Table I
SFUR Eigenmode Equation Parameters and Kernels
Systems | Γℓ,k | C1 | C2 | BℓKℓ(C1, C2, r, r′, r″)dA | s | t |
---|
Linear | Γℓ × Γk |
|
|
| −1 | 1 |
Radial | Γℓ,k |
|
| (−1)ℓ+1C1C2Jℓ(C2rr′) Jℓ(C1r′r″)r′r″dr′dr″ | 0 | 1 |
Table II
CSR Elgenmode Equation Parameters and Kernels
Systems | γℓ,k | bℓ | c | kℓ(c, r, r′) | dL | s | t | dk,ℓ |
---|
Linear | βk |
|
|
| dx′ | −1 | 1 | 1 |
Radial | γℓ,k | iℓ+1 |
| cJℓ(crr′) | r′dr′ | 0 | 1 | 1 |
Table III
Discretized Integral Equation Parameters
Systems | Dℓ,k | Sm,p |
---|
Linear |
|
|
Radial | C2(−1)ℓ+1 |
|
Table IV
Egenvalues for Linear Prolates and Linear SFUR (M = −2, −5)
| Linear Eigenvalues |
---|
|
|
---|
k | Prolates βk | M = −2 SFUR Γk | M = −5 SFUR Γk |
---|
0 | 0.99048 | 0.80201 | 0.52416 |
1 | 0.86581 | 0.37333 | 0.10362 |
2 | 0.49355 | 0.05635 | 0.00618 |
3 | 0.15699 | 0.00239 | 9.95 × 10−5 |
4 | 0.03265 | 4.86 × 10−5 | 7.95 × 10−7 |
5 | 5.24 × 10−3 | 6.15 × 10−7 | 4.00 × 10−9 |
6 | 6.93 × 10−4 | 5.35 × 10−9 | 1.39 × 10−11 |
7 | 7.83 × 10−5 | 3.41 × 10−11 | 3.50 × 10−14 |
Table V
Prolate Series Expansion Coefficients for Linear SFUR and Adjoint Linear SFUR (M = −2)
| Linear Expansion Coefficients (M = −2) |
---|
|
|
---|
| Even SFUR Eigenmodes | Even SFUR Adjoint Modes |
---|
|
|
|
---|
n | k = | 0 | 2 | 4 | 0 | 2 | 4 |
---|
0 | | 0.96544 | 0.13187 | 0.00078 | 0.99110 | 0.26017 | 0.02353 |
2 | | 0.26027 | 0.98213 | 0.00922 | 0.13314 | 0.96552 | 0.13846 |
4 | | 0.01345 | 0.13422 | 0.99634 | 0.00046 | 0.00873 | 0.99009 |
6 | | 0.00015 | 0.00360 | 0.08494 | 1 × 10−7 | 5 × 10−6 | 0.00179 |
8 | | 4 × 10−7 | 0.00003 | 0.00197 | 3 × 10−12 | 5 × 10−10 | 5 × 10−7 |
| Odd SFUR Eigenmodes | Odd SFUR Adjoint Modes |
|
|
|
n | k = | 1 | 3 | 5 | 1 | 3 | 5 |
|
1 | | 0.97878 | 0.03770 | 0.00009 | 0.99928 | 0.20478 | 0.01545 |
3 | | 0.20480 | 0.99387 | 0.00350 | 0.03791 | 0.97880 | 0.10451 |
5 | | 0.00632 | 0.10387 | 0.99742 | 0.00004 | 0.00341 | 0.99440 |
7 | | 0.00005 | 0.00261 | 0.07170 | 5 × 10−9 | 1 × 10−6 | 0.00107 |
9 | | 3 × 10−8 | 0.00002 | 0.00153 | 3 × 10−14 | 1 × 10−10 | 2 × 10−7 |
Table VI
Prolate Series Expansion Coefficients for Linear SFUR and Adjoint Linear SFUR (M = −5)
| Linear Expansion Coefficients (M = −5) |
---|
|
|
---|
| Even SFUR Eigenmodes | Even SFUR Adjoint Modes |
---|
|
|
|
---|
n | k = | 0 | 2 | 4 | 0 | 2 | 4 |
---|
0 | | 0.94884 | 0.16081 | 0.00113 | 0.98660 | 0.31478 | 0.03396 |
2 | | 0.31494 | 0.97305 | 0.01153 | 0.16318 | 0.94911 | 0.17265 |
4 | | 0.02250 | 0.16513 | 0.99410 | 0.00077 | 0.01066 | 0.98439 |
6 | | 0.00046 | 0.00662 | 0.10780 | 3 × 10−7 | 9 × 10−6 | 0.00227 |
8 | | 5 × 10−6 | 0.00011 | 0.00356 | 4 × 10−11 | 2 × 10−9 | 8 × 10−7 |
| Odd SFUR Eigenmodes | Odd SFUR Adjoint Modes |
|
|
|
n | k = | 1 | 3 | 5 | 1 | 3 | 5 |
|
1 | | 0.96759 | 0.04680 | 0.00013 | 0.99888 | 0.25220 | 0.02214 |
3 | | 0.25223 | 0.99029 | 0.00442 | 0.04722 | 0.96767 | 0.13192 |
5 | | 0.01196 | 0.13078 | 0.99581 | 0.00007 | 0.00426 | 0.99101 |
7 | | 0.00022 | 0.00477 | 0.09127 | 2 × 10−8 | 2 × 10−6 | 0.00136 |
9 | | 2 × 10−6 | 0.00008 | 0.00273 | 2 × 10−12 | 3 × 10−10 | 4 × 10−7 |
Table VII
Egenvalue Comparison for Circular Prolates and Circular SFUR (M = −2, −5)
| Circular Eigenvalues |
---|
|
|
---|
k | Prolates γ0,k | M = −2 SFUR Γ0,k | M = −5 SFUR Γ0,k |
---|
0 | 0.98350 | 0.64768 | 0.27715 |
1 | 0.47184 | 0.03867 | 0.00274 |
2 | 0.03516 | 4.03 × 10−5 | 4.18 × 10−7 |
3 | 9.36 × 10−4 | 6.93 × 10−9 | 1.14 × 10−11 |
4 | 1.36 × 10−5 | 3.67 × 10−13 | 3.20 × 100−15 |
Table VIII
Prolate Series Expansion Coefficients for Circular SFUR and Adjoint Circular SFUR (M = −2)
| Circular Expansion Coefficients (M = −2) |
---|
|
|
---|
| SFUR Eigenmodes | SFUR Adjoint Modes |
---|
|
|
|
---|
n | k = | 0 | 1 | 2 | 0 | 1 | 2 |
---|
0 | | 0.94038 | 0.16794 | 0.00144 | 0.98535 | 0.33910 | 0.03987 |
1 | | 0.33930 | 0.97107 | 0.01344 | 0.17057 | 0.94067 | 0.17858 |
2 | | 0.02350 | 0.16960 | 0.99316 | 0.00088 | 0.01224 | 0.98311 |
3 | | 0.00040 | 0.00655 | 0.11595 | 4 × 10−7 | 0.00001 | 0.00306 |
4 | | 2 × 10−6 | 0.00009 | 0.00386 | 3 × 10−11 | 3 × 10−9 | 1 × 10−6 |
5 | | 3 × 10−10 | 5 × 10−7 | 0.00005 | 4 × 10−17 | 1 × 10−13 | 2 × 10−10 |
6 | | 4 × 10−11 | 1 × 10−11 | 4 × 10−7 | 4 × 10−20 | 2 × 10−20 | 8 × 10−15 |
Table IX
Prolate Series Expansion Coefficients for Circular SFUR and Adjoint Circular SFUR (M = −5)
| Circular Expansion Coefficients (M = −5) |
---|
|
|
---|
| SFUR Eigenmodes | SFUR Adjoint Modes |
---|
|
|
|
---|
n | k = | 0 | 1 | 2 | 0 | 1 | 2 |
---|
0 | | 0.91420 | 0.20243 | 0.00208 | 0.97831 | 0.40324 | 0.05732 |
1 | | 0.40350 | 0.95740 | 0.01674 | 0.20716 | 0.91498 | 0.22117 |
2 | | 0.03785 | 0.20563 | 0.98902 | 0.00145 | 0.01464 | 0.97354 |
3 | | 0.00112 | 0.01152 | 0.14662 | 1 × 10−6 | 0.00002 | 0.00384 |
4 | | 0.00002 | 0.00029 | 0.00686 | 3 × 10−10 | 8 × 10−9 | 3 × 10−6 |
5 | | 2 × 10−7 | 4 × 10−6 | 0.00015 | 2 × 10−14 | 1 × 10−12 | 5 × 10−10 |
6 | | 4 × 10−11 | 1 × 10−11 | 4 × 10−7 | 4 × 10−20 | 2 × 10−20 | 8 × 10−15 |
Table X
Partial Sum of Squares of Coefficients from Eqs. (D1) and (D2) Compared with Elgenvalues from Direct Discretization
Sum of Squares of Coefficients [see Eqs. (D1) and (D2)] |
---|
|
---|
Number of Terms in Series | M = −2 Linear SFUR Refer to Table V | M = −5 Circular SFUR Refer to Table IX |
---|
|
|
---|
Eigenmode | Adjoint | Eigenmode | Adjoint |
---|
1 | 0.93208 | 0.98227 | 0.83575 | 0.95708 |
2 | 0.99982 | 0.99999 | 0.99857 | 0.99999 |
3 | 0.99999 | 1.00000 | 0.99999 | 1.00000 |
4 | 1.00000 | 1.00000 | 1.00000 | 1.00000 |
5 | 1.00000 | 1.00000 | 1.00000 | 1.00000 |