8.1 One-Dimensional Flow
401
The radiation boundary conditions (8.7) and (8.8) have the following alternative
derivation and interpretation as one-way wave equations. The general solution for
the perturbation velocity in the linearized shallow-water system may be expressed
as
U = Fr[x - (U + e)t] + FI[x - (U - e)t].
The first component of the general solution, Fr , represents a wave traveling to
the right, and the second component represents a wave traveling to the left. The
partial differential equation (8.7) imposed at the boundary is satisfied by solutions
of the form Fr but does not admit solutions of the form FI. All reflection at the
right boundary may be eliminated by ensuring that no leftward-propagating waves
are present at the boundary, and this may be achieved by imposing (8.7) and (8.8)
at the right boundary. Because all the solutions to (8.7) propagate in the same
direction, that equation is sometimes known as a one-way wave equation. Oneway wave equations are particularly useful in problems involving several spatial
dimensions.
8.1.3 Time-Dependent Boundary Data
In some applications it is important to allow changes in the surrounding fluid to influence the interior solution while simultaneously radiating outward-propagating
waves through the boundary without reflection. Suppose that U e (x, t) and TJe (x, t)
taining the subdomain °: : : : x :::: L, and that information about the solution in the
are the velocity and the scaled free-surface displacement in a large domain conlarge domain is to be used to generate boundary conditions for a simulation of
the linearized shallow-water equations inside the subdomain. It is not generally
possible simply to set u(L , t) = ue(L, t) and TJ(L, t) = TJe(L, t) without generating spurious reflections in those waves attempting to pass outward through the
boundary at x = L. Well-posed boundary conditions are obtained by specifying
the value ofthe incomiug Rien-anninvariant. At the right boundary, d(L, t) is set
to de(L, t), where
de(x, t) = ue(x, t) - TJe(x, t)fe,
and at the left boundary, e(O, t) is set to ee(O, t) ,where
ee(x, t) = ue(x, t) + TJe(x, t)fe.
As an alternative to the direct specification of the incoming Riemann invariants,
the data from the large domain can be incorporated in the one-way wave equations
for U and TJ as folIows. For all t > 0, the value of the incoming Riemann invariant
in a small neighborhood of x = L will equal de(x, t). Thus, in this neighborhood
d(x, t) == u(x, t) -
TJ(x, t)
- - = de(x , r),
e
401
The radiation boundary conditions (8.7) and (8.8) have the following alternative
derivation and interpretation as one-way wave equations. The general solution for
the perturbation velocity in the linearized shallow-water system may be expressed
as
U = Fr[x - (U + e)t] + FI[x - (U - e)t].
The first component of the general solution, Fr , represents a wave traveling to
the right, and the second component represents a wave traveling to the left. The
partial differential equation (8.7) imposed at the boundary is satisfied by solutions
of the form Fr but does not admit solutions of the form FI. All reflection at the
right boundary may be eliminated by ensuring that no leftward-propagating waves
are present at the boundary, and this may be achieved by imposing (8.7) and (8.8)
at the right boundary. Because all the solutions to (8.7) propagate in the same
direction, that equation is sometimes known as a one-way wave equation. Oneway wave equations are particularly useful in problems involving several spatial
dimensions.
8.1.3 Time-Dependent Boundary Data
In some applications it is important to allow changes in the surrounding fluid to influence the interior solution while simultaneously radiating outward-propagating
waves through the boundary without reflection. Suppose that U e (x, t) and TJe (x, t)
taining the subdomain °: : : : x :::: L, and that information about the solution in the
are the velocity and the scaled free-surface displacement in a large domain conlarge domain is to be used to generate boundary conditions for a simulation of
the linearized shallow-water equations inside the subdomain. It is not generally
possible simply to set u(L , t) = ue(L, t) and TJ(L, t) = TJe(L, t) without generating spurious reflections in those waves attempting to pass outward through the
boundary at x = L. Well-posed boundary conditions are obtained by specifying
the value ofthe incomiug Rien-anninvariant. At the right boundary, d(L, t) is set
to de(L, t), where
de(x, t) = ue(x, t) - TJe(x, t)fe,
and at the left boundary, e(O, t) is set to ee(O, t) ,where
ee(x, t) = ue(x, t) + TJe(x, t)fe.
As an alternative to the direct specification of the incoming Riemann invariants,
the data from the large domain can be incorporated in the one-way wave equations
for U and TJ as folIows. For all t > 0, the value of the incoming Riemann invariant
in a small neighborhood of x = L will equal de(x, t). Thus, in this neighborhood
d(x, t) == u(x, t) -
TJ(x, t)
- - = de(x , r),
e