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Question #42633

Difficulty: 8/10 | 🟢 Free

The Question

A transverse wave propagates on a string of linear mass density $\rho$ and tension $T$ . The string is embedded in a non-linear elastic medium, such that the potential energy density (energy per unit length) of the medium is given by $u_p = \frac{1}{2}\rho \omega_0^2 y^2 + \alpha y^3$ , where $y(x,t)$ is the transverse displacement. Two independent wave sources emit waves which, if the medium were purely linear ( $\alpha=0$ ), would result in the steady-state superposition $y_{lin}(x,t) = A \sin(kx - \omega t) + A \sin(kx - \omega t + \pi/3)$ . Due to the non-linear term, the actual displacement $y(x,t)$ contains a local time-independent shift $y_0$ . Assuming $\alpha$ is small, what is the magnitude of the constant shift $y_0$ to leading order in $A$ ?

Options

A.
$\frac{9\alpha A^2}{2\rho \omega_0^2}$
B.
$\frac{3\alpha A^2}{\rho \omega_0^2}$
C.
$\frac{3\alpha A^2}{2\rho \omega_0^2}$
D.
$\frac{6\alpha A^2}{\rho \omega_0^2}$

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

Superposition of SHMs|Energy Density in Transverse Waves|Taylor Series Expansion

Why This Question Is Tricky

Common Mistakes

This question tests the failure of the superposition principle in non-linear media (anharmonicity).

Students commonly assume that the net effect of two waves is simply the sum of their individual effects.

If they calculate the DC shift for each wave independently ( $y_{0,single} = \frac{3\alpha A^2}{2\rho \omega_0^2}$ ) and add them, they get $\frac{3\alpha A^2}{\rho \omega_0^2}$ , which is incorrect.

The non-linear cubic term in the potential energy ( $U \propto y^3$ ) creates a quadratic term in the restoring force ( $F \propto y^2$ ).

When squaring the superimposed waves, the cross-term $2y_1 y_2$ has a non-zero time average, generating an additional DC shift ('Harmonic Generation' and 'Frequency Mixing').

This demonstrates that in non-linear dynamics, energy and forces cannot be linearly superimposed.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Oscillations - Class 11 Part 2
Concept	Superposition of SHMs
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free