BELIEVE ME NOT!   A SKEPTIC's GUIDE
Next: Speed of Propagation
Up: Wave Phenomena
Previous: Wave Phenomena
How do we represent this behaviour mathematically?
Well, is a function of position
and time : .
At any fixed position ,
oscillates in time at a frequency .
We can describe this statement mathematically by saying that
the entire time dependence of is contained in
[the real part of]
a factor
(that is, the amplitude at any fixed position obeys
SHM).^{14.2}
The oscillation with respect to position
at any instant of time is given by the analogous
factor
where
is the wave vector;^{14.3}it points in the direction of propagation of the wave
and has a magnitude (called the "wavenumber") given by

(14.1) 
where is the wavelength.
Note the analogy between and

(14.2) 
where is the period of the oscillation in time
at a given point. You should think of
as the "period in space."
We may simplify the above description by
choosing our coordinate system so that the axis
is in the direction of ,
so that^{14.4}
.
Then the amplitude no longer depends on or ,
only on and .
We are now ready to give a full description of the
function describing this wave:
or, recalling the multiplicative property of the exponential function,
,

(14.3) 
To achieve complete generality we can restore the vector version:

(14.4) 
This is the preferred form for a general description of a PLANE WAVE,
but for present purposes the scalar version (3) suffices.
Using Eqs. (1) and (2) we can also write the
plane wave function in the form

(14.5) 
but you should strive to become completely comfortable with
and  we will be seeing a lot of them in Physics!
Footnotes
 . . . SHM).^{14.2}
 Note that
would have worked just as well, since the real part is the same
as for
. The choice of sign
does matter, however, when we write down the combined
time and space dependence in Eq. (4), which see.
 . . . vector;^{14.3}
 The name "wave vector"
is both apt and inadequate  apt because the term vector
explicitly reminds us that its direction defines the direction of
propagation of the wave; inadequate because the essential inverse
relationship between and the wavelength
[see Eq. (1)] is not suggested by the name. Too bad.
It is at least a little more descriptive than the name given to
the magnitude of ,
namely the "wavenumber."
 . . . that^{14.4}
 In general
.
If
then and ,
giving
.
Next: Speed of Propagation
Up: Wave Phenomena
Previous: Wave Phenomena
Jess H. Brewer 
Last modified: Sun Nov 15 17:57:49 PST 2015