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It is a bit confusing, because when I say "wave equation" the connotation it invokes in my mind is if I apply the equation I will find the function of the wave I'm exploring.

Of course I know now this is wrong.. it is the other way around, I have to know my wave-function in advance and substitute it into the "wave equation".

Tell me if so far I'm on the right track ?

My next question is what do I get back ?

I can see there second-partial-derivative of position OR second-partial-derivative of time, but what is their meaning.. what is the idea ?

For example in the Newton law I can understand the second-derivative of position against time is acceleration, but in the wave equation we have second-partial-derivative of position against "frozen"-time , what does this mean ?

And then similar for time ?

Help me make the connection, thank you ?