376
Figure 1: Characteristic curves in a depression wave.
A B
Figure 2: Depression wave.
Surface waves that can be simulated with a shallow water model are essentially
(nonlinear) combinations of depression waves and bores, which are, respectively, the
analogous of rarefaction waves and shocks in compressible fluids. Let us describe these
kind of waves in a simple case.
Once again, we consider fluid at rest occupying the region x :2: 0, with a piston in
x = ° that starts to move in the negative direction at constant speed w. Using relations
(33), (34) it is not difficult to obtain the solution of (25), (26) representing this situation.
In this case the characteristics of the family {Xl} are straight lines along each of which
u and 17 are constant (see Fig.I). The solution at a time t = to consists of a depression
wave connecting two region of differents constants states (see Fig.2).
If the piston moves now in the positive direction, it can be shown again that the
family {xd is composed of straight lines along which u. and 17 are constants. But, in
this case, the characteristics will cut each other and, as a consequence, continuous solutions of the equations can not exist.. Solut.ions present.ing discontinuities and, therefore,
satisfying the equations in a weaker sense (as the distributions sense) have to be considered. Let us suppose that such a solution has a discontinuity along the curve x = crt),
connecting two smooth regions. If conservation of mass and momentum have to be
preserved, the following relations must hold:
P(17- + H)
PC'/]+ + H)
p(l)+ + H)u+(u+ - X') - p(l)- + H)u-(u- - X')
~pg(l)+ + H)2 - ~pg(l)- + H)2
(35)
where + represents limits to the right of the discontinuity and -, limits to the left. These
are the Rankine-Hugoniot conditions for this particular case.
Figure 1: Characteristic curves in a depression wave.
A B
Figure 2: Depression wave.
Surface waves that can be simulated with a shallow water model are essentially
(nonlinear) combinations of depression waves and bores, which are, respectively, the
analogous of rarefaction waves and shocks in compressible fluids. Let us describe these
kind of waves in a simple case.
Once again, we consider fluid at rest occupying the region x :2: 0, with a piston in
x = ° that starts to move in the negative direction at constant speed w. Using relations
(33), (34) it is not difficult to obtain the solution of (25), (26) representing this situation.
In this case the characteristics of the family {Xl} are straight lines along each of which
u and 17 are constant (see Fig.I). The solution at a time t = to consists of a depression
wave connecting two region of differents constants states (see Fig.2).
If the piston moves now in the positive direction, it can be shown again that the
family {xd is composed of straight lines along which u. and 17 are constants. But, in
this case, the characteristics will cut each other and, as a consequence, continuous solutions of the equations can not exist.. Solut.ions present.ing discontinuities and, therefore,
satisfying the equations in a weaker sense (as the distributions sense) have to be considered. Let us suppose that such a solution has a discontinuity along the curve x = crt),
connecting two smooth regions. If conservation of mass and momentum have to be
preserved, the following relations must hold:
P(17- + H)
PC'/]+ + H)
p(l)+ + H)u+(u+ - X') - p(l)- + H)u-(u- - X')
~pg(l)+ + H)2 - ~pg(l)- + H)2
(35)
where + represents limits to the right of the discontinuity and -, limits to the left. These
are the Rankine-Hugoniot conditions for this particular case.