead of $\omega / \text{Re}(k)$ , which leads to an incorrect second-order term (Option C).

Another lethal misconception is drawing a false analogy with damped harmonic oscillators (where temporal frequency $\omega$ decreases), leading students to incorrectly assume the spatial frequency $k$ decreases, which would erroneously increase the phase velocity (Option B).

The problem rigorously demands the correct application of the binomial expansion to the complex dispersion relation and proper identification of the real and imaginary components.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Waves - Class 11 Part 2
Concept	Wave Equation Sign Convention
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free