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GATE 2022 EC – Question 30

Electromagnetics · Maxwell's Equations · 1 mark · Multiple select

Consider the following wave equation,

$$\frac{\partial^2f(x,t)}{\partial t^2}=10000\,\frac{\partial^2f(x,t)}{\partial x^2}.$$

Which of the given options is/are solution(s) to the given wave equation?

  1. $f(x,t)=e^{-(x-100t)^2}+e^{-(x+100t)^2}$
  2. $f(x,t)=e^{-(x-100t)}+0.5e^{-(x+1000t)}$
  3. $f(x,t)=e^{-(x-100t)}+\sin(x+100t)$
  4. $f(x,t)=e^{j100\pi(-100x+t)}+e^{j100\pi(100x+t)}$

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Show answer and explanation

Correct answer: (A) $f(x,t)=e^{-(x-100t)^2}+e^{-(x+100t)^2}$; (C) $f(x,t)=e^{-(x-100t)}+\sin(x+100t)$

Explanation

The equation has wave speed $c=\sqrt{10000}=100$, so its solutions are sums of functions of $(x-100t)$ and $(x+100t)$. A and C fit this form. In B the second term depends on $x+1000t$ (speed 1000). In D the terms depend on $x\mp t/100$ (speed $0.01$).