What characteristics define a wave for a physicist? 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 (a solution of wave equation with $f_2=0$) $$y(x,t)=f_1=A\exp\left[-\frac{(x-vt)}{L}\right]; y(x,t)=f_1=A(x-vt)^2$$ which satisfies the one-dimensional wave equation but nothing is "waving" or "repeating" for this functions. Are these examples qualify as waves?
 A: The definition of wave used in a introductory course often runs along the lines of 

A wave is a travelling disturbance.

A single pulse qualifies within that definition without trouble, and we distinguish between general waves, periodic waves and harmonic waves (periodic and sinusoidal).
Later you define a wave 

A wave is a solution to a wave equation, 

and yes, a single pulse can still be a solution. 
Now, a single pulse (or indeed any non-harmonic solution) won't have a single frequency, which means that in dispersive media it won't hold its shape as it propagates, but that doesn't change the fact that it qualifies under either kind of definition.
A: I am going to consider travelling waves since your question gives the equation of $f(x-vt)$ and a travelling disturbance like a crest seems "wavey".
Any travelling disturbance can be seen such that the parameter of the wave at a certain instant at a particular time is copied at the adjacent position at a later time.This would mean a crest would keep moving as time goes on. It is such that the parameter at a specific location at this instant is taken by the next location at a different instant. So, we have $$y=f(x,t)=f(x+\Delta x, t+\Delta t)$$ 
So, we can say $$f(x,t)=f(0,t-x/v)=g(vt-x)$$ 
The definition of wave itself of $f(x-vt)$ itself means that the wave is going to look like a travelling wave. I don't see why a travelling wave has to repeat.  
