What is the meaning of the wave equation .in English?

[yungman]

What is the meaning of the wave equation...in English??!

Everybody knows one dimensional wave equation $$\frac{\partial^2u}{\partial t^2} = c^2 \frac{\partial^2u}{\partial x^2}$$

This together with verious boundary and initial condition give various solution of u(x,t). Also it can be transform by D'Alembert solution into two waves traveling forward and backward. From D'Alembert, it shows that "c" is the PROPAGATING velocity along x-axis in this case.

1) But what is .$$\frac{\partial^2u}{\partial t^2} = c^2 \frac{\partial^2u}{\partial x^2}$$. really mean in physical world.


From study of Electromagnetics, my understanding is wave equation represent a transverse wave because u(x,t) is orthogonal to the direction of propagation.


2) What is Poisson's equation .$$\nabla^2 u = A$$. mean in physical world? I know it is a steady state function.

 

[Hootenanny]

Everybody knows one dimensional wave equation $$\frac{\partial^2u}{\partial t^2} = c^2 \frac{\partial^2u}{\partial x^2}$$

This together with verious boundary and initial condition give various solution of u(x,t). Also it can be transform by D'Alembert solution into two waves traveling forward and backward. From D'Alembert, it shows that "c" is the PROPAGATING velocity along x-axis in this case.

1) But what is .$$\frac{\partial^2u}{\partial t^2} = c^2 \frac{\partial^2u}{\partial x^2}$$. really mean in physical world.


From study of Electromagnetics, my understanding is wave equation represent a transverse wave because u(x,t) is orthogonal to the direction of propagation.


2) What is Poisson's equation .$$\nabla^2 u = A$$. mean in physical world? I know it is a steady state function.

Firstly, let me say that an equation, by itself has no physical interpretation. In this case, the physical interpretation depends on what the function u represents. The mere fact that u satisifes that wave equation doesn't give it a physical interpretation anymore than the fact that u is differentiable does.

One could argue that if the field u satisfies that wave equation, then it behaves like a wave. This is indeed true, but it does not give you any physical insite into u without knowing what is represents. Moreover, if a function u satisfies that wave equation doesn't necesserily mean that it has a physical interpretation. For example, the function

u = const. ,​

clearly satisfies the wave equation, but equally doesn't have a physical interpretation. One cannot make a physical interpretation of u unless one attaches a physical meaning to u.

I would also point out that solutions to the wave equation can represent any type of wave, be it transverse, longitudinal, spherical etc.