Let u(x, t) be a smooth solution to the wave equation:

​$$\frac{\partial^2 u}{\partial t^2} - \frac{\partial^2 u}{\partial x^2} = 0, \quad \text{for } (x,t) \in \mathbb{R}^2$$​​

Which of the following is FALSE?

Correct option is C

Results:

(i) If u(x, t) is a solution of the wave equation, then U(x − a, t − b) is also asolution for all a, b ∈ R.

(ii) If u(x, t) is a solution of the wave equation, then U(ax, bt) is also a solutionif a = b.

From these results we can say that u(x,t) given n options (A), and (D) solvesthe equation

so these statements are true.and we know that if u satisfies wave equation then so does $$\frac{\partial u}{\partial x}$$​ .

so,statement given in option (B) is also true.Now, in Option (C) u(3x, 9t) is given where 3 $$\neq$$​ 9

so, from result (ii) , thisstatement is $$\textbf{False}$$​ and we have to find the false statement.

==> Option (C) is correct.