It is a curious fact that the Schrodinger wave equation is not really like any wave equation, but rather like the heat equation --which describes the diffusion of heat through a medium. A wave equation is taking a value for the acceleration of a string and saying that equals the curvature. That is, the second partial  derivative with respect to time equals the second partial derivative with respect to position. The Schrodinger equation on the other hand, is the first partial derivative with respect to time times i equals the second partial derivative with respect to position times i^2. What is the relation between diffusion of heat with the wave?     The first partial derivative with respect to time ought to be the velocity, and in the heat equation in fact tells us the speed the heat spreads. But that is not a wave equation.    In a string it would tell us the velocity of the string equals the curvature.

Schrodinger went to find such a equation because de Broglie had said in his PhD Thesis that an electron ought to have a corresponding wave equation based on E=mc^2 and velocity of a wave = wavelength times frequency, and energy = h times frequency But he never suggested what the equation for that wave.

So the Schrodinger wave equation is the heat equation with an "i" thrown in. Take heat and thrown in an ''i'', and you get an electron wave?     And after all, what is  heat? Kinetic energy of moving particles. Kinetic energy times ''i'' gives you a wave? Or entropy times ''i'' gives you a wave? 

Note: Particles have kinetic energy. So  Schrodinger's wave equation is describing particle's KE  spreading through a medium, i.e a complex medium.