Most general `wave' equation

Prototype: \(u(x,t)=f(x-vt)\) 

$$g(x,t)=f\big(x(t_0)\big)=f\left(x(t)\pm\sum_{n=1}^{\infty}\frac{f^{(n)}(0)}{n!} x^{n}\right)$$

(Taylor series)

or, Prototype: \(u(x,t)=f(v-at)\) 

$$\frac{d^{n+1} x}{dt^{n+1}}(t)=0$$

then

$$p\big(\frac{d^n x}{dt^n},t\big)=h\left(\frac{d^n x}{dt^n}(t)\pm t\frac{d}{dt}\frac{d^{n-1} x}{dt^{n-1}}(t)\right).$$

(Taylor approximation)