Plane Waves in Two and Three Dimensions

From Strings to Fields

Imagine a stretched drumhead. When you strike it, the disturbance does not travel in one direction only; it spreads out in all directions across the surface. In one dimension, a wave is a simple function of position \(x\) and time \(t\). In two dimensions, the displacement \(u\) depends on \(x\), \(y\), and \(t\). In three dimensions, it depends on \(x\), \(y\), \(z\), and \(t\). The core idea remains the same: the acceleration of a point in the medium is proportional to the curvature of the displacement field around that point. This curvature is measured by the Laplacian operator, which generalises the second derivative with respect to position. A plane wave in this context is a wave whose crests and troughs are flat planes, moving perpendicular to those planes. It is the simplest possible wave in higher dimensions, analogous to a pure sine wave in one dimension.

The Multi-Dimensional Wave Equation

For a homogeneous, isotropic medium, the wave equation in \(n\) spatial dimensions is given by:

where \(c\) is the wave speed and \(\nabla^2\) is the Laplacian operator. In three Cartesian coordinates, the Laplacian is defined as:

A plane wave travelling in the direction of a unit vector \(\hat{\mathbf{n}}\) can be written as a function of the projection of the position vector \(\mathbf{r}\) onto \(\hat{\mathbf{n}}\). Let \(\mathbf{k} = k \hat{\mathbf{n}}\) be the wave vector, where \(k\) is the wavenumber. The phase is \(\phi = \mathbf{k} \cdot \mathbf{r} - \omega t\). A general plane wave solution is:

Substituting this into the wave equation shows that it is a valid solution provided the dispersion relation \(\omega = c k\) holds. This confirms that in a non-dispersive medium, the speed \(c\) is independent of frequency and direction.

Wave Vectors and Direction

The wave vector \(\mathbf{k}\) encodes both the spatial frequency and the direction of propagation. Its magnitude \(k = 2\pi/\lambda\) determines the wavelength, while its direction \(\hat{\mathbf{n}}\) indicates the normal to the wavefronts. In two dimensions, if a wave travels at an angle \(\theta\) to the \(x\)-axis, we can decompose \(\mathbf{k}\) into components \(k_x = k \cos \theta\) and \(k_y = k \sin \theta\). The phase becomes \(\phi = k_x x + k_y y - \omega t\). This decomposition is crucial for understanding interference and diffraction, where multiple plane waves with different \(\mathbf{k}\) vectors superpose to form complex patterns. The surfaces of constant phase are planes defined by \(\mathbf{k} \cdot \mathbf{r} = \text{constant}\), which are perpendicular to \(\mathbf{k}\).

Dispersion in Higher Dimensions

While the 1D analysis of dispersion is intuitive, it generalises directly to \(N\) dimensions. The dispersion relation \(\omega = \omega(\mathbf{k})\) may depend on the magnitude \(k = |\mathbf{k}|\) and, in anisotropic media, on the direction of \(\mathbf{k}\). For an isotropic medium, \(\omega\) depends only on \(k\). The phase velocity is still \(v_p = \omega/k\). However, the group velocity, which represents the energy transport direction, is given by the gradient of \(\omega\) with respect to \(\mathbf{k}\):

In an isotropic medium, this vector points in the same direction as \(\mathbf{k}\), and its magnitude is \(v_g = d\omega/dk\). If the medium is dispersive, \(v_g \neq v_p\), and the wave packet spreads out as it propagates. This spreading is a fundamental feature of wave motion in higher dimensions and is responsible for phenomena such as chromatic dispersion in optics.