   Next: Three-Dimensional Wave Equation Up: Multi-Dimensional Waves Previous: Multi-Dimensional Waves

# Plane Waves

As we saw in the previous chapter, a sinusoidal wave of amplitude , wavenumber , and angular frequency , propagating in the positive -direction, can be represented in terms of a wavefunction of the form (529)

This type of wave is conventionally termed a one-dimensional plane wave. It is one-dimensional because its associated wavefunction only depends on a single Cartesian coordinate. Furthermore, it is a plane wave because the wave maxima, which are located at (530)

where is an integer, consist of a series of parallel planes, normal to the -axis, that are equally spaced a distance apart, and propagate along the -axis at the fixed phase velocity , where is the characteristic wave speed. These conclusions follow because Equation (530) can be re-written in the form (531)

where . Moreover, (531) 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 (532)

where is a unit vector directed along the -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 Equation (532) can be re-interpreted as the general equation of a plane (Fitzpatrick 2008). As before, the plane is normal to , and its distance of closest approach to the origin is . (See Figure 38.) This observation allows us to write the three-dimensional equivalent to the wavefunction (529) as (533)

where the constant vector is known as the wavevector. The wave represented in the previous expression 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 (534)

or (535)

where , and . The wavenumber, , is the magnitude of the wavevector, . That is, . It follows, by comparison with Equation (532), that the wave maxima consist of a series of parallel planes, normal to the wavevector, that are equally spaced a distance apart, and propagate in the -direction at the fixed phase velocity . (See Figure 39.) Hence, the direction of the wavevector corresponds to the direction of wave propagation. The most general expression for the wavefunction of a three-dimensional plane wave is , where is a constant phase angle. As is readily demonstrated, the inclusion of a non-zero phase angle in the wavefunction shifts all the wave maxima a distance in the -direction. In the following, whenever possible, is set to zero, for the sake of simplicity.    Next: Three-Dimensional Wave Equation Up: Multi-Dimensional Waves Previous: Multi-Dimensional Waves
Richard Fitzpatrick 2013-04-08