Deriving the operator version of the classical wave equation

From the book i studying from to ‘derive’ the Schrodinger equation this is part of the process:

Two differential operators, $hat{p}$ and $hat{E}$, representing the classical momentum and energy observable, p and E:

$p→p ̂= frac{ℏ}{i} frac{∂}{∂x}$ and $E→E ̂=iℏ frac{∂}{∂t}$

When acting on the plane wave solution, the operators have simple effect that they return the classical numerical values:
$p ̂ϕ(x,t)=frac{ℏ}{i} frac{∂}{∂x} [exp⁡(ifrac{(px-Et)}{ℏ})]=pϕ(x,t)$

and

$E ̂ϕ(x,t)=iℏ frac{∂}{∂t}[exp⁡(ifrac{(px-Et)}{ℏ})]=Eϕ(x,t)$

Using this formalism, we can write: $frac{∂^2 ϕ(x,t)}{∂t^2 }=c^2 frac{∂^2ϕ(x,t)}{∂x^2}$ –> $E ̂^2 ϕ(x,t)=(p ̂c)^2 ϕ(x,t)$

Which is a new operator version of the classical wave equation.

I do not understand the the last step, how using the wave equation we got the new operator version of the classical wave equation.

Also can anyone give a clearer explanation of ‘When acting on the plane wave solution, the operators have simple effect that they return the classical numerical values:’