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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 ...
Print/export Download as PDF; Printable version; In other projects ... d´Alembert's formula is the general solution to the one-dimensional wave equation: ...
By comparison with vector wave equations, the scalar wave equation can be seen as a special case of the vector wave equations; in the Cartesian coordinate system, the scalar wave equation is the equation to be satisfied by each component (for each coordinate axis, such as the x component for the x axis) of a vector wave without sources of waves ...
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 ...
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.
The KP equation was first written in 1970 by Soviet physicists Boris B. Kadomtsev (1928–1998) and Vladimir I. Petviashvili (1936–1993); it came as a natural generalization of the KdV equation (derived by Korteweg and De Vries in 1895). Whereas in the KdV equation waves are strictly one-dimensional, in the KP equation this restriction is ...
The Airy wave train is the only dispersionless wave in one dimensional free space. [20] In higher dimensions, other dispersionless waves are possible. [21] The Airy wave train in phase space. Its shape is a series of parabolas with the same axis, but oscillating according to the Airy function.