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Question #43571

Difficulty: 8/10 | 🟢 Free

The Question

A stationary source at $x=0$ continuously emits a composite wave consisting of two harmonic components with angular frequencies $\omega_1 = \omega_0 + \Delta \omega$ and $\omega_2 = \omega_0 - \Delta \omega$ , where $\Delta \omega \ll \omega_0$ . The waves propagate in the $+x$ direction through a non-linear dispersive medium. The dispersion relation of the medium is given by $k(\omega) = \frac{\omega}{v_0} + \beta \omega^2$ , where $v_0$ and $\beta$ are positive constants. At any given instant $t$ , what is the spatial distance $\Delta x$ between two adjacent nodes of the beat envelope?

Options

A.
$\frac{\pi}{\Delta \omega \left( \frac{1}{v_0} + 2\beta \omega_0 \right)}$
B.
$\frac{\pi}{\Delta \omega \left( \frac{1}{v_0} + \beta \omega_0 \right)}$
C.
$\frac{2\pi}{\Delta \omega \left( \frac{1}{v_0} + 2\beta \omega_0 \right)}$
D.
$\frac{\pi v_0}{\Delta \omega}$

Check Your Answer

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

Superposition Principle|Beats and Frequency Difference|Approximation (Binomial)

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to distinguish between phase velocity and group velocity in a dispersive medium when analyzing beat patterns.

A common misconception is to assume that the spatial beat length can be calculated using $\Delta x = v_p T_{beat}$ , where $v_p = \omega_0/k_0$ is the phase velocity (leading to Option B).

However, the beat envelope's spatial structure is governed by the group velocity $v_g = \Delta \omega / \Delta k$ .

By forcing the derivation of the spatial distance between envelope nodes directly from the superposition principle and the non-linear dispersion relation, the trap of using the standard non-dispersive kinematic formula is exposed.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Wave Optics - Class 12
Concept	Superposition Principle
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free