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

$$\psi(x,t)=\psi_0\,\cos(\omega\,t-k\,x).$$ (7.1)

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

$$\omega\,t -k\,x = j\,2\pi,$$ (7.2)

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 (7.2) can be rewritten in the form

$$x= d,$$ (7.3)

where $d=v\,t-j\,\lambda$. Moreover, Equation (7.3) is the equation of a plane, normal to the $x$-axis, whose distance of closest approach to the origin is $d$.

Figure 7.1: The solution of ${\bf n}\cdot {\bf r} = d$ is a plane.

The previous equation can also be written in the coordinate-free form

$${\bf n}\cdot{\bf r} = d,$$ (7.4)

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. Because 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 (7.4) 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 7.1. This observation allows us to write the three-dimensional equivalent to the wavefunction (7.1) as

$$\psi(x,y,z,t)=\psi_0\,\cos(\omega\,t-{\bf k}\cdot{\bf r}),$$ (7.5)

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

$$\omega\,t-{\bf k}\cdot{\bf r} = j\,2\pi,$$ (7.6)

or

$${\bf n}\cdot{\bf r} = v\,t-j\,\lambda,$$ (7.7)

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 (7.4), 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 7.2. 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 7.2: Wave maxima associated with a plane wave.