Back to de Broglie's thesis

 

$$u(x,t)=f\big(\gamma(x-vt)\big)$$

$$x'=\gamma(x-vt)$$

$$u(x,t)=f(x')$$

$$x'=\gamma (x-vt)$$

Since

$$t'=\gamma\big(t-\frac{vx}{c^2}\big)$$

Is it possible to have some other phenomenon which is essentially relativistic 

$$w(x,t)=w(t')=w\left(\gamma\big(t-\frac{vx}{c^2}\big)\right)$$

We can see that $w$ satisfies the following PDE

$$\boxed{(\frac{c^2}{v})^2\frac{\partial^2 w}{\partial x^2}=\frac{\partial^2 w}{\partial t^2}}$$

whose solution is

$$w(x,t)=g\big(t-\frac{x}{v'}\big)$$

where

$$v'=\frac{c^2}{v}$$ so the harmonic solution $$w(x,t)=e^{i(\omega t - kx)}$$ where with the dispersion relation $$\boxed{v=\frac{kc^2}{\omega}}$$

On the other hand

$$v=\frac{p}{m}$$

so

$$p=\frac{mc^2}{\omega}k$$