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Plane Waves

As we saw in the previous chapter, a sinusoidal wave of amplitude $ \psi_0>0$ , wavenumber $ k>0$ , and angular frequency $ \omega>0$ , propagating in the positive $ x$ -direction, can be represented in terms of a wavefunction of the form

$\displaystyle \psi(x,t)=\psi_0\,\cos(\omega\,t-k\,x).$ (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

$\displaystyle \omega\,t -k\,x = j\,2\pi,$ (530)

where $ j$ is an integer, consist of a series of parallel planes, normal to the $ x$ -axis, that are equally spaced a distance $ \lambda=2\pi/k$ apart, and propagate along the $ x$ -axis at the fixed phase velocity $ \omega/k=v$ , where $ v$ is the characteristic wave speed. These conclusions follow because Equation (530) can be re-written in the form

$\displaystyle x= d,$ (531)

where $ d=v\,t-j\,\lambda$ . Moreover, (531) is the equation of a plane, normal to the $ x$ -axis, whose distance of closest approach to the origin is $ d$ .

Figure 38: The solution of $ {\bf n}\cdot {\bf r} = d$ is a plane.
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The previous equation can also be written in the coordinate-free form

$\displaystyle {\bf n}\cdot{\bf r} = d,$ (532)

where $ {\bf n} = (1,\,0,\,0)$ is a unit vector directed along the $ x$ -axis, and $ {\bf r}=(x,\,y,\,z)$ represents the vector displacement of a general point from the origin. Since there is nothing special about the $ x$ -direction, it follows that if $ {\bf n}$ 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 $ {\bf n}$ , and its distance of closest approach to the origin is $ d$ . (See Figure 38.) This observation allows us to write the three-dimensional equivalent to the wavefunction (529) as

$\displaystyle \psi(x,y,z,t)=\psi_0\,\cos(\omega\,t-{\bf k}\cdot{\bf r}),$ (533)

where the constant vector $ {\bf k} = (k_x,\,k_y,\,k_z)=k\,{\bf n}$ 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, $ \psi(x,y,z,t)$ , depends on all three Cartesian coordinates. Moreover, it is a plane wave because the wave maxima are located at

$\displaystyle \omega\,t-{\bf k}\cdot{\bf r} = j\,2\pi,$ (534)

or

$\displaystyle {\bf n}\cdot{\bf r} = v\,t-j\,\lambda,$ (535)

where $ \lambda=2\pi/k$ , and $ \omega/k=v$ . The wavenumber, $ k$ , is the magnitude of the wavevector, $ {\bf k}$ . That is, $ k= \vert{\bf k}\vert$ . 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 $ \lambda$ apart, and propagate in the $ {\bf k}$ -direction at the fixed phase velocity $ \omega/k=v$ . (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 $ \psi = \psi_0\,\cos(\omega\,t-{\bf k}\cdot{\bf r}-\phi)$ , where $ \phi$ 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 $ -(\phi/2\pi)\,\lambda$ in the $ {\bf k}$ -direction. In the following, whenever possible, $ \phi$ is set to zero, for the sake of simplicity.

Figure 39: Wave maxima associated with a plane wave.
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next up previous
Next: Three-Dimensional Wave Equation Up: Multi-Dimensional Waves Previous: Multi-Dimensional Waves
Richard Fitzpatrick 2013-04-08