Let F be some property of the fluid at position
at time t.
If the equations governing F are linear and contain
coefficients which are independent of t (as in Ch. I, sec. 4.2)
then progressive wave solutions of simple harmonic behavior
in time are possible, i.e.,
where 
is the frequency (rad/sec).
is a spatially variable amplitude and 
is the spatially variable part of the phase function
If the coefficients of the linear equations controlling F
are independent of , then solutions for which A is
constant and S is linear in are possible.
Such solutions are called plane waves since
the phase surfaces 
= constant are planes, i.e.,
where 
is a wave number vector with Cartesian
components 
, 
, and 
.
Another possible plane wave is
for which the wave number vector is horizontal and the
plane phase surfaces are vertical.
A surface gravity wave is an example.
More generally may be non-linear in and hence
the phase surfaces are curved and/or non-uniformly spaced.
In any event a local wave number vector can be
identified with 
. Thus
The rate of displacement of the phase surface = constant in
the direction of is defined as the phase velocity
.
It's magnitude 
is obtained from (1.5) with
= 0:
where 
and 
.
This is the speed at which a wave crest (maximum in F) advances
in a direction normal to the surface on which = constant.
As a vector
The components of are thus
These are to be contrasted to the trace speeds
, 
, and 
which
correspond to the rate of advance of the intercept of a phase
plane with one of the coordinate axes.
The trace speeds always exceed the value of the
phase speed .