\partial_{x} = \partial_{ξ} + \partial_{η}

and

\partial_{t} = c (\partial_{ξ} - \partial_{η})

. Hence

ζ_{tt} - c^{2} ζ_{xx} = - 4 c^{2} ζ_{ξ η} = 0.

Integrating this mixed derivative gives

ζ = F (ξ) + G (η)

, a sum of oppositely traveling profiles. The initial data require

F + G = u_{0}

and

c (F^{'} - G^{'}) = v_{0}

, so

F^{'} = \frac{1}{2} (u_{0}^{'} + \frac{v _{0}}{c}), G^{'} = \frac{1}{2} (u_{0}^{'} - \frac{v _{0}}{c}) .

Integrating the profiles and matching their total additive constant gives the D'Alembert formula with initial velocity:

ζ (x, t) = \frac{u _{0} ( x + c t ) + u _{0} ( x - c t )}{2} + \frac{1}{2 c} \int_{x - c t}^{x + c t} v_{0} (s) d s .

Here

u_{0}

u

appearing in the original shallow water equations. The formula verifies both initial conditions directly.

 Ancestors (10)

 Discussion (0)

There are no discussions about this article yet.

There are currently no matching articles.