Question

The general solution of one dimensional wave equation \[ \frac{\partial^2v}{\partial t^2} = c^2 \frac{\partial^2v}{\partial x^2} \] is \(u(x,t)=\)

• A. \(f(x+ct)\,g(x-ct)\)
• B. \(f(x+ct)+g(x-ct)\)
• C. \(f(x-ct)+g(x-ct)\)
• D. \(f(x+ct)+g(x+ct)\)

Hint:

D'Alembert's solution is \[ \boxed{ u(x,t) = f(x-ct) + g(x+ct), } \] representing two waves travelling in opposite directions with speed \(c\).

Answer 1

The Correct Option is B

Step 1: Recall D'Alembert's solution of the wave equation. The one-dimensional wave equation \[ \frac{\partial^2u}{\partial t^2} = c^2 \frac{\partial^2u}{\partial x^2} \] has the general solution \[ \boxed{ u(x,t) = f(x-ct) + g(x+ct), } \] where \(f\) and \(g\) are arbitrary functions.

Step 2: Compare with the given options. Since the names of arbitrary functions can be interchanged, \[ f(x-ct)+g(x+ct) \] is equivalent to \[ f(x+ct)+g(x-ct). \] Hence, \[ \boxed{ u(x,t)=f(x+ct)+g(x-ct). } \] Therefore, \[ \boxed{(B)} \] is the correct answer.