§3.3. 2 Separation of spherical variables: zonal harmonics Christopher Crawford PHY 416 2014-10-29.
-
Upload
rolf-higgins -
Category
Documents
-
view
219 -
download
1
Transcript of §3.3. 2 Separation of spherical variables: zonal harmonics Christopher Crawford PHY 416 2014-10-29.
![Page 1: §3.3. 2 Separation of spherical variables: zonal harmonics Christopher Crawford PHY 416 2014-10-29.](https://reader035.fdocuments.in/reader035/viewer/2022062422/56649e795503460f94b798d7/html5/thumbnails/1.jpg)
§3.3.2 Separation of spherical variables: zonal harmonics
Christopher CrawfordPHY 416
2014-10-29
![Page 2: §3.3. 2 Separation of spherical variables: zonal harmonics Christopher Crawford PHY 416 2014-10-29.](https://reader035.fdocuments.in/reader035/viewer/2022062422/56649e795503460f94b798d7/html5/thumbnails/2.jpg)
Outline• Separation of variables in different coordinate systems
Cartesian, cylindrical, and spherical coordinatesBoundary conditions: external and internal
• Plane wave functions in different coordinates Linear waves: Circular harmonics (sin, cos, exp) (x,y,z)Azimuthal waves: Cylindrical (sectoral) harmonics (φ)Polar waves: Legendre poly/fns: zonal harmonics (θ)Angular waves: Spherical (tesseral) harmonics (θ,φ)Radial waves: 2d Bessel (s), 3d spherical Bessel (r)Laplacian: planar (s,φ), solid harmonics (r,θ,φ)
• Putting it all togetherGeneral solutions to Laplace equation
2
![Page 3: §3.3. 2 Separation of spherical variables: zonal harmonics Christopher Crawford PHY 416 2014-10-29.](https://reader035.fdocuments.in/reader035/viewer/2022062422/56649e795503460f94b798d7/html5/thumbnails/3.jpg)
Helmholtz equation: free wave• k2 = curvature of wave; k2=0 [Laplacian]
3
![Page 4: §3.3. 2 Separation of spherical variables: zonal harmonics Christopher Crawford PHY 416 2014-10-29.](https://reader035.fdocuments.in/reader035/viewer/2022062422/56649e795503460f94b798d7/html5/thumbnails/4.jpg)
Review: external boundary conditions• Uniqueness theorem – difference between any two solutions of
Poisson’s equation is determined by values on the boundary
• External boundary conditions:
4
![Page 5: §3.3. 2 Separation of spherical variables: zonal harmonics Christopher Crawford PHY 416 2014-10-29.](https://reader035.fdocuments.in/reader035/viewer/2022062422/56649e795503460f94b798d7/html5/thumbnails/5.jpg)
Internal boundary conditions• Possible singularities (charge, current) on the interface between two materials• Boundary conditions “sew” together solutions on either side of the boundary• External: 1 condition on each side Internal: 2 interconnected conditions
• General prescription to derive any boundary condition:
5
![Page 6: §3.3. 2 Separation of spherical variables: zonal harmonics Christopher Crawford PHY 416 2014-10-29.](https://reader035.fdocuments.in/reader035/viewer/2022062422/56649e795503460f94b798d7/html5/thumbnails/6.jpg)
Linear wave functions – exponentials
6
![Page 7: §3.3. 2 Separation of spherical variables: zonal harmonics Christopher Crawford PHY 416 2014-10-29.](https://reader035.fdocuments.in/reader035/viewer/2022062422/56649e795503460f94b798d7/html5/thumbnails/7.jpg)
Circular waves – Bessel functions
7
![Page 8: §3.3. 2 Separation of spherical variables: zonal harmonics Christopher Crawford PHY 416 2014-10-29.](https://reader035.fdocuments.in/reader035/viewer/2022062422/56649e795503460f94b798d7/html5/thumbnails/8.jpg)
Polar waves – Legendre functions
8
![Page 9: §3.3. 2 Separation of spherical variables: zonal harmonics Christopher Crawford PHY 416 2014-10-29.](https://reader035.fdocuments.in/reader035/viewer/2022062422/56649e795503460f94b798d7/html5/thumbnails/9.jpg)
Angular waves – spherical harmonics
9
![Page 10: §3.3. 2 Separation of spherical variables: zonal harmonics Christopher Crawford PHY 416 2014-10-29.](https://reader035.fdocuments.in/reader035/viewer/2022062422/56649e795503460f94b798d7/html5/thumbnails/10.jpg)
Radial waves – spherical Bessel fn’s
10
![Page 11: §3.3. 2 Separation of spherical variables: zonal harmonics Christopher Crawford PHY 416 2014-10-29.](https://reader035.fdocuments.in/reader035/viewer/2022062422/56649e795503460f94b798d7/html5/thumbnails/11.jpg)
Solid harmonics
11
![Page 12: §3.3. 2 Separation of spherical variables: zonal harmonics Christopher Crawford PHY 416 2014-10-29.](https://reader035.fdocuments.in/reader035/viewer/2022062422/56649e795503460f94b798d7/html5/thumbnails/12.jpg)
General solutions to Laplace eq’nor: All I really need to know I learned in PHY311
•Cartesian coordinates – no general boundary conditions!
•Cylindrical coordinates – azimuthal continuity
•Spherical coordinates – azimuthal and polar continuity
12