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Question #42850

Difficulty: 8/10 | 🟢 Free

The Question

A high-frequency periodic thermal source generates a temperature wave in a semi-infinite rod (Rod 1). The wave travels towards a junction at $x=0$ where Rod 1 is perfectly joined to another semi-infinite rod (Rod 2). The amplitude of the incident temperature wave at the junction is $T_0 = 60 \text{ K}$ . At the junction, both the temperature and the total heat current are continuous. The properties of the rods are given as:

Rod 1: Cross-sectional area $A_1 = A$ , thermal conductivity $\kappa_1 = 2\kappa$ , volumetric heat capacity $(\rho c)_1 = 5C$ . Rod 2: Cross-sectional area $A_2 = 2A$ , thermal conductivity $\kappa_2 = 5\kappa$ , volumetric heat capacity $(\rho c)_2 = 8C$ .

Assuming the thermal wave propagates as $T(x,t) \propto e^{-\gamma x} \cos(\omega t - \gamma x)$ where $\gamma = \sqrt{\omega \rho c / 2\kappa}$ , what is the amplitude of the transmitted temperature wave in Rod 2 at the junction?

Options

A.
24 \text{ K}
B.
20 \text{ K}
C.
40 \text{ K}
D.
36 \text{ K}

Check Your Answer

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

Boundary Condition Reflection|Thermal Conductivity in Series and Parallel|Small Angle Approximation

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to distinguish between steady-state heat conduction and transient thermal wave propagation.

In steady-state problems, students correctly use static thermal conductance $K = \kappa A$ to evaluate heat flow.

However, for a high-frequency thermal wave, the spatial derivative of the temperature profile introduces the wave vector $\gamma$ , which depends on the thermal diffusivity.

Consequently, the dynamic thermal impedance (or effusivity) governing wave reflection and transmission depends on $A\sqrt{\rho c \kappa}$ .

Students who mistakenly apply the static conductance ratio will choose 20 K, while those who ignore the cross-sectional area mismatch will choose 40 K.

Calculating the reflected amplitude instead of the transmitted one leads to 36 K.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Waves - Class 11 Part 2
Concept	Boundary Condition Reflection
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free