Pdf electromagnetic fields and waves in fractional dimensional. A fast and accurate solution was obtained by using the orthogonal function expansion method. The solution of the wave equation in two dimensions can be obtained by solving the three dimensional wave equation in the case where the initial data depends only on xand y, but not z. Calculating fractal dimension hausdorffbesicovitch dimension traditional. So a fractal landscape made up of a large hill covered with tiny mounds would be close to the second dimension, while a rough surface composed of many mediumsized hills. This demonstration shows the solution of the twodimensional wave equation subjected to an instantaneous hammer hit centered at the source point location with zero initial displacement and velocity. But for a fractal, the weight scales as a fractional usually noninteger power of its length, and that fractional value is the fractal dimension. Twodimensional regular and irregular waves are simulated, with the capability. Fractal solids, product measures and fractional wave equations.
Continuum mechanics, fractals, fractal derivative, fractional integrals, product measures, waves fractal solids, product measures and. Pdf on the wave propagation in isotropic fractal media. The spherical shell domain in two views along with the spatial. Twodimensional wave equations and wave characteristics. The threedimensional current and surface wave equations. Analysis of fractal wave equations by local fractional. We prove that the wave equation for noninteger dimensional space is similar to equation of waves in non fractal medium with heterogeneity of powerlaw type. The theory of quaternionvalued hyperholomorphic functions i. Free and smooth boundaries in 2d fd schemes transient elastic waves. Mandelbrot and nature clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line. Two fractal dimensions can be employed to characterize the urban form.
In section 2, we investigate full analytical cylindrical wave solution to the wave equation in ddimensional fractional space, where the parameter d is used to describe the measure distribution of space. Boundary value problems for hyperholomorphic solutions of two. The book also investigates elastodynamic problems involving onespace dimension in governing boundaries, equations, and initial conditions. As in the one dimensional situation, the constant c has the units of velocity. We consider the twodimensional water wave problem in an infinitely. The first case is the equation with classical derivative. The variational method for the twodimensional wave equation. Two dimensional waves have the ability to travel around corners. This paper builds on the recently begun extension of continuum thermomechanics to fractal porous media that are specified by a mass or spatial fractal dimension d, a surface fractal dimension d and a resolution length scale r. Two dimensional wave equation since the modeling here will be similar to that of sec. Fractals and dynamics in mathematics, science, and the arts. Dec 19, 20 for the love of physics walter lewin may 16, 2011 duration. This important property is easy to interpret visually. This theory is required in order to analyze changes in the characteristics of a wave as it propagates from the deep sea to the shore.
So, for example, as two traveling waves moving along the string in opposite. General solution of the onedimensional wave equation. Mapping physical problems on fractals onto boundary value. The moving boundary method is adopted in this model to generate water waves. A new technique for nonlinear twodimensional wave equations. This book introduces background and recent progress on these problems, from both. Likewise, a hilly fractal scene will reach a dimension somewhere between two and three. A natural next step is to consider extensions of the methods for various variants of the onedimensional wave equation to twodimensional 2d and threedimensional 3d versions of the wave equation. Finite difference methods for 2d and 3d wave equations. Therefore, the general solution, 2, of the wave equation, is the sum of a rightmoving wave and a leftmoving wave. The movement of an interfacial boundary of two fluids, even with severe freesurface deformation, is tracked by using the level set method, while an. The analytical solutions are obtained by using local fractional fourier series method. Two dimensional wave equations with fractal boundaries article in applicable analysis 9034. In contrast to the fractional di usionwave equation, the fractional wave equation contains fractional derivatives of the same order.
Equally important is its two dimensional analog, namely, the motion of an elastic membrane, such. Fractal dimension of the boundary of a fractal mathematics. A threedimensional numerical model with an ltype wave. Fractional differential equations fdes 12345 are powerful tools for. In this paper, a multidimensional fractional wave equation that describes propagation of the damped waves is introduced and analyzed. Twodimensional growth simulations are described for a random rain model, where the candidates for sticking approach the growing cluster along random straight lines. Dalemberts solution compiled 30 october 2015 in this lecture we discuss the one dimensional wave equation. A threedimensional numerical wave tank was developed based on reynolds averaged navierstokes equations and the volume of fluid method. A practicing coastal engineer must have a basic and relatively easy to use theory that defines the important characteristics of twodimensional waves. Detailed treatment of this theory, which is in the same relation to the two dimensional helmholtz equation as the usual one dimensional complex analysis is to the laplace equation in r 2, can be. One is form dimension, which is defined with the areascale scaling based on a 2 dimensional fractal initiator. This paper focuses on two cases of two dimensional wave equations with fractal boundaries.
Mathematical analysis of the kuznetsov equation tel. The 2d wave equation separation of variables superposition examples the two dimensional wave equation ryan c. The fractal form indicates the selfsimilar shape and structure of a city. For the derivation of the wave equation from newtons second law, see exercise 3. Helmholtz equation in a network seen as a limit of a twodimensional. The book then compares two dimensional problems as being subject to more precise mathematical analysis compared to three dimensional situations by using scalar wave equations. A new numerical technique for solving the local fractional. We derive some dimensionlike properties of this cocycle attractor such as fractal. More precisely, we study the following nonhomogeneous hyperbolic initialboundary value problem ibvp. So i also expect the boundary of a fractal to be defined in a different way. In this paper, a new technique, namely, the new homotopy perturbation method nhpm is applied for solving a nonlinear twodimensional wave equation. Finally, the one, two and three dimensional wave equations are developed, showing that the continuum mechanics approach is consistent with that obtained via variational energy principles. The 2d wave equation separation of variables superposition examples remarks. In this paper, a new technique, namely, the new homotopy perturbation method nhpm is applied for solving a nonlinear two dimensional wave equation.
Twodimensional wave equations with fractal boundaries. This demonstration shows the solution of the two dimensional wave equation subjected to an instantaneous hammer hit centered at the source point location with zero initial displacement and velocity. Siam journal on mathematical analysis siam society for. Twodimensional freesurface flow modeling for wavestructure. In this case, the three dimensional solution consists of cylindrical waves. Aug 25, 2014 using the two dimensional wave interference pattern shown and the two equations involving path difference, complete the following.
A threedimensional numerical model with an ltype wavemaker. Mandelbrot fractal based on newtons method for finding roots applied to. Book of abstracts equadiff 2019 mathematical institute. The homogenization through which this fractal wave equation is derived. Fractal foundation online course chapter 1 fractals in nature. Newton fractals are examples of convergent fractal equations. The fractal wave equations with local fractional derivatives are investigated in this paper. Finite difference methods for 2d and 3d wave equations examples on wave equations written out in 2d3d boundary and initial conditions example. This paper focuses on two cases of twodimensional wave equations with fractal boundaries. Twodimensional wave equations with fractal boundaries article in applicable analysis 9034. Wave equation on a two dimensional rectangle in these notes we are concerned with application of the method of separation of variables applied to the wave equation in a two dimensional rectangle. A set of fractal dimension equations and its application to. An introduction to fractals written by paul bourke may 1991 philosophy is written in this grand book i mean universe which stands continuously open to our gaze, but which cannot be understood unless one first learns to comprehend the language in which it is written. Both isotropic growth from a central seed and growth on a base line on to which the atoms fall obliquely from a parallel line are studied.
The velocity of the energy on the boundaries in normal direc. Electromagnetic waves in noninteger dimensional spaces and. A practicing coastal engineer must have a basic and relatively easy to use theory that defines the important characteristics of two dimensional waves. Advanced local fractional calculus and its applications. Finally, the one, two and threedimensional wave equations are developed, showing that the continuum mechanics approach is consistent with that obtained via variational energy principles. The boundaryvalue problems for the regions with the moving boundaries are not. When twodimensional fractals are iterated many times, the perimeter of the fractal increases up to infinity, but the area may never exceed a certain value. Li ostojastarzewski fractal solids, product measures and fractional wave equations 93abstract. Here we have taken an excerpt from the book flatland a romance of many dimensions, where. When two dimensional fractals are iterated many times, the perimeter of the fractal increases up to infinity, but the area may never exceed a certain value. The mapping from the fractal to the fractional dimensional space with. The seismic wave equation x 1 x 2 x 3 t x 1 t x 1 dx 1 dx 2 dx 3 figure 3. A theory of quaternionvalued functions, called hyperholomorphic, of two real variables has long been established. The speed of electromagnetic waves and the effective refractive index of noninteger dimensional spaces and fractals are discussed.
Presented here is a list of fractals ordered by increasing hausdorff dimension, with the purpose of visualizing what it means for a fractal to have a low or a high dimension. Two dimensional wave equations with fractal boundaries 2327 remark 2. Siam journal on numerical analysis society for industrial. We solve the damped wave equation 2 in the region 0 0. Pdf this book presents the concept of fractional dimensional space applied to the. Pistontype wavemakers are mimicked for the total replication of the physical wave tank conditions.
For the love of physics walter lewin may 16, 2011 duration. The present method is very efficient and accurate to process a class of local fractional differential equations. In this case, the threedimensional solution consists of cylindrical waves. Unit 1 waves many waves have the ability to travel in more than one dimension. In 3, we show example two dimension wave equations with fractal boundaries, and in this note, other examples, as well as an idea to. The wave equation can thus be compactly written in terms of the laplacian as 1 v.
Boundary value problems for hyperholomorphic solutions of. This theory is in the same relation to the two dimensional helmholtz equation as the usual one dimensional complex analysis is to the laplace equation in r 2. The boundary conditions on fractal boundaries can be formulated by. Nov 15, 20 this paper focuses on two cases of two dimensional wave equations with fractal boundaries. The two most important steps in application of the new homotopy perturbation method are to construct a suitable homotopy equation and to choose a suitable initial guess. The more the flat fractal fills a plane, the closer it approaches two dimensions. Stokes equation twophase flow solver, to investigate wavestructure interactions and induced motions of floating bodies in two dimensions. To calculate a fractals dimension, we simply extend the formula for. Stokes equation two phase flow solver, to investigate wave structure interactions and induced motions of floating bodies in two dimensions. One is form dimension, which is defined with the areascale scaling based on a 2dimensional fractal initiator. Twodimensional wave equation since the modeling here will be similar to that of sec.
Jul 26, 2006 2018 higherorder accurate two step finite difference schemes for the many dimensional wave equation. Analysis, probability and mathematical physics on fractals fractals. A plane wave in two or three dimensions is like a sine wave in one dimension except that crests and troughs arent points, but form lines 2d or planes 3d perpendicular to the direction of wave propagation. Benoit mandelbrot has stated that a fractal is by definition a set for which the hausdorffbesicovitch dimension strictly exceeds the topological dimension. Fractal foundation online course chapter 1 fractals in.
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