Sinusoidal plane wave

A plane sinusoidal wave may also be expressed in terms of the complex exponential function

${\displaystyle e^{{\boldsymbol {i}}z}=\exp({\boldsymbol {i}}z)=\cos z+{\boldsymbol {i}}\sin z}$

where ${\displaystyle e}$ is the base of the natural exponential function, and ${\displaystyle {\boldsymbol {i}}\,}$ is the imaginary unit, defined by the equation ${\displaystyle {\boldsymbol {i}}^{2}=-1}$. With those tools, one defines the complex exponential plane wave as

${\displaystyle U({\vec {x}},t)\;=\;A\exp[{\boldsymbol {i}}(2\pi \nu ({\vec {x}}\cdot {\vec {n}}-ct)+\varphi )]\;=\;A\exp[{\boldsymbol {i}}(2\pi {\vec {x}}\cdot {\vec {v}}-\omega t+\varphi )]}$

where ${\displaystyle A,\nu ,{\vec {n}},c,{\vec {v}},\omega ,\varphi }$ are as defined for the (real) sinusoidal plane wave. This equation gives a field ${\displaystyle U({\vec {x}},t)}$ whose value is a complex number, or a vector with complex coordinates. To obtain the

${\displaystyle F({\vec {x}},t)={\text{Re}}[U({\vec {x}},t)]}$

To appreciate this equation's relationship to the earlier ones, below is this same equation expressed using sines and cosines. Observe that the first term equals the real form of the plane wave just discussed.

${\displaystyle U({\vec {x}},t)=A\cos(2\pi \nu {\vec {n}}\cdot {\vec {x}}-\omega t+\varphi )+{\boldsymbol {i}}A\sin(2\pi \nu {\vec {n}}\cdot {\vec {x}}-\omega t+\varphi )}$

${\displaystyle U({\vec {x}},t)=\qquad \ \ F({\vec {x}},t)\qquad \qquad +{\boldsymbol {i}}A\sin(2\pi \nu {\vec {n}}\cdot {\vec {x}}-\omega t+\varphi )}$

The introduced complex form of the plane wave can be simplified by using a complex-valued amplitude ${\displaystyle C\,}$ substitute the real valued amplitude ${\displaystyle A\,}$.
Specifically, since the complex form

${\displaystyle \exp[{\boldsymbol {i}}(2\pi {\vec {x}}\cdot {\vec {v}}-\omega t+\varphi )]\;=\;\exp[{\boldsymbol {i}}(2\pi \nu {\vec {n}}\cdot {\vec {x}}-\omega t)]\,e^{{\boldsymbol {i}}\varphi }}$

one can absorb the phase factor ${\displaystyle e^{{\boldsymbol {i}}\varphi }}$ into a complex amplitude by letting ${\displaystyle C=Ae^{{\boldsymbol {i}}\varphi }}$, resulting in the more compact equation

${\displaystyle U({\vec {x}},t)=C\exp[{\boldsymbol {i}}(2\pi {\vec {x}}\cdot {\vec {v}}-\omega t)]}$

While the complex form has an imaginary component, after the necessary calculations are performed in the complex plane, its real value can be extracted giving a real valued equation representing an actual plane wave.

${\displaystyle \operatorname {Re} [U({\vec {x}},t)]=F({\vec {x}},t)=A\cos(2\pi \nu {\vec {n}}\cdot {\vec {x}}-\omega t+\varphi )}$

The main reason one would choose to work with complex exponential form of plane waves is that complex exponentials are often algebraically easier to handle than the trigonometric sines and cosines. Specifically, the angle-addition rules are extremely simple for exponentials.

Additionally, when using Fourier analysis techniques for waves in a lossy medium, the resulting attenuation is easier to deal with using complex Fourier coefficients. If a wave is traveling through a lossy medium, the amplitude of the wave is no longer constant, and therefore the wave is strictly speaking no longer a true plane wave.

In quantum mechanics the solutions of the Schrödinger wave equation are by their very nature complex-valued and in the simplest instance take a form identical to the complex plane wave representation above. The imaginary component in that instance however has not been introduced for the purpose of mathematical expediency but is in fact an inherent part of the “wave”.

In special relativity, one can utilize an even more compact expression by using four-vectors.

The four-position ${\displaystyle {\vec {x}}=(ct,{\vec {x}})}$

The four-wavevector ${\displaystyle 2\pi \nu {\vec {n}}=\left({\frac {\omega }{c}},2\pi \nu {\vec {n}}\right)}$

The scalar product ${\displaystyle 2\pi \nu {\vec {n}}\cdot {\vec {x}}=\omega t-2\pi \nu {\vec {n}}\cdot {\vec {x}}}$

Thus,

${\displaystyle U({\vec {x}},t)=C\exp[{\boldsymbol {i}}(2\pi \nu {\vec {n}}\cdot {\vec {x}}-\omega t)]}$

becomes

${\displaystyle U({\vec {x}})=C\exp[-{\boldsymbol {i}}(2\pi \nu {\vec {n}}\cdot {\vec {x}})]}$

The equations describing electromagnetic radiation in a homogeneous dielectric medium admit as special solutions that are sinusoidal plane waves. In electromagnetism, the field ${\displaystyle F}$ is typically the electric field, magnetic field, or vector potential, which in an isotropic medium is perpendicular to the direction of propagation ${\displaystyle {\vec {n}}}$. The amplitude ${\displaystyle A}$ is then a vector of the same nature, equal to the maximum-strength field. The propagation speed ${\displaystyle c}$ will be the speed of light in the medium.

The equations that describe vibrations in a homogeneous elastic solid also admit solutions that are sinusoidal plane waves, both transverse and longitudinal. These two types have different propagation speeds, that depend on the density and the Lamé parameters of the medium.

The fact that the medium imposes a propagation speed means that the parameters ${\displaystyle \omega }$ and ${\displaystyle k}$ must satisfy a dispersion relation characteristic of the medium. The dispersion relation is often expressed as a function, ${\displaystyle \omega (k)}$. The ratio ${\displaystyle \omega /|k|}$ gives the magnitude of the phase velocity, and the derivative ${\displaystyle \partial \omega /\partial k}$ gives the group velocity. For electromagnetism in an isotropic medium with index of refraction ${\displaystyle r}$, the phase velocity is ${\displaystyle c/r}$, which equals the group velocity if the index is not frequency-dependent.

In linear uniform media, a general solution to the wave equation can be expressed as a superposition of sinusoidal plane waves. This approach is known as the angular spectrum method. The form of the planewave solution is actually a general consequence of translational symmetry. More generally, for periodic structures having discrete translational symmetry, the solutions take the form of Bloch waves, most famously in crystalline atomic materials but also in photonic crystals and other periodic wave equations. As another generalization, for structures that are only uniform along one direction x (such as a waveguide along the x direction), the solutions (waveguide modes) are of the form exp[i(kx-ωt)] multiplied by some amplitude function a(y,z). This is a special case of a separable partial differential equation.