wave vector (Rev #3) in nLab

Contents

Idea

A wave vector is a vector that encodes wavelength and direction of a plane wave.

Definition

Let n∈ℕn \in \mathbb{N} and write ℝ n\mathbb{R}^n the Cartesian space of dimension nn. Thinking of ℝ n\mathbb{R}^n as a vector space, then each point in it is a vector x→∈ℝ n\vec x \in \mathbb{R}^n and hence a smooth function f:ℝ n→ℂf \colon \mathbb{R}^n \to \mathbb{C} may be thought of as a function of these “position vectors”.

If ff is a function with rapidly decreasing partial derivatives, then its Fourier transform f^:ℝ n→ℂ\hat f \;\colon\; \mathbb{R}^n \to \mathbb{C} exists. By the Fourier inversion theorem, this function is such that it expresses ff as a superposition of “plane wave” functions x→↦e 2πix→⋅k→\vec x \mapsto e^{2\pi i \vec x \cdot \vec k} as

f(x→)=∫k→∈ℝ nf^(k)e 2πik→⋅x→dk→. f(\vec x) \;=\; \underset{\vec k \in \mathbb{R}^n}{\int} \hat f(k) \, e^{2 \pi i \vec k \cdot \vec x} \, d \vec k \,.

Here the vector k→∈ℝ n\vec k \in \mathbb{R}^n determines

1. the wavelength λ≔1/|k→|\lambda \coloneqq 1/{\vert \vec k\vert} (the inverse of the norm of k→\vec k);

2. the direction k→|k→|∈S(ℝ n)\frac{\vec k}{{\vert \vec k\vert }} \in S(\mathbb{R}^n) (the corresponding unit vector in the unit sphere)

of the “plane wave” x→↦e 2πix→⋅k→\vec x \mapsto e^{2 \pi i \vec x \cdot \vec k}.

If here ℝ n≃ℝ p,1\mathbb{R}^n \simeq \mathbb{R}^{p,1} is identified with Minkowski spacetime with canonical coordinates denoted (x 0,x 1,⋯,x p)(x^0, x^1, \cdots, x^p), then the 0-component of the wave vector

ν≔k 0 \nu \coloneqq k_0

is called the frequency of the corresponding plane wave (in the chosen coordinate system).

• plane wave

• Fourier transform

• wave

• frequency, amplitude

References

See also

• Wikipedia, Wave vector