The wave vector (denoted by the symbol k) is a mathematical vector that represents the spatial and directional properties of a wave. It provides essential information about the wave's propagation direction, magnitude, wavelength, and phase. The wave vector is typically measured in meters^-1 (m^-1).

The magnitude of the wave vector is called the wave number and is given by, 

Where λ is the wavelength in the medium

The wave vector is always perpendicular to the wavefronts. In isotropic media, the wave vector has the same direction as the direction of wave propagation. But in the case of a non-isotropic media, the energy flow direction, represented by the Poynting vector, may exhibit a certain degree of deviation from the wave vector's direction. This occurrence is commonly referred to as spatial walk-off. i.e., When a wave traverses an anisotropic medium, like light waves passing through an asymmetric crystal or sound waves propagating through a sedimentary rock, the wave vector may deviate from the exact direction of wave propagation.

Spatial walk-off happens because the wave experiences different refractive indices in different spatial directions within the anisotropic medium. This variation in refractive index causes the wave vector to deviate from the direction of wave propagation. Consequently, the energy flow, represented by the Poynting vector (S), may exhibit a certain degree of deviation from the wave vector's direction. The Poynting vector describes the direction and magnitude of energy flow associated with the wave.

Mathematically, the Poynting vector is given by the cross product of the electric field (E) and magnetic field (B) vectors:

In certain materials that can absorb or amplify light or other waves, the wave vector can have some special properties. For example, it may have complex components which means that it has both real and imaginary parts.

An evanescent wave is a special type of wave that exists near the surface of a material and doesn't travel far. In the presence of an evanescent wave, the wave vector can be purely imaginary, meaning it only has an imaginary part and no real part.

These complex components and imaginary parts in the wave vector are associated with how the wave interacts with the material. They are essential for understanding the behavior of waves in such materials and are important in various applications, including optical devices like waveguides and fibers. 

Applications of Wave Vector

The wave vector has various applications across different fields. In solid-state physics, the wave vector is crucial for studying the electronic band structure of materials. It helps to determine the dispersion relation, which describes how waves propagate through a crystal lattice. This information is important for understanding the behavior of electrons and their interactions with the crystal lattice, leading to insights into electrical conductivity, magnetism, and optical properties of materials. 

In optics, the wave vector plays a central role in analyzing the behavior of light waves. It helps define the direction, phase, and polarization of light, enabling the design and optimization of optical devices such as lenses, mirrors, and fiber optics. 

In acoustics, the wave vector helps characterize the propagation of sound waves, aiding in the understanding of the behavior of sound in different media and the design of acoustic systems and devices. Also, the wave vector finds applications in fields like quantum mechanics, signal processing, and wave phenomena in fluids, contributing to advancements in various scientific and technological domains.