r r 0000088812 00000 n 1 ) , 2 / x a ) a ( 2 (22) produces = {\displaystyle y=r\sin {\mathrm {\theta } }} r = 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. [ Overview and Motivation: While Cartesian coordinates are attractive because of . 2 endobj Solving the Wave Equation in Polar Coordinates. r r r n We don't need to prove that the wave travels as ejz again since the differentiation in z for the Laplacian is the same in cylindrical coordinates as it is in rectangular coordinates (@2=@z2). , m b a ( The energy density is displayed by varying colors. = b Suppose that the curved portion of the bounding surface corresponds to , while the two flat portions correspond to and , respectively. cos + ) This latter solution represents a wave travelling in the -z direction. . b However for simplicity, the wave mode solution can be formed in cylindrical coordinates that must satisfy the following scalar wave equation: r2(;`;z)+2(;`;z) = 0 (10) The goal is to solve for the temperature u ( x, t). 2 m b / Biot's theory is used to derive the Christoffel equation for the propagation of cylindrical waves in an anisotropic fluid-saturated porous material. Cylindrical and Spherical wave equations have also been expressed in fractals [14, 15]. + 0000007578 00000 n ) 1 {\displaystyle {\begin{aligned}{\mathrm {\partial } }r/{\mathrm {\partial } }x=r_{x}=x/r=\cos {\mathrm {\theta } }=b,\\{\mathrm {\partial } }r/{\mathrm {\partial } }y=r_{y}=y/r=\sin {\mathrm {\theta } }=a;\\{\mathrm {\partial } }\theta /{\mathrm {\partial } }x=\theta _{x}=\left[{\frac {\mathrm {\partial } }{{\mathrm {\partial } }x}}\left(y/x\right)\right]/[1+\left(y/x\right)^{2}]=-{\frac {y}{x^{2}}}\left({\frac {1}{1+({\frac {y}{x}})^{2}}}\right)\\=-\left(\sin \theta \right)/r=-a/r,\\{\mathrm {\partial } }\theta /{\mathrm {\partial } }y=\theta _{y}=\left[{\frac {\mathrm {\partial } }{{\mathrm {\partial } }y}}\left(y/x\right)\right]/[1+\left(y/x\right)^{2}]={\frac {1}{x}}\left({\frac {1}{1+({\frac {y}{x}})^{2}}}\right)\\=\left(\cos \theta \right)/r=b/r.\end{aligned}}}. / ) a 0000009085 00000 n r x r endobj 2 1 Then, field outside the cylinder will be. r r Therefore a cylindrical wave expression must be Seismology and the Earth's Deep Interior The elastic wave equation Solutions to the wave equation -Solutions to the wave equation - ggeneraleneral Let us consider a region without sources 2=c2 t Where n could be either dilatation or the vector potential and c is either P- or shear-wave velocity. + ( = r ( ] r / y y ( ( a n r / r ) 2 n ] 2 ) z / + + r a The energy flows along the channel in the positive direction. n = x r m n << /S /GoTo /D (Outline0.1.3.34) >> ) y / z ( Standard calculus and physics textbooks contain the formulae for x r 2 r (2) then the Helmholtz differential equation becomes. m b 2 / = 1 ) / b 2 ) 2 2 r %PDF-1.3 % [ ( a / a sin x y z [ z y << /S /GoTo /D (Outline0.1.2.10) >> 2 n ( b 2 + However, the full system in polar coordinates is not as easy to solve, mostly because of the $1/r$ factors in the flux derivatives w.r.t. z ( stream r r 1 y / r ) This is Bessel's equation of order zero. b I was expecting that by plugging a function of the form u ( r, t) = 1 r f ( r, t) I would arrive with something a nice plane wave form for this f ( r, t) but I don't. / / m . 1 16 0 obj ( 1 a (5) returns exactly to the first equation in Eq. n / r endobj (b) Propagating along an arbitrary direction in three-dimensional space repeat themselves in space after traveling an interval of in the direction of vector k. The spatially r The particular geometry I am interested in is the initial condition of a toroidal magnetic flux loop, which is to say, a magnetic field loop situated on a plane, concentrated between a minor and major radius. / 2 + x 2 ) r 2 0000010106 00000 n r + r {\displaystyle m} b sin endobj ( 1 1 n the mentioned article (1)), e.g. ) (a) Propagating along the z-axis. z 1 ) 2 r z z r = ( Wave Equation in Cylindrical Coordinates. / 0000004424 00000 n / 13 0 obj y m Field lines near the dipole are not shown. r b 2 ( , = / 1 0000009469 00000 n r 2 In cylindrical S wave, u is nonzero. r cos 1 2 ) r / We thus have: The Cylindrical Helmholtz Equation, Separated d d dR d + h (k ) 2 n i R = 0 d2 d2 + n2 = 0 d2Z dz2 + k2 z Z = 0 k2 + k 2 z = k 2 The rst of these equations is calledBessel's Equation; the others are familiar. 2 / n The TM 01 mode pattern is shown in Figure (12.5.12 (b)). n = sin As pointed out above, the divergence of is zero, so the wave equation reduces to. 00962795525052. a {\displaystyle r} / x a r x a + a r {\displaystyle {\begin{aligned}\nabla ^{2}\psi &=\psi _{rr}\left(a^{2}m^{2}+a^{2}n^{2}+b^{2}\right)+\psi _{r\theta }\;\left[{\frac {2ab\left(m^{2}+n^{2}\right)}{r}}-{\frac {2ab}{r}}\right]\\&\quad +\psi _{r}\left[{\frac {\left(b^{2}n^{2}+m^{2}\right)+\left(1-a^{2}m^{2}\right)+a^{2}}{r}}\right]\\&\quad +\psi _{\theta \theta }\;\left({\frac {b^{2}n^{2}+b^{2}m^{2}+a^{2}}{r^{2}}}\right)\\&\quad +\psi _{\theta }\;\left[{\frac {\left(bm^{2}/a-2abm^{2}\right)+\left(-2abm^{2}+bn^{2}/a+2ab\right)}{r}}\right]\\&\quad +\psi _{\phi \phi }\left[(-m/ar)^{2}+\left(n/ar\right)^{2}\right]\\&=\psi _{rr}+\left({\frac {2}{r}}\right)\psi _{r}+\left({\frac {1}{r^{2}}}\right)\psi _{\theta \theta }\;+\left({\frac {\cot \theta }{r^{2}}}\right)\psi _{\theta }\;+\left({\frac {1}{a^{2}r^{2}}}\right)\psi _{\phi \phi }.\end{aligned}}}, Substituting the values of r z 1 r sin 2 terms to zero. endobj 2 lecture we move on to studying the wave equation in spherical-polar coordinates. 0000088607 00000 n x (TEz and TMz Modes) tan / + m 2 We have x = r cos {\displaystyle x=r\cos {\mathrm {\theta } }} , y = r sin {\displaystyle y=r\sin {\mathrm {\theta } }} , z = z {\displaystyle z=z} , and r 2 = x 2 + y 2 {\displaystyle r^{2}=x^{2}+y^{2}} , = tan 1 ( y / x ) {\displaystyle {\mathrm {\theta } }=\tan ^{-1}\left(y/x\right)} . a 2 1 n 2 ) y 2 r b 54 0 obj << a a with respect to time t (holding x constant): y/t = - v(2/) A cos[(2/)(x - vt)] Wich gives the y-component of the velocity of an element. m r . y ) r m n b r ( Show that the wave equation (2.5a) can be written in cylindrical coordinates (see Figure 2.6a) as, We shall solve by direct substitution. a + ) 1 r r 0000002141 00000 n 2 + 2 , x r m b ) y r r / = / 1 ) + (1). stream The third equation is just an acknowledgement that the z z -coordinate of a point in Cartesian and polar coordinates is the same. / ( r 2 r ] r , etc. ( 1 Theory of Circular (Cylindrical) waveguides Bessel equations and Bessel Functions: A Bessel equation Problems in Exploration Seismology and their Solutions, http://dx.doi.org/10.1190/1.9781560801733, Sum of waves of different frequencies and group velocity, Interrelationships among elastic constants, Magnitude of disturbance from a seismic source, Potential functions used to solve wave equations, Boundary conditions at different types of interfaces, Boundary conditions in terms of potential functions, Far- and near-field effects for a point source, Directional geophone responses to different waves, https://wiki.seg.org/index.php?title=Wave_equation_in_cylindrical_and_spherical_coordinates&oldid=119888, Problems in Exploration Seismology & their Solutions, the Creative Commons Attribution-ShareAlike 3.0 Unported License (CC-BY-SA). cos