Longuet-Higgins (Proc. Roy. Soc. A, 1964) considered
the problem of reflection of non-divergent
Rossby waves from a straight boundary having general orientation
relative to the meridians. We consider here only the case
of a meridional boundary, but treat the divergent Rossby
waves discussed in the previous sections.
Unlike the case of a gravity wave for which the wave length of
the reflected wave equals that of the incident wave, the Rossby
waves experience a change in wave length on reflection.
The essential quantitities which must be conserved on reflection
for any type wave are frequency and energy. As shown
in the accompanying sketch of 
vs. 
for
given 
using (4.12), for any 
there exist two possible which are candidate values
for incident and reflected waves. The reflection process is best
described by reference to the wave number plane, where the
locus of all 
consistent with the
same is a circle (see Section 4.3).
For a meridional boundary on the west side of a basin
The value of (i.e., the component of 
projected onto the boundary) must be the same for reflected
and incident waves and, of course, , but the
components differ (the difference being large
if 
).
In any event, the waves reflected from a boundary on the
west are reduced in wave length ( 
),
while the opposite is true for a boundary on the east.
Both incident and reflected waves have westward components of
phase speed. However, the x-components of their group speed
have opposite sign (that of the incident wave being towards
the boundary while the reflected wave is away from the
boundary). The group speeds differ in magnitude with the
energy flux having equal and opposite value for the
component normal to the wall as required by energy conservation
considerations.
The fluid velocity is geostrophic. In order that there be no
net velocity normal to the wall, the pressure anomaly must
vanish along the wall. The requires that the amplitude A of
p for the reflected wave be equal but opposite in sign
to that of the incident wave ( 
).
However, the energy per unit area for the reflected and
incident wave differ since the 
differ, i.e.
.