What characteristics define a wave for a physicist? Any superposition of two arbitrary functions $f_1(x-vt)$ and $f_2(x+vt)$, satisfies the wave equation in one-dimension. Will it be called a wave if the function $y(x,t)$ doesn't have any periodicity? For example, consider the aperiodic functions $y(x,t)=A\exp\left[-\frac{(x-vt)}{L}\right]$ or $y=A(x-vt)^2$ which satisfies the one-dimensional wave equation but nothing is "waving" or "repeating" for this functions.