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As we have just seen, a wave of amplitude , wavenumber , angular frequency , and phase
angle ,
propagating in the positive direction, is represented by the following wavefunction:

(29) 
Now, the type of wave represented above is conventionally termed a onedimensional plane wave. It is onedimensional
because its associated wavefunction only depends on the single Cartesian coordinate .
Furthermore, it is a plane wave because the wave maxima, which are located at

(30) 
where is an integer, consist of a series of parallel planes, normal to the axis, which are equally spaced a distance
apart, and propagate along the
positive axis at the velocity .
These conclusions follow because Eq. (30) can be rewritten in the form

(31) 
where
. Moreover, as is wellknown, (31)
is the equation of a plane, normal to the axis, whose distance of closest approach to the
origin is .
Figure 1:
The solution of
is a plane.

The previous equation can also be written in the coordinatefree form

(32) 
where
is a unit
vector directed along the positive axis, and
represents the vector displacement of a general point from the origin. Since there is nothing special about the direction, it follows that if is reinterpreted as a
unit vector pointing in an arbitrary direction then (32) can be reinterpreted as the general equation of a plane.
As before, the plane is normal to
, and its distance of closest approach to the origin is . See Fig. 1. This observation allows us to write the threedimensional
equivalent to the wavefunction (29) as

(33) 
where the constant vector
is called the wavevector. The wave represented above is conventionally termed
a threedimensional plane wave. It is threedimensional because its wavefunction, , depends on all
three Cartesian coordinates. Moreover, it is a plane wave because the wave maxima are located at

(34) 
or

(35) 
where
, and . Note that the wavenumber, , is the
magnitude of the wavevector, : i.e.,
.
It follows, by comparison with Eq. (32), that the
wave maxima consist of a series of parallel planes, normal to the wavevector, which are equally spaced a distance apart, and which propagate in the direction at the velocity . See Fig. 2. Hence, the direction of the wavevector specifies the wave propagation direction, whereas its magnitude determines the wavenumber, , and, thus, the wavelength,
.
Figure 2:
Wave maxima associated with a threedimensional plane wave.

Next: Representation of Waves via
Up: WaveParticle Duality
Previous: Wavefunctions
Richard Fitzpatrick
20100720