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Question #43524

Difficulty: 8/10 | 🟢 Free

The Question

A point source fixed at the origin $O(0,0)$ of the laboratory frame emits a short sound pulse at $t=0$ . A uniform wind blows with a steady velocity $\vec{v}_w = v_0 \hat{i}$ relative to the ground. The speed of sound in still air is $c$ (where $c > v_0$ ). A stationary observer is located at a point $P(0, d)$ on the positive y-axis. All distances are in $\text{m}$ and speeds in $\text{m s}^{-1}$ . When the wavefront reaches the observer at $P$ , what is the phase speed of the wavefront (defined as the speed of the wavefront along its normal direction) as measured in the laboratory frame?

Options

A.
$\frac{c^2 - v_0^2}{c}$
B.
$\sqrt{c^2 - v_0^2}$
C.
c
D.
$\sqrt{c^2 + v_0^2}$

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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

Frame of Reference Dependency|Wave Propagation Direction|Vector Kinematics

Why This Question Is Tricky

Common Mistakes

This question tests the critical distinction between ray velocity (energy propagation) and phase velocity (wavefront propagation) in a moving medium.

Students often incorrectly assume that the speed of the wavefront reaching the observer is simply the magnitude of the ray velocity $\sqrt{c^2 - v_0^2}$ or mistakenly apply scalar addition.

The trap exploits the geometric definition of a wavefront: while the pulse travels along the y-axis in the lab frame, the wavefront itself is tilted.

The phase speed must be calculated by projecting the resultant lab-frame velocity onto the invariant normal vector of the wavefront, demonstrating a deep understanding of Galilean transformations applied to wave equations.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Motion in a Straight Line - Class 11 Part 1
Concept	Frame of Reference Dependency
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free