Accelerating `waves'

 Waves travel only at constant speed. We can see this in light of argument of $$u(x,t)=f(x-vt)$$ which is $$x=vt.$$

The fact that relativity and wave equation are so harmonious together is related to this: $x=vt$ is first-order both in space and time. 

From $u(x,t)=f(x-vt)$ one can easily arrive at the 1D wave equation and from there it suffices to enter the 3D generalization of second derivative, Laplacian. 

Recently motivated by Milgrom's constant $$a_0\approx \frac{cH_0}{2\pi}$$ I was led to ask what is this acceleration $a_0$?

It is not hard to speculate that it is the acceleration of light. But light is a wave (we are not indulging quantum mechanics yet) and a wave cannot accelerate. 

So how do we make a `wave' accelerate? Quite easy in fact: Pursuing the same method, we begin in 1D $$\boxed{u(x,t)=f(x-\frac{1}{2}at^2 - vt)}$$  

This yields $$\boxed{\frac{1}{v+at}\frac{\partial}{\partial t}\left(\frac{1}{v+at}\frac{\partial \phi}{\partial t}\right)=\nabla^2 \phi}$$