You have learnt that a travelling wave in one dimension is represented by a function $y=f(x, t)$ where $x$ and $t$ must appear in the combination $x-vt$ or $x+vt$,i.e.,$y=f(x \pm vt)$. Is the converse true? Examine if the following functions for $y$ can possibly represent a travelling wave:
$(a)$ $(x-vt)^2$
$(b)$ $\log[(x+vt)/x_0]$
$(c)$ $1/(x+vt)$

(NONE) The converse is not true. $A$ function $y=f(x \pm vt)$ represents a travelling wave only if it remains finite for all values of $x$ and $t$.
$(a)$ For $y = (x-vt)^2$: As $x \to \infty$ or $t \to \infty$,$y \to \infty$. Since the function does not remain finite for all $x$ and $t$,it does not represent a physically possible travelling wave.
$(b)$ For $y = \log[(x+vt)/x_0]$: As $x+vt \to 0$,$y \to -\infty$. Since the function does not remain finite for all $x$ and $t$,it does not represent a physically possible travelling wave.
$(c)$ For $y = 1/(x+vt)$: As $x+vt \to 0$,$y \to \infty$. Since the function does not remain finite for all $x$ and $t$,it does not represent a physically possible travelling wave.