For 
treated as constants a solution of (5.3)
is 
where
.
Substitution into (5.3) gives
An alternative form for this dispersion equation is
where
This shows that a surface of uniform 
in wave number
space is an ellipsoid of revolution about an axis parallel to
whose intercept on the 
plane
is located at 
, 
. THe
intersection of the surface on is a circle
of radius b, while the intersection on the
surface is an ellipse with minor semi-axis b
and major semi-axis bN/f. Contours of in the
-plane (i.e., ) are shown
in the accompanying figure 1. If one intersects the
ellipsoids with a plane at 
= constant the resulting
contours in such a 
-plane resemble
those for the modal dispersion relation (namely offset
circles within circles).
Referring to the accompanying diagram we that for
upward progagating phase, the energy must have a
downward component and vice verse, since
and
.
This is qualitatively similar to the behavior of internal
gravity waves.
The reflective properties on a sloping sea bed have characteristics also reminiscent of internal gravity waves, in that the waves can reflect forward as in A or B of Fig. 2 or backward as in C. Also, waves change length on reflection.