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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:
![\begin{displaymath}
\psi(x,t)=A \cos(k x-\omega t+\varphi).
\end{displaymath}](img170.png) |
(29) |
Now, the type of wave represented above is conventionally termed a one-dimensional plane wave. It is one-dimensional
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
![\begin{displaymath}
k x-\omega t+\varphi = j 2\pi,
\end{displaymath}](img171.png) |
(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 re-written in the form
![\begin{displaymath}
x= d,
\end{displaymath}](img172.png) |
(31) |
where
. Moreover, as is well-known, (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.
![\begin{figure}
\epsfysize =2.5in
\centerline{\epsffile{Chapter03/fig01.eps}}
\end{figure}](img176.png) |
The previous equation can also be written in the coordinate-free form
![\begin{displaymath}
{\bf n}\cdot{\bf r} = d,
\end{displaymath}](img177.png) |
(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 re-interpreted as a
unit vector pointing in an arbitrary direction then (32) can be re-interpreted 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 three-dimensional
equivalent to the wavefunction (29) as
![\begin{displaymath}
\psi(x,y,z,t)=A \cos({\bf k}\cdot{\bf r}-\omega t+\varphi),
\end{displaymath}](img181.png) |
(33) |
where the constant vector
is called the wavevector. The wave represented above is conventionally termed
a three-dimensional plane wave. It is three-dimensional because its wavefunction,
, depends on all
three Cartesian coordinates. Moreover, it is a plane wave because the wave maxima are located at
![\begin{displaymath}
{\bf k}\cdot{\bf r} -\omega t +\varphi= j 2\pi,
\end{displaymath}](img184.png) |
(34) |
or
![\begin{displaymath}
{\bf n}\cdot{\bf r} = (j-\varphi/2\pi) \lambda + v t,
\end{displaymath}](img185.png) |
(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 three-dimensional plane wave.
![\begin{figure}
\epsfysize =2.5in
\centerline{\epsffile{Chapter03/fig02.eps}}
\end{figure}](img189.png) |
Next: Representation of Waves via
Up: Wave-Particle Duality
Previous: Wavefunctions
Richard Fitzpatrick
2010-07-20