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.

 

[PJC01]

Or is there a typo and it should be ∂tη = \dot{η_{0}}(x) = 0? Please clarify.

Assuming that the initial velocity condition is indeed ∂tη = \dot{η_{0}}(x) = 0, D'Alembert's solution of the wave equation provides a way to solve for the displacement of a wave at any given time and position. This solution is based on the principle that the solution at any point in space and time is the sum of two waves traveling in opposite directions.

By using the substitution u = x - ct and v = x + ct, we can transform the wave equation into a simpler form where the displacement is a function of only u and v. This allows us to solve for the displacement at any given time and position by finding the values of u and v that satisfy the initial conditions.

D'Alembert's solution is a powerful tool in understanding and predicting the behavior of waves in various physical systems. It has applications in fields such as acoustics, electromagnetics, and fluid dynamics. By providing a mathematical solution to the wave equation, D'Alembert's solution allows us to make predictions and analyze the behavior of waves in a wide range of scenarios.