# Can we obtain the wave velocity and particle velocity relation from the wave equation?

Let the wave equation be $$u(x,t)$$. If it is an equation of a wave $$\Rightarrow u(x,t) = u(x+c\triangle t,t+\triangle t)$$

$$\Rightarrow \frac{\partial u}{\partial t} = -c\frac{\partial u}{\partial x}$$(particle velocity and wave velocity relation)

The above relation was derived from the fact that wave looks like the same, it just shifts towards the direction of propagation as time passes.

Any function $$u(x,t)$$ satisfying the wave equation $$\frac{\partial^2 u}{\partial t^2} = c^2\frac{\partial^2 u}{\partial x^2}$$ must describe an equation of a wave $$\Rightarrow$$ it also satisfies $$\frac{\partial u}{\partial t} = -c\frac{\partial u}{\partial x}$$. I think there might be a way to show that $$\frac{\partial^2 u}{\partial t^2} = c^2\frac{\partial^2 u}{\partial x^2} \Rightarrow \frac{\partial u}{\partial t} = -c\frac{\partial u}{\partial x}$$. I do not have much knowledge of partial derivatives so i am unable to show the above. Can someone show me how to do it?

• Hint: solutions to the wave equation can travel either forwards or backwards at velocity $c$. Jul 21 at 20:43
• @jacob1729 Yes we can find the general solution of the wave equation and see that the fit $\frac{\partial u}{\partial t} = '+' or '-'c\frac{\partial u}{\partial x}$ which would show that wave equation $\Rightarrow$ particle velocity and wave velocity relation, but is there any other way to do that which does not involve finding solutions to wave equation, Jul 28 at 10:08