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?

• You can get $y(x,t)$ by the superposition of sine and cosine functions; which is periodic functions. Is this where you are confused? Oct 13 '16 at 8:15
• @SRS"Any superposition , of two arbitrary function and of satisfies the wave equation in one-dimension." I don't understand this part. Oct 13 '16 at 8:40
– SRS
Oct 13 '16 at 8:43
• No, not necessarily Schrödinger's equation. Solutions of Maxwell's equations, for example, also would be hard to understand if they were unbounded at infinity. Same for elastic waves, sound waves, water sufrace waves, most of other types of waves. Oct 13 '16 at 9:36
• There is a class of nonlinear equations that have soliton (also called solitary wave) solutions. These are sometimes nonperiodic. Oct 13 '16 at 15:13

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.

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.