Before You Begin

Question #42204

Difficulty: 8/10 | 🟢 Free

The Question

Two long, thin cylindrical rods, Rod 1 and Rod 2, are joined end-to-end at $x=0$ . Both rods have the same uniform cross-sectional area and the same uniform density $\rho$ . The materials of both rods have an identical shear modulus $G$ . Rod 1 has a bulk modulus $B_1 = \frac{2}{3}G$ . Rod 2 is made of an ideal incompressible material ( $B_2 \to \infty$ ). A longitudinal displacement wave $s_i(x,t) = A_i \sin(\omega t - k_1 x)$ travels through Rod 1 towards the junction. If the reflected wave is $s_r(x,t) = A_r \sin(\omega t + k_1 x)$ , what is the amplitude reflection coefficient $R = \frac{A_r}{A_i}$ ?

Options

A.
$2\sqrt{6} - 5$
B.
-1
C.
$5 - 2\sqrt{6}$
D.
$-\frac{1}{5}$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Intuition_Breaker

Train your brain to resist "obvious" answers that are designed to deceive your intuition.

Concepts Behind This Question

Poisson's Ratio Limits|Boundary Condition Reflection|Waves

Why This Question Is Tricky

Common Mistakes

This question tests the understanding of longitudinal wave propagation in bounded media (thin rods) versus bulk media.

A common misconception is that an incompressible material ( $B \to \infty$ ) implies an infinite wave speed, leading students to incorrectly conclude that the acoustic impedance $Z_2 \to \infty$ and the reflection coefficient $R = -1$ (total reflection).

However, in a thin rod, lateral expansion (Poisson effect) allows longitudinal waves to propagate at a finite speed governed by Young's modulus $Y$ .

Even as $B \to \infty$ , $Y$ approaches a finite limit ( $3G$ ), resulting in a finite impedance and partial reflection.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Mechanical Properties of Solids - Class 11 Part 2
Concept	Poisson's Ratio Limits
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free