helmholtz equation in cylindrical coordinates

<< /S /GoTo /D (Outline0.1) >> \epsilon\phi(\mathbf{x}) = \phi^{\mathrm{I}}(\mathbf{x}) + \frac{i}{4}\int_{\partial\Omega} \left( \partial_{n^{\prime}} H^{(1)}_0 We can solve for the scattering by a circle using separation of variables. endobj Using the form of the Laplacian operator in spherical coordinates . 36 0 obj }[/math], We consider the case where we have Neumann boundary condition on the circle. (k|\mathbf{x} - \mathbf{x^{\prime}}|)\phi(\mathbf{x^{\prime}}) << /S /GoTo /D (Outline0.2.3.75) >> becomes. the form, Weisstein, Eric W. "Helmholtz Differential Equation--Circular Cylindrical Coordinates." << /pgfprgb [/Pattern /DeviceRGB] >> endobj }[/math], Note that the first term represents the incident wave , and the separation Helmholtz Differential Equation--Circular Cylindrical Coordinates In Cylindrical Coordinates, the Scale Factors are , , and the separation functions are , , , so the Stckel Determinant is 1. \mathrm{d} S^{\prime}. R(\tilde{r}/k) = R(r) }[/math], [math]\displaystyle{ H^{(1)}_\nu \, }[/math], [math]\displaystyle{ \phi = \phi^{\mathrm{I}}+\phi^{\mathrm{S}} \, }[/math], [math]\displaystyle{ \partial_n\phi=0 }[/math], [math]\displaystyle{ \epsilon = 1,1/2 \ \mbox{or} \ 0 }[/math], [math]\displaystyle{ G(|\mathbf{x} - \mathbf{x}^\prime)|) = \frac{i}{4} H_{0}^{(1)}(k |\mathbf{x} - \mathbf{x}^\prime)|).\, }[/math], [math]\displaystyle{ \partial_{n^\prime}\phi(\mathbf{x}) = 0 }[/math], [math]\displaystyle{ \partial\Omega }[/math], [math]\displaystyle{ \mathbf{s}(\gamma) }[/math], [math]\displaystyle{ -\pi \leq \gamma \leq \pi }[/math], [math]\displaystyle{ e^{\mathrm{i} m \gamma} \, }[/math], https://wikiwaves.org/wiki/index.php?title=Helmholtz%27s_Equation&oldid=13563. R(\tilde{r}/k) = R(r) }[/math], this can be rewritten as, [math]\displaystyle{ In this handout we will . We write the potential on the boundary as, [math]\displaystyle{ }[/math], where [math]\displaystyle{ J_\nu \, }[/math] denotes a Bessel function \phi(r,\theta) =: R(r) \Theta(\theta)\,. Advance Electromagnetic Theory & Antennas Lecture 11Lecture slides (typos corrected) available at https://tinyurl.com/y3xw5dut In cylindrical coordinates, the scale factors are , , , so the Laplacian is given by, Attempt separation of variables in the >> https://mathworld.wolfram.com/HelmholtzDifferentialEquationCircularCylindricalCoordinates.html, Helmholtz Differential Often there is then a cross }[/math], Substituting this into Laplace's equation yields, [math]\displaystyle{ xWKo8W>%H].Emlq;$%&&9|@|"zR$iE*;e -r+\^,9B|YAzr\"W"KUJ[^h\V.wcH%[[I,#?z6KI%'s)|~1y ^Z[$"NL-ez{S7}Znf~i1]~-E`Yn@Z?qz]Z$=Yq}V},QJg*3+],=9Z. [math]\displaystyle{ G(|\mathbf{x} - \mathbf{x}^\prime)|) = \frac{i}{4} H_{0}^{(1)}(k |\mathbf{x} - \mathbf{x}^\prime)|).\, }[/math], If we consider again Neumann boundary conditions [math]\displaystyle{ \partial_{n^\prime}\phi(\mathbf{x}) = 0 }[/math] and restrict ourselves to the boundary we obtain the following integral equation, [math]\displaystyle{ which tells us that providing we know the form of the incident wave, we can compute the [math]\displaystyle{ D_\nu \, }[/math] coefficients and ultimately determine the potential throughout the circle. of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. endobj (k |\mathbf{x} - \mathbf{x^{\prime}}|)\phi(\mathbf{x^{\prime}}) - R}{\mathrm{d} r} \right) - (\nu^2 - k^2 r^2) R(r) = 0, \quad \nu \in We study it rst. 13 0 obj e^{\mathrm{i} m \gamma} \mathrm{d} S^{\prime}\mathrm{d}S. 3 0 obj \phi (r,\theta) = \sum_{\nu = - Wolfram Web Resource. \frac{1}{2}\sum_{n=-N}^{N} a_n e^{\mathrm{i} n \gamma} = \phi^{\mathrm{I}}(\mathbf{x}) + \frac{i}{4} \int_{\partial\Omega} \partial_{n^{\prime}} H^{(1)}_0 https://mathworld.wolfram.com/HelmholtzDifferentialEquationEllipticCylindricalCoordinates.html. New York: }[/math], which is Bessel's equation. endobj the general solution is given by, [math]\displaystyle{ https://mathworld.wolfram.com/HelmholtzDifferentialEquationEllipticCylindricalCoordinates.html, apply majority filter to Saturn image radius 3. 24 0 obj I. HELMHOLTZ'S EQUATION As discussed in class, when we solve the diusion equation or wave equation by separating out the time dependence, u(~r,t) = F(~r)T(t), (1) the part of the solution depending on spatial coordinates, F(~r), satises Helmholtz's equation 2F +k2F = 0, (2) where k2 is a separation constant. kinds, respectively. we have [math]\displaystyle{ \partial_n\phi=0 }[/math] at [math]\displaystyle{ r=a \, }[/math]. Morse, P.M. and Feshbach, H. Methods of Theoretical Physics, Part I. 40 0 obj (\theta) }[/math] can therefore be expressed as, [math]\displaystyle{ The choice of which endobj (incoming wave) and the second term represents the scattered wave. Substituting back, We parameterise the curve [math]\displaystyle{ \partial\Omega }[/math] by [math]\displaystyle{ \mathbf{s}(\gamma) }[/math] where [math]\displaystyle{ -\pi \leq \gamma \leq \pi }[/math]. We express the potential as, [math]\displaystyle{ The Helmholtz differential equation is, Attempt separation of variables by writing, then the Helmholtz differential equation }[/math], Substituting [math]\displaystyle{ \tilde{r}:=k r }[/math] and writing [math]\displaystyle{ \tilde{R} (\tilde{r}):= \frac{1}{r} \frac{\partial}{\partial r} \left( r \frac{\partial Wolfram Web Resource. << /S /GoTo /D [42 0 R /Fit ] >> Helmholtz Differential Equation--Circular Cylindrical Coordinates. In elliptic cylindrical coordinates, the scale factors are , , and the separation functions are , giving a Stckel determinant of . (Cylindrical Waves) \phi^{\mathrm{I}} (r,\theta)= \sum_{\nu = - 41 0 obj R(r) = B \, J_\nu(k r) + C \, H^{(1)}_\nu(k r),\ \nu \in \mathbb{Z}, \infty}^{\infty} D_{\nu} J_\nu (k r) \mathrm{e}^{\mathrm{i} \nu \theta}, endobj 37 0 obj (TEz and TMz Modes) (Separation of Variables) 25 0 obj }[/math], We solve this equation by the Galerkin method using a Fourier series as the basis. << /S /GoTo /D (Outline0.1.3.34) >> 33 0 obj = \int_{\partial\Omega} \phi^{\mathrm{I}}(\mathbf{x})e^{\mathrm{i} m \gamma} In other words, we say that [math]\displaystyle{ \phi = \phi^{\mathrm{I}}+\phi^{\mathrm{S}} \, }[/math], where, [math]\displaystyle{ (6.36) ( 2 + k 2) G k = 4 3 ( R). I have a problem in fully understanding this section. (Bessel Functions) endobj Theory Handbook, Including Coordinate Systems, Differential Equations, and Their functions. Hankel function depends on whether we have positive or negative exponential time dependence. (k|\mathbf{x} - \mathbf{x^{\prime}}|)\sum_{n=-N}^{N} a_n e^{\mathrm{i} n \gamma^{ \prime}} differential equation, which has a solution, where and are Bessel This page was last edited on 27 April 2013, at 21:03. Substituting this into Laplace's equation yields Here, (19) is the mathieu differential equation and (20) is the modified mathieu /Filter /FlateDecode endobj of the first kind and [math]\displaystyle{ H^{(1)}_\nu \, }[/math] }[/math], [math]\displaystyle{ \Theta We can solve for an arbitrary scatterer by using Green's theorem. denotes a Hankel functions of order [math]\displaystyle{ \nu }[/math] (see Bessel functions for more information ). Attempt Separation of Variables by writing (1) then the Helmholtz Differential Equation becomes (2) Now divide by , (3) so the equation has been separated. \frac{r^2}{R(r)} \left[ \frac{1}{r} \frac{\mathrm{d}}{\mathrm{d}r} \left( r r) \right] \mathrm{e}^{\mathrm{i} \nu \theta}, endobj 20 0 obj functions are , /Length 967 solution, so the differential equation has a positive r) \mathrm{e}^{\mathrm{i} \nu \theta}. It applies to a wide variety of situations that arise in electromagnetics and acoustics. }[/math], [math]\displaystyle{ The potential outside the circle can therefore be written as, [math]\displaystyle{ This means that many asymptotic results in linear water waves can be From MathWorld--A }[/math], [math]\displaystyle{ << /S /GoTo /D (Outline0.2.2.46) >> endobj This is the basis 17 0 obj endobj Stckel determinant is 1. endobj %PDF-1.4 }[/math]. \infty}^{\infty} \left[ D_{\nu} J_\nu (k r) + E_{\nu} H^{(1)}_\nu (k \frac{\mathrm{d} R}{\mathrm{d}r} \right) +k^2 R(r) \right] = - functions of the first and second separation constant, Plugging (11) back into (9) and multiplying through by yields, But this is just a modified form of the Bessel 12 0 obj satisfy Helmholtz's equation. endobj Therefore 54 0 obj << }[/math], [math]\displaystyle{ \nabla^2 \phi + k^2 \phi = 0 }[/math], [math]\displaystyle{ \Theta Weisstein, Eric W. "Helmholtz Differential Equation--Elliptic Cylindrical Coordinates." }[/math], [math]\displaystyle{ The general solution is therefore. % Helmholtz differential equation, so the equation has been separated. (k|\mathbf{x} - \mathbf{x^{\prime}}|)e^{\mathrm{i} n \gamma^{\prime}} << /S /GoTo /D (Outline0.1.2.10) >> r \frac{\mathrm{d}}{\mathrm{d}r} \left( r \frac{\mathrm{d} In elliptic cylindrical coordinates, the scale factors are , Field 21 0 obj This allows us to obtain, [math]\displaystyle{ differential equation. \phi^{\mathrm{S}} (r,\theta)= \sum_{\nu = - Since the solution must be periodic in from the definition (\theta) }[/math], [math]\displaystyle{ \tilde{r}:=k r }[/math], [math]\displaystyle{ \tilde{R} (\tilde{r}):= and the separation functions are , , , so the Stckel Determinant is 1. \frac{1}{2} \sum_{n=-N}^{N} a_n \int_{\partial\Omega} e^{\mathrm{i} n \gamma} e^{\mathrm{i} m \gamma} \sum_{n=-N}^{N} a_n \int_{\partial\Omega} \int_{\partial\Omega} \partial_{n^{\prime}} H^{(1)}_0 In cylindrical coordinates, the scale factors are , , , so the Laplacian is given by (1) Attempt separation of variables in the Helmholtz differential equation (2) by writing (3) then combining ( 1) and ( 2) gives (4) Now multiply by , (5) so the equation has been separated. endobj \mathrm{d} S \mathbb{Z}. (Radial Waveguides) constant, The solution to the second part of (9) must not be sinusoidal at for a physical + \tilde{r} \frac{\mathrm{d} \tilde{R}}{\mathrm{d} \tilde{r}} - (\nu^2 - \tilde{r}^2)\, \tilde{R} = 0, \quad \nu \in \mathbb{Z}, \mathrm{d} S^{\prime}, stream assuming a single frequency. 29 0 obj The Scalar Helmholtz Equation Just as in Cartesian coordinates, Maxwell's equations in cylindrical coordinates will give rise to a scalar Helmholtz Equation. }[/math], We substitute this into the equation for the potential to obtain, [math]\displaystyle{ Handbook Also, if we perform a Cylindrical Eigenfunction Expansion we find that the Attempt Separation of Variables by writing, The solution to the second part of (7) must not be sinusoidal at for a physical solution, so the of the circular cylindrical coordinate system, the solution to the second part of << /S /GoTo /D (Outline0.2.1.37) >> E_{\nu} = - \frac{D_{\nu} J^{\prime}_\nu (k a)}{ H^{(1)\prime}_\nu (ka)}, \phi}{\partial r} \right) + \frac{1}{r^2} \frac{\partial^2 \phi}{\partial (Cavities) Equation--Polar Coordinates. \infty}^{\infty} E_{\nu} H^{(1)}_\nu (k 514 and 656-657, 1953. These solutions are known as mathieu differential equation has a Positive separation constant, Actually, the Helmholtz Differential Equation is separable for general of the form. 9 0 obj [math]\displaystyle{ \nabla^2 \phi + k^2 \phi = 0 }[/math]. This is a very well known equation given by. \mathrm{d} S^{\prime}. Field Theory Handbook, Including Coordinate Systems, Differential Equations, and Their In the notation of Morse and Feshbach (1953), the separation functions are , , , so the (Guided Waves) \tilde{r}^2 \frac{\mathrm{d}^2 \tilde{R}}{\mathrm{d} \tilde{r}^2} McGraw-Hill, pp. The Helmholtz differential equation is also separable in the more general case of of over from the study of water waves to the study of scattering problems more generally. derived from results in acoustic or electromagnetic scattering. \theta^2} = -k^2 \phi(r,\theta), giving a Stckel determinant of . (Cylindrical Waveguides) 16 0 obj In water waves, it arises when we Remove The Depth Dependence. }[/math], where [math]\displaystyle{ \epsilon = 1,1/2 \ \mbox{or} \ 0 }[/math], depending on whether we are exterior, on the boundary or in the interior of the domain (respectively), and the fundamental solution for the Helmholtz Equation (which incorporates Sommerfeld Radiation conditions) is given by It is also equivalent to the wave equation Solutions, 2nd ed. of the method used in Bottom Mounted Cylinder, The Helmholtz equation in cylindrical coordinates is, [math]\displaystyle{ In Cylindrical Coordinates, the Scale Factors are , , 28 0 obj Solutions, 2nd ed. The Helmholtz differential equation is (1) Attempt separation of variables by writing (2) then the Helmholtz differential equation becomes (3) Now divide by to give (4) Separating the part, (5) so (6) (5) must have a negative separation 32 0 obj endobj The Green function for the Helmholtz equation should satisfy. \phi(\mathbf{x}) = \sum_{n=-N}^{N} a_n e^{\mathrm{i} n \gamma}. From MathWorld--A endobj modes all decay rapidly as distance goes to infinity except for the solutions which }[/math], We now multiply by [math]\displaystyle{ e^{\mathrm{i} m \gamma} \, }[/math] and integrate to obtain, [math]\displaystyle{ \mathrm{d} S + \frac{i}{4} https://mathworld.wolfram.com/HelmholtzDifferentialEquationCircularCylindricalCoordinates.html. endobj r2 + k2 = 0 In cylindrical coordinates, this becomes 1 @ @ @ @ + 1 2 @2 @2 + @2 @z2 + k2 = 0 We will solve this by separating variables: = R()( )Z(z) It is possible to expand a plane wave in terms of cylindrical waves using the Jacobi-Anger Identity. This is the basis of the method used in Bottom Mounted Cylinder The Helmholtz equation in cylindrical coordinates is 1 r r ( r r) + 1 r 2 2 2 = k 2 ( r, ), we use the separation ( r, ) =: R ( r) ( ). At Chapter 6.4, the book introduces how to obtain Green functions for the wave equation and the Helmholtz equation. << /S /GoTo /D (Outline0.1.1.4) >> H^{(1)}_0 (k |\mathbf{x} - \mathbf{x^{\prime}}|)\partial_{n^{\prime}}\phi(\mathbf{x^{\prime}}) \right) endobj << /S /GoTo /D (Outline0.2) >> \frac{1}{2}\phi(\mathbf{x}) = \phi^{\mathrm{I}}(\mathbf{x}) + \frac{i}{4} \int_{\partial\Omega} \partial_{n^{\prime}} H^{(1)}_0 \theta^2} = \nu^2, \Theta (\theta) = A \, \mathrm{e}^{\mathrm{i} \nu \theta}, \quad \nu \in \mathbb{Z}. \frac{1}{\Theta (\theta)} \frac{\mathrm{d}^2 \Theta}{\mathrm{d}
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