Plane wave

A plane wave is a wave whose wavefront is a plane everywhere. These occur when the wavefunction is only dependent on the coordinate the wave propagates in. The simplest and most important form of plane wave is a monochromatic or sinusoidal plane wave (often just called a sine wave), whose general equation is

u(r,t)=Asin⁡(k⋅r−ωt+φ)u(\mathbf{r},t)=A\sin(\mathbf{k}\cdot \mathbf{r}-\omega t+\varphi)

where

  • AA is constant interpreted as the amplitude of the wave.
  • k\mathbf{k} is the wavevector. This is constant in a monochromatic wave.
  • ω\omega is the angular frequency. This is constant in a monochromatic wave.
  • φ\varphi is the phase.

Moreover, a nondispersive sine wave has dispersion relation ω=∣k∣v\omega=\lvert \mathbf{k} \rvert v and therefore a phase velocity of v=ω/∣k∣v=\omega/\lvert \mathbf{k} \rvert.

Often, the sine wave is given as a complex wave using complex exponentials:

u~(r,t)=A~ei[k⋅r−ωt+φ]\tilde{u}(\mathbf{r},t)=\tilde{A}e^{i[\mathbf{k}\cdot \mathbf{r}-\omega t+\varphi]}

This is achieved by adding a new imaginary term to the sine wave and then using Euler's formula. The real part of the complex wave is the previous equation: Re[u~(r,t)]=u(r,t)\text{Re}[\tilde{u}(\mathbf{r},t)]=u(\mathbf{r},t). The benefit of this formulation is that it's often a lot easier to work with complex exponentials that it is to deal with sines and cosines. The constant part of this representation, A~eiφ\tilde{A}e^{i\varphi}, is called a phasor.

The importance of plane waves comes from the fact that every wave can be seen as being locally plane at small enough scales, due to any geometric Surface being approximately plane for small enough regions. The importance of sine waves in particular is different though. It comes from the field of Fourier analysis, where the Fourier series guarantees that, under some conditions, almost any function can be represented as a (possibly infinite) sum of sines. As such, pretty much any wave can be rewritten as a big sum of sine waves and so, by just knowing how these work, we know in principle how any wave works.

Definition

The condition for a plane wave is that the wavefront must be a plane at any given time. To start, consider a plane Surface oriented in no particular way in three dimensional space. Call k\mathbf{k} the normal vector to the surface, r\mathbf{r} and r0\mathbf{r}_{0} vectors pointing to two points on the surface.

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In Cartesian coordinates we can represent the vectors as

r=xx^+yy^+zz^,r0=x0x^+y0y^+z0z^\mathbf{r}=x \hat{\mathbf{x}}+y \hat{\mathbf{y}}+ z \hat{\mathbf{z}},\qquad\mathbf{r}_{0}=x_{0} \hat{\mathbf{x}}+y_{0} \hat{\mathbf{y}}+ z_{0} \hat{\mathbf{z}}

Their difference is

r−r0=(x−x0)x^+(y−y0)y^+(z−z0)z^\mathbf{r}-\mathbf{r}_{0}=(x-x_{0}) \hat{\mathbf{x}}+(y-y_{0}) \hat{\mathbf{y}}+ (z-z_{0}) \hat{\mathbf{z}}

For the surface to be a plane, the difference must always be perpendicular to the normal (and so parallel to the plane). Our condition then is

(r−r0)⋅k=0(\mathbf{r}-\mathbf{r}_{0})\cdot \mathbf{k}=0

so

(x−x0)kxx^+(y−y0)kyy^+(z−z0)kzz^=0(x-x_{0})k_{x} \hat{\mathbf{x}}+(y-y_{0})k_{y} \hat{\mathbf{y}}+ (z-z_{0})k_{z} \hat{\mathbf{z}}=0

We can regroup the terms as

xkxx^+ykyy^+zkzz^⏟r⋅k−(x0kxx^+y0kyy^+z0kzz^)⏟r0⋅k≡a=0\underbrace{ xk_{x} \hat{\mathbf{x}}+yk_{y} \hat{\mathbf{y}}+zk_{z}\hat{\mathbf{z}} }_{ \mathbf{r}\cdot \mathbf{k} }-\underbrace{ (x_{0}k_{x} \hat{\mathbf{x}}+y_{0}k_{y} \hat{\mathbf{y}}+z_{0}k_{z} \hat{\mathbf{z}}) }_{ \mathbf{r}_{0}\cdot \mathbf{k}\equiv a }=0

With this, we get

r⋅k=a=constant\boxed{\mathbf{r}\cdot \mathbf{k}=a=\text{constant}}

The connection to waves should be readily apparent now: a wave is plane if the position vector is always orthogonal to the wavevector.