The frequency and wave-vector are in direct proportion through
the constant c. To understand this, we consider a typical standing
wave . Inserting into the wave
equation, each spatial derivative gives a factor of k and each time
derivative gives a factor of . Thus, we'll find , and so . We also understand that c, therefore, is
the wave-speed because then
.
(Eq. )
-
. Inserting into the wave
equation, each spatial derivative gives a factor of k and each time
derivative gives a factor of 
. Thus, we'll find 
, and so 
.