Evenif wave number is, by definition, number of waves in unit distance (space),
we may account that direction of space when we want to specify the direction
of wave propagation.
The 'k' and 'l' are wave numbers in X and Y directions.
For a 2D wave, if we measure the wave characterestics along two axes
(i.e. X and Y) we get two wave speed. They may or may not be equal,
according to the geometry of the axis.
Phase line of a wave is a "line through constant wave
proporties", for example a line through constant height. It is equivalent
to the "wave front".
A figure is attached. It is partially redrawn from Pedlosky book. In that
pciture, three items are shown. (1) A set of waves radiating from a
source at the center and its phase line (b) A measure of wave number
in XY plane at one part of the wave (c) A Physical view of wave propagation
and "lines of constant height" (called phase lines).
The phase speed along X-direction of the reference frame is the
speed at which the "intersection of phase lines" at X-axis
advances in time. Similiarly for the phase speed along Y-axis also.
Now try to project the mean Wave Vector "K" onto X-axis and
try to find its speed. That will be different from the former calculation.
k = 2.pi./lamda_x
l = 2.pi./lamda_y
K = sqrt (k^2+l^2)
C = w/K
Cx = w/k = C.K/k (where BIG K=mean wave vector, BIG C=mean phase speed)
Cy = w/l = C.K/l