Mathematical definition of wavefront in case of non-harmonic waves What is the general mathematical definition of wavefront?

Wavefront is the surface where, at fixed time, the phase is constant

But for non-harmonic waves we cannot talk about phase as the function $\phi(\vec{r},t)=\vec{k} \cdot \vec{r}-\omega t+ \delta$, so what can be the mathematical way to define a wavefront in the case of a generic wavefunction $\xi (\vec{r},t)$ (not necessarily sinusoidal)?
 A: A progressive wave is a function of space and time where the dependence on space and time can be modeled by a function of one variable, composed with an “argument” function, which combines space and time and describes the geometry of the wave. Wavefronts are the loci where the “argument” function is constant.
Let’s give a few examples.
Plane wave
If $A(\vec r,t)$ is a plane wave of velocity $c$ propagating in the direction of unit vector $\vec u$, it can be expressed as $A(\vec r,t)=f(\vec r·\vec u-ct)$. What I call the “argument” function is the argument of $f$, namely $\vec r·\vec u-ct$. The wavefront is defined by $\vec r·\vec u-ct=\text{const.}$, which indeed is a plane (orthogonal to $\vec u$) moving at speed $c$.
Spherical wave
If $B(\vec r,t)$ is a spherical wave of velocity $c$, the source of which is at $\vec r=\vec 0$, it can be expressed (e.g.) as $B(\vec r,t)=g(r-ct)/r$. The “argument” function here is $r-ct$ (not $1/r$, which only describes the amplitude decay in space). The wavefront is thus defined by $r-ct=\text{const.}$, which indeed is a sphere (centered on $\vec r=\vec 0$) of radius growing at speed $c$.
Other waves
As long as the wave is progressive, in principle it can be put under the form $a(r)×f(w(\vec r,t))$ where $a$ describes the amplitude in space, $f$ is the wave form, and $w$ is the “argument” function describing the propagation.

As you can see, I made no hypothesis on the wave form. “Harmonic” refers to the form of $f$ or $g$ above, namely they must be sinusoidal functions, not to the front geometry. The latter, as any locus, is given by an equation like $w(\vec r,t)=\text{const.}$
