The normal equation we get, for waves on a string or on water,
relates the second space derivative to the second time derivative.
The Schrödinger equation uses only the first time derivative, however,
the addition of the
relates the real part of the wave function to the
imaginary part, in effect shifting the phase by 90 degrees as the 2nd derivative would do.
When Dirac tried to make a relativistic version of the equation, where the energy relation is a bit more complicated, he discovered new physics.
Gasiorowicz Chapter 3
Griffiths Chapter 1
Cohen-Tannoudji et al. Chapter