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The square wave in mathematics has many definitions, which are equivalent except at the discontinuities: It can be defined as simply the sign function of a sinusoid: = () = () = () = (), which will be 1 when the sinusoid is positive, −1 when the sinusoid is negative, and 0 at the discontinuities.
Functional approximation of square wave using 5 harmonics Functional approximation of square wave using 25 harmonics Functional approximation of square wave using 125 harmonics. The Gibbs phenomenon is a behavior of the Fourier series of a function with a jump discontinuity and is described as the following:
A modulated wave resulting from adding two sine waves of identical amplitude and nearly identical wavelength and frequency. A common situation resulting in an envelope function in both space x and time t is the superposition of two waves of almost the same wavelength and frequency: [2]
Where a square wave pattern is used (simple black and white lines) not only is there more risk of aliasing, but account must be taken of the fact that the fundamental component of a square wave is higher than the amplitude of the square wave itself (the harmonic components reduce the peak amplitude).
A square wave (represented as the blue dot) is approximated by its sixth partial sum (represented as the purple dot), formed by summing the first six terms (represented as arrows) of the square wave's Fourier series. Each arrow starts at the vertical sum of all the arrows to its left (i.e. the previous partial sum).
Consequently, the wave equation is approximated in the SVEA as: + = . It is convenient to choose k 0 and ω 0 such that they satisfy the dispersion relation: = . This gives the following approximation to the wave equation, as a result of the slowly varying envelope approximation:
In this case, the wave functions are square integrable. One can initially take the function space as the space of square integrable functions, usually denoted L 2. The displayed functions are solutions to the Schrödinger equation. Obviously, not every function in L 2 satisfies the Schrödinger equation for the hydrogen atom.
The Helmholtz equation has a variety of applications in physics and other sciences, including the wave equation, the diffusion equation, and the Schrödinger equation for a free particle. In optics, the Helmholtz equation is the wave equation for the electric field. [1] The equation is named after Hermann von Helmholtz, who studied it in 1860. [2]