travellingwave

A traveling wave is a wave that propagates through space or a medium, transporting energy and information without requiring the medium itself to undergo a net displacement. Its waveform moves along a direction and, in one dimension, can be described by a function of x − ct for a wave moving in the positive x direction or x + ct for a wave moving in the negative x direction, where c is the propagation speed.

In harmonic form, a simple one-dimensional traveling wave is A cos(kx − ωt + φ) or A e^{i(kx − ωt)}. The phase velocity v_p = ω/k; the group velocity v_g = dω/dk. In nondispersive media ω = c k, so v_p = v_g = c. In dispersive media the two can differ, causing the waveform to distort as it travels.

Traveling waves arise across many domains, including acoustics, electromagnetism, optics, fluid dynamics, and quantum mechanics. They can take the shape of plane waves, spherical waves, or more complex forms. Boundary conditions and the properties of the medium determine the forward and backward traveling components, which can be treated by linear superposition.

Real traveling waves often experience attenuation due to absorption and scattering, and may undergo dispersion that alters their shape over distance. In nonlinear media, solitary traveling waves called solitons can occur, maintaining shape through a balance between nonlinearity and dispersion.

A standing wave can be viewed as the result of the interference of two traveling waves moving in opposite directions with equal amplitude and frequency, illustrating the close relationship between the two concepts. Traveling-wave analysis underpins much of communication theory, antenna design, and waveguide modeling.