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# Plane Waves

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 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 (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 (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 . The previous equation can also be written in the coordinate-free 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 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 (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 (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, .    Next: Representation of Waves via Up: Wave-Particle Duality Previous: Wavefunctions
Richard Fitzpatrick 2010-07-20