2.2: Plane-Waves

As we have just seen, a wave of amplitude $A$, wavenumber $k$, angular frequency $\omega$, and phase angle $\varphi$, propagating in the positive $x$-direction, is represented by the following wavefunction:

$$\label{e10.1} \psi(x,t)=A\,\cos(k\,x-\omega\,t+\varphi).$$ This type of wave is conventionally termed a one-dimensional plane-wave. It is one-dimensional because its associated wavefunction only depends on the single Cartesian coordinate, $x$. Furthermore, it is a plane-wave because the wave maxima, which are located at

$$\label{e10.2} k\,x-\omega\,t+\varphi = j\,2\pi,$$ 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 positive $x$-axis at the velocity $v=\omega/k$. These conclusions follow because Equation (2.2.2) can be rewritten in the form

$$\label{e10.3} x= d,$$ where $d=(j-\varphi/2\pi)\,\lambda + v\,t$. Moreover, as is well known, Equation (2.2.3) is the equation of a plane, normal to the $x$-axis, whose distance of closest approach to the origin is $d$.

Figure 1: The solution of $\begin{equation}\mathbf{n} \cdot \mathbf{r}=d\end{equation}$ is a plane.

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

$$\label{e10.4} {\bf n}\cdot{\bf r} = d,$$ where ${\bf n} = (1,\,0,\,0)$ is a unit vector directed along the positive $x$-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 reinterpreted as a unit vector pointing in an arbitrary direction then Equation (2.2.4) can be reinterpreted as the general equation of a plane. As before, the plane is normal to ${\bf n}$, and its distance of closest approach to the origin is $d$. See Figure [f10.1]. This observation allows us to write the three-dimensional equivalent to the wavefunction (2.2.1) as

$$\label{e10.5} \psi({\bf r},t)=A\,\cos({\bf k}\cdot{\bf r}-\omega\,t+\varphi),$$

where the constant vector ${\bf k} = (k_x,\,k_y,\,k_z)=k\,{\bf n}$ is called the wavevector. The wave represented previously is conventionally termed a three-dimensional plane-wave. It is three-dimensional because its wavefunction, $\psi({\bf r},t)$, depends on all three Cartesian coordinates. Moreover, it is a plane-wave because the wave maxima are located at $${\bf k}\cdot{\bf r} -\omega\,t +\varphi= j\,2\pi,$$ or $${\bf n}\cdot{\bf r} = (j-\varphi/2\pi)\,\lambda + v\,t,$$ where $\lambda=2\pi/k$, and $v=\omega/k$. Note that the wavenumber, $k$, is the magnitude of the wavevector, ${\bf k}$: that is, $k\equiv |{\bf k}|$. It follows, by comparison with Equation (2.2.4), that the wave maxima consist of a series of parallel planes, normal to the wavevector, that are equally spaced a distance $\lambda$ apart, and that propagate in the ${\bf k}$-direction at the velocity $v$. See Figure [f10.2]. Hence, the direction of the wavevector specifies the wave propagation direction, whereas its magnitude determines the wavenumber, $k$, and, thus, the wavelength, $\lambda=2\pi/k$.

Figure 2: Wave maxima associated with a three-dimensional plane wave.

Contributors and Attributions

• Richard Fitzpatrick (Professor of Physics, The University of Texas at Austin)

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