Wave properties from the differential equation of a wave

[sanpokhrel]

How can we work out all the properties of wave from differential equation? And what really does differential equation of wave implies?

 

[olivermsun]

Which differential equation do you have in mind? There are many such equations for various types of waves.

 

[sophiecentaur]

How can we work out all the properties of wave from differential equation? And what really does differential equation of wave implies?

You don't 'work out' all the properties, exactly; the properties are really what you impose on the conditions. What you do is to write down the Equation of Motion (or the equivalent in electromagnetic terms). This will have second order derivatives (or more) and you can solve it. But you also need to know the boundary conditions and you impose the condition that the solution has the form of a wave (i.e. repeats in space and time). There will be many possible solutions but you choose a simple one for a start - like assuming a sine wave. That doesn't imply that sine waves are the only solution. We know that there are huge possibilities for the shapes of waves.

 

[olivermsun]

I'd say it's very much an iterative process to "work out" the properties between the physics and the mathematics. You notice how a physical system seems to behave, model the physical system with some equation(s), the mathematical properties predict some physical behaviors, you see if the predicted behaviors are actually observed, then go back and decide whether the equations need to be modified, and so on.