Question

Consider the following wave equation \[ \frac{\partial^2 f(x,t)}{\partial t^2} = 10000 \frac{\partial^2 f(x,t)}{\partial x^2} \] Which of the given options is/are solution(s) to the given wave equation?

• A. $f(x,t)= e^{-(x-100t)^2}+e^{-(x+100t)^2}$
• B. $f(x,t)= e^{-(x-100t)} + 0.5e^{-(x+1000t)}$
• C. $f(x,t)= e^{-(x-100t)} + \sin(x+100t)$
• D. $f(x,t)= e^{j100\pi (-100x+t)} + e^{j100\pi(100x+t)}$

Hint:

A wave equation solution must depend only on $(x-ct)$ or $(x+ct)$, where $c$ is the wave speed.

Answer 1

The Correct Option is A, C

The general solution of the wave equation \[ f_{tt} = c^2 f_{xx}, \quad c=100 \] is: \[ f(x,t) = F(x-100t) + G(x+100t). \] Option (A): Both terms are of the form \(F(x-100t)\) and \(G(x+100t)\). Thus valid. Option (B): The second term contains \(x+1000t\). The wave speed is 100, not 1000. Thus not a solution. Option (C): Both terms contain \(x-100t\) and \(x+100t\). Satisfies the form. Thus valid. Option (D): Exponents contain \(100\pi\), implying wave-speed \(100\pi\), which does not match 100. Not a solution. Final Answer: (A), (C)