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 $$$$\mathbf{n} \cdot \mathbf{r}=d$$$$ 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.
