D'Alembert's solution of the wave equation

[Gear300]

For a wave equation η(u,v) = f₁(u) + f₂(v) where u = x - ct and v = x + ct, consider an initial displacement η = η₀(x) and an initial velocity ∂_tη = \dot{η_{0}}(x).

I'm a little confused with the velocity initial condition; shouldn't the time derivative of η₀(x) be 0?

 

[Ken G]

For a wave equation η(u,v) = f₁(u) + f₂(v) where u = x - ct and v = x + ct, consider an initial displacement η = η₀(x) and an initial velocity ∂_tη = \dot{η_{0}}(x).

I'm a little confused with the velocity initial condition; shouldn't the time derivative of η₀(x) be 0?

I think the problem you are having is that the explicit time derivative of a function of only x is zero. But note the explicit time derivative is not applied to the η_{0}(x) function, it is applied to the η(x,t) function at t=0, and that is just given the name \dot{η_{0}}(x). You could call it anything-- call it h(x) if you like. The key point is, it is a constraint on the η(x,t) function, without which you cannot get a unique solution. Think of the two constraints as being on η(x,t) at t=0 and on ∂_tη(x,t) at t=0.

 

[Gear300]

Thanks for the reply. I figured the dot was implying the time derivation of the first constraint.

 

[Ken G]

So do you see now that this is not the case? There are two independent constraints there, they have no connection to each other other than that they are both taken as constraints on the full solution.