438
8. Nonreflecting Boundary Conditions
4. Show that the second-order Higdon boundary condition (8.35) can be expressed in a form similar to the second-order Engquist and Majda condition
(8.31) as
(l + coser cosaz)-z + e(cosa\ + cosaz)-- - e
Z -
z = O.
ot
otox
oy
5. Show that if eis the shallow-water gravity-wave phase speed and IUI < c,
then
oZ1/l
-+(U+e)
atZ
( oZ1/l
e oZ1/l)
- - - -
otox 2 oyZ
=0
is a one-way wave equation that admits only waves with x-component
group velocities greater than zero. How do wave solutions to this equation
compare with the solutions to the two-dimensional shallow-water equations
linearized about a constant basic-state flow U parallel to the x-axis,
(
0
- + U -
ot
0 )Z 1/1 -
OX
e
Z ( oZ
-
ox z
oZ )
+ - oyZ
1/1 = O.
Consider the dispersion relations for each system. Also discuss the dependence of the sign of the x-component group velocity on U and the horizontal wave numbers.
6. Derive the approximate one-way wave equation (8.67) for the right-moving
wave satisfying the Rayleigh damped shallow-water equations (8.65) and
(8.66) with v = 0 and R constant. Describe the conditions under which this
is an accurate approximation.
7. Derive the conditions under which the vertical trace speed (w/m) of internal gravity waves satisfying the dispersion relation (8.42) is in the same
direction as the vertical group velocity.
8. Consider linear Boussinesq flow in the x-z plane . Show that a hydrostatic
internal gravity wave propagating relative to the mean flow in the positive
x-direction satisfies the relation P = Nw/lml, where the hat denotes a
Fourier transform along the z coordinate and m is the vertical wave number.
Use these results to derive a partial differential equation relating P to u that
could, in principle, be evaluated using the grid-point values of pressure and
velocity.
9. Suppose that the linearized Boussinesq equations are to be solved in the
domain -L
x
L. The relation derived in Problem 9 might serve as a
radiation boundary condition along the boundary at x = L when the basicstate horizontal wind speed is zero. Explain how this boundary condition
might prove unsatisfactory if
8. Nonreflecting Boundary Conditions
4. Show that the second-order Higdon boundary condition (8.35) can be expressed in a form similar to the second-order Engquist and Majda condition
(8.31) as
(l + coser cosaz)-z + e(cosa\ + cosaz)-- - e
Z -
z = O.
ot
otox
oy
5. Show that if eis the shallow-water gravity-wave phase speed and IUI < c,
then
oZ1/l
-+(U+e)
atZ
( oZ1/l
e oZ1/l)
- - - -
otox 2 oyZ
=0
is a one-way wave equation that admits only waves with x-component
group velocities greater than zero. How do wave solutions to this equation
compare with the solutions to the two-dimensional shallow-water equations
linearized about a constant basic-state flow U parallel to the x-axis,
(
0
- + U -
ot
0 )Z 1/1 -
OX
e
Z ( oZ
-
ox z
oZ )
+ - oyZ
1/1 = O.
Consider the dispersion relations for each system. Also discuss the dependence of the sign of the x-component group velocity on U and the horizontal wave numbers.
6. Derive the approximate one-way wave equation (8.67) for the right-moving
wave satisfying the Rayleigh damped shallow-water equations (8.65) and
(8.66) with v = 0 and R constant. Describe the conditions under which this
is an accurate approximation.
7. Derive the conditions under which the vertical trace speed (w/m) of internal gravity waves satisfying the dispersion relation (8.42) is in the same
direction as the vertical group velocity.
8. Consider linear Boussinesq flow in the x-z plane . Show that a hydrostatic
internal gravity wave propagating relative to the mean flow in the positive
x-direction satisfies the relation P = Nw/lml, where the hat denotes a
Fourier transform along the z coordinate and m is the vertical wave number.
Use these results to derive a partial differential equation relating P to u that
could, in principle, be evaluated using the grid-point values of pressure and
velocity.
9. Suppose that the linearized Boussinesq equations are to be solved in the
domain -L
x
L. The relation derived in Problem 9 might serve as a
radiation boundary condition along the boundary at x = L when the basicstate horizontal wind speed is zero. Explain how this boundary condition
might prove unsatisfactory if