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The wave equation is a second-order linear partial differential equation for the description of waves or standing wave fields such as mechanical waves (e.g. water waves, sound waves and seismic waves) or electromagnetic waves (including light waves).
Download as PDF; Printable version; In other projects Wikidata item; ... d´Alembert's formula is the general solution to the one-dimensional wave equation: ...
The one-way equation and solution in the three-dimensional case was assumed to be similar way as for the one-dimensional case by a mathematical decomposition (factorization) of a 2nd order differential equation. [15] In fact, the 3D One-way wave equation can be derived from first principles: a) derivation from impedance theorem [3] and b ...
1-dimensional corollaries for two sinusoidal waves The following may be deduced by applying the principle of superposition to two sinusoidal waves, using trigonometric identities. The angle addition and sum-to-product trigonometric formulae are useful; in more advanced work complex numbers and fourier series and transforms are used.
Intuitively, one can think of the inhomogeneous problem as a set of homogeneous problems each starting afresh at a different time slice t = t 0. By linearity, one can add up (integrate) the resulting solutions through time t 0 and obtain the solution for the inhomogeneous problem. This is the essence of Duhamel's principle.
where + is the right-going wave and is the left-going wave. It can be seen from this representation that sampling the function at a given point and time merely involves summing two delayed copies of its traveling waves. These traveling waves will reflect at boundaries such as the suspension points of vibrating strings or the open or closed ends ...
In this theory, the wave equations and the wave functions have their place, but in a somewhat different guise. The main objects of interest are not the wave functions, but rather operators, so called field operators (or just fields where "operator" is understood) on the Hilbert space of states (to be described next section). It turns out that ...
The dynamic wave is the full one-dimensional Saint-Venant equation. It is numerically challenging to solve, but is valid for all channel flow scenarios. The dynamic wave is used for modeling transient storms in modeling programs including Mascaret (EDF), SIC (Irstea) , HEC-RAS , [ 18 ] InfoWorks_ICM Archived 2016-10-25 at the Wayback Machine ...