Dictionary:Wave

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A disturbance that is propagated through the body or on the surface of a medium without involving net movement of material. Waves are usually characterized by periodicity (Figure W-2). The general expression for a plane wave in rectangular coordinates is

${\displaystyle f(lx+my+nz-Vt)+g(lx+my+nz+Vt)}$,

where f and g are any functions and l, m, n are direction cosines for the direction of travel.

For a spherical wave, the general expression is

${\displaystyle {\frac {1}{r}}f(r-Vt)+{\frac {1}{r}}g(r+Vt),}$

where r is the distance from the point source. Wave amplitude is usually defined as the maximum displacement from the equilibrium or null position. The rms amplitude is the square root of the mean of the squares of the displacements, which is ${\displaystyle {\frac {\sqrt {2}}{2}}}$ times the peak amplitude for sinusoidal waves. A wave peak or crest is a point at which the displacement is greater (in the positive direction) than at adjacent points, and a wave trough is a point which is displaced farther than adjacent points in the negative sense. The wave height is the difference in displacement between successive peaks and troughs. The wavelength (${\displaystyle \lambda }$) is the distance perpendicular to the wavefront between successive similar points on the wavetrain. The wavenumber is the number of cycles per unit distance, the reciprocal of the wavelength (sometimes defined as ${\displaystyle {\frac {2\pi }{\lambda }}}$, which is given the symbol ${\displaystyle \kappa }$ ). Body waves propagate through the body of the medium, and surface or interface waves propagate along a boundary. Body waves may be either P-waves (q.v.) or S-waves (q.v.). Wave energy that has traveled partly as a P-wave and partly as an S-wave is called a converted wave. Surface waves (or interface waves) may travel by several modes, the most common of which are Rayleigh waves (q.v.) or pseudo-Rayleigh waves. Other surface waves include Love waves (q.v.), hydrodynamic waves, coupled waves, and Stoneley waves. A tube wave (q.v.) is a surface wave which travels along the surface of a borehole. See also wave notation. Wave motion at a point is often described mathematically in terms of harmonic components:

${\displaystyle f(t)=\sum \left(A_{n}cos(2\pi nft)+B_{n}sin(2\pi nft)\right)=\sum C_{n}cos(2\pi nft-\gamma _{n})}$,

or in complex notation:

${\displaystyle f(t)=\sum C_{n}e^{j(2\pi nft-\gamma _{n})}}$,

where Cn is amplitude, n is an integer, f is frequency, ${\displaystyle \gamma _{n}}$ is phase angle, and ${\displaystyle j={\sqrt {-1}}}$.