Green's function for the lossy wave equation

Abstract

Using an integral representation for the first kind Hankel (Hankel-Bessel Integral Representation) function we obtain the so-called Basset formula, an integral representation for the second kind modified Bessel function. Using the Sonine-Bessel integral representation we obtain the Fourier cosine integral transform of the zero order Bessel function. As an application we present the calculation of the Green's function associated with a second-order partial differential equation, particularly a wave equation for a lossy two-dimensional medium. This application is associated with the transient electromagnetic field radiated by a pulsed source in the presence of dispersive media, which is of great importance in the theory of geophysical prospecting, lightning studies and development of pulsed antenna systems. Keywords: Sonine-Bessel, integral representation, dissipative wave equation.Usando uma representação integral para a função de Hankel de primeira espécie (representação integral de Hankel-Bessel) obtemos a chamada fórmula de Basset, uma representação integral para a função de Bessel modificada de segunda espécie. A partir de uma representação integral de Sonine-Bessel obtemos a transformada de Fourier em co-senos da função de Bessel de ordem zero. Como uma aplicação, apresentamos o cálculo da função de Green associada a uma equação diferencial parcial de segunda ordem, a saber, a equação da onda em um meio dissipativo de dimensão dois. Esta aplicação está associada ao campo eletromagnético transiente irradiado por uma fonte tipo pulso na presença de meios dispersivos, o qualé de grande importância na teoria de prospecção geofísica, estudos sobre luz e desenvolvimento de sistemas de antenas tipo pulso. Palavras-chave: Sonine-Bessel, representação integral, equação da onda dissipativa.

Energetic boundary element method for accurate solution of damped waves hard scattering problems

“…is the fundamental solution of the 2D damped wave operator, with H [•] the Heaviside distribution, and r := r 2 = x − ξ 2 . Definition ( 5) is an extension of fundamental solutions found in [24][25][26], where only the viscous damping coefficient was taken into account, and it has been verified with the use of Mathematica software.…”

“…Proof The limit of F 2 (r, t h , t k ) can be evaluated straightforwardly from (24) and it turns out to be − (t h −t k ) 2 4π . For what concerns the limit of r F 1 (r, t h , t k ), we cannot perform, as already remarked, time integrations in (23) analytically; anyway, since we are interested in the behavior of the integrand function for vanishing r , we observe that in this case the time integration, which for r > 0 must handle a weak singularity due to the presence of [c 2 (t − τ ) 2 − r 2 ] −1/2 , shows a stronger singularity in the neighborhood of τ = t. Hence, if we expand the whole integrand function in (23) w.r.t.…”

Fundamentals of Diffusion-Based Molecular Communication in Nanonetworks

“…The Green's function g d ( x, t) of the Telegraph Equation in (14), which is analytically equivalent to the wave equation in a lossy medium [56,57], is the analytical solution for the concentration evolution in space and time when r( x, t) is a Dirac delta function both in time t and in space S : r( x, t) = δ( x)δ(t). It is analytically expressed as…”

El método de la función de green en el cálculo del campo eléctrico bidimensional para una corriente eléctrica en una guía de onda rectangular

“…Este método encuentra aplicación aún en los casos de ecuaciones diferenciales ordinarias homogéneas y constituye un método alternativo a los de Fourier, variación de parámetros o coeficientes indeterminados (Challis, 2003;Sepulveda, 2009). En el caso general, la función de Green es una distribución, que fue introducida por Green en el electromagnetismo, y más tarde usada por Neumann en la teoría del potencial newtoniano (Strauss, 1992) Helmholtz en acústica (Aleixo & Oliveira, 2008) y al igual que Richard Feynman en la teoría de campos cuánticos con un nombre diferente, propagador (De La Peña, 2014).…”