high freq limits of fm,gm and hm REVIEWED

High Freq Limits of fm, gm and hm PhL 5.16.14 This was early work on doing these large ω limits. They are now derived inside lines doc D.10 in a derivation that was first done below relating to the Bessel function of large argument. I also confirmed these limits with at least one mws files in this folder. Unlike the low ω results, I think my high ω results are accurate. (10.8.14) 1. Large ω limit of fm Here is the fm case taken from lines doc Fact: In the extreme skin effect regime, fm(r) = -2j [ (a/r)1/2 e-(a-r)/δ ej[(r-a)/δ] fm(a) = -2j . (6.5.22) Proof: From NIST 10.7.8 we find that for large x argument, Jm(x) ≈ (2/πx)1/2 cos(x-mπ/2-π/4) . NIST 10.7.8 Then for x = βr = (j-1)(r/δ) from (6.5.9) and with δ << r ≤a we keep only the larger exponential term so that Jm(x) ≈ (2/πx)1/2 cos(x-mπ/2-π/4) ≈ (2/πx)1/2 (1/2) e+r/δ ej[(r/δ)+mπ/2+π/4] = (1/πβr)1/2 e+r/δ ej[(r/δ)+mπ/2+π/4] . (6.5.23) From (6.5.14), fm(r) ≡ [ - ] . (6.5.14) We then examine the two terms = = (a/r)1/2 e-(a-r)/δ ej[(r-a)/δ-π/2] = (a/r)1/2 e-(a-r)/δ ej[(r-a)/δ+π/2] . (6.5.24) Then fm = [ - ] = (a/r)1/2 e-(a-r)/δ ej[(r-a)/δ [e-jπ/2 - e+jπ/2] = (a/r)1/2 e-(a-r)/δ ej[(r-a)/δ [-j - j] = -2j [ (a/r)1/2 e-(a-r)/δ ej[(r-a)/δ] . which then verifies (6.5.22). QED 2. Large ω limit of gm Use this result from the previous section Jm(x) ≈ (1/πβr)1/2 e+r/δ ej[(r/δ)+mπ/2+π/4] Now, from (***), gm ≡ [ + - ]. We then examine the three terms. We borrow this result from the previous section = (a/r)1/2 e-(a-r)/δ ej[(r-a)/δ+π/2] = j (a/r)1/2 e-(a-r)/δ ej[(r-a)/δ] so then the first term becomes = j (a/r)1/2 e-(a-r)/δ ej[(r-a)/δ] The second term is = = (a/r)1/2 e-(a-r)/δ ej[(r-a)/δ and the third term is = = (a/r)1/2 e-(a-r)/δ ej[(r-a)/δ] ejπ = - (a/r)1/2 e-(a-r)/δ ej[(r-a)/δ] Combining the three terms we get gm ≡ [ + - ] = j (a/r)1/2 e-(a-r)/δ ej[(r-a)/δ] + (a/r)1/2 e-(a-r)/δ ej[(r-a)/δ - [- (a/r)1/2 e-(a-r)/δ ej[(r-a)/δ The last two terms are the same, so then gm = { j + 2 } (a/r)1/2 e-(a-r)/δ ej[(r-a)/δ] But since we are doing x = βr >> 1 in this limit, this becomes just gm = 2 (a/r)1/2 e-(a-r)/δ ej[(r-a)/δ] where we ignore that fact that m can get large in distant partial waves. 3. Large ω limit of hm Same as previous section except hm ≡ [- + + ]. so now the last two terms will cancel leaving just the first term (assuming we can ignore correction terms in the asymptotic expansion), so hm = - { j } (a/r)1/2 e-(a-r)/δ ej[(r-a)/δ] 4. Implication for the Appendix D electric fields: Ez(r,m) = (1/4) ηm I Rdc (aβ) fm Er(r,m) = (j/4) ηm I Rdc (aβd) gm Eθ(r,m) = (1/4) ηm I Rdc (aβd) hm fm = -2j [ (a/r)1/2 e-(a-r)/δ ej[(r-a)/δ] gm = 2 (a/r)1/2 e-(a-r)/δ ej[(r-a)/δ] hm = - { j } (a/r)1/2 e-(a-r)/δ ej[(r-a)/δ] Comments: (a) Notice that for x>> 1 we get hm ≈ 0 and then Eθ ≈ 0 which I like in general terms. (b) Both fm and gm are independent of m. (c) Recall from lines doc that Jz(r,θ) = (-2j) [(a/r)1/2 e-(a-r)/δ ej[(r-a)/δ ](I/πa2)(βa/4) * [ 1 + 2 Σm=1∞ (-1)m e-m|ξ|cos(mθ) ] . (6.5.25) and also n1(ξ1,θ) = (q/2π)[ 1 + 2 !Syntax Error, I (-1)m e-m|ξ| cos(mθ) ] . Bipolar (A.13) (6.5.5) allowing us to write Jz(r,θ) = (-2j) [(a/r)1/2 e-(a-r)/δ ej[(r-a)/δ ] (I/πa2)(βa/4) (2π/q) n1(ξ1,θ) We can make then a similar conclusion about Jr(r,θ) by replacing -2j by 2, so Jr(r,θ) = 2 [(a/r)1/2 e-(a-r)/δ ej[(r-a)/δ ] (I/πa2)(βa/4) (2π/q) n1(ξ1,θ) This shows that both Jz and Jr "track n(θ)" in the extreme skin effect regime. (d) From this last equation we can state that Er(r,θ) ≈ Er(a,θ) e(r-a)/δ ej(r-a)/δ (6.5.18) and this then justifies the assumption I made in (6.5.18) !! So I can improve that section with these extra facts. Maple Computations Start with NIST 10.17.2, keeping a few leading terms in each inverse power expansion : Jm(x) ≈ (2/πx)1/2 [ cos(w)*(1-a2(m)/x2) - sin(w)*(a1(m)/x) ] where w = x - mπ/2 -π/4 e-w = e-j(x-mπ/2-π/4) = e-j(z-π/4) ejmπ/2 But x = βr = ej3π/4 (/δ) r which has a large positive imaginary part for small δ, so w has a positive imaginary part. Then cos(w) = [ ejw + e-jw]/2 ≈ (1/2) e-jw sin(w) = [ ejw - e-jw]/2j ≈ -(1/2j) e-jw = (j/2)e-jw Then our expansion above becomes Jm(x) ≈ (1/2) (2/πx)1/2 [e-jw (1-a2/x2) - j e-jw (a1/x) ] = (1/2πx)1/2 e-jw [ 1 -j a1(m) (1/x)- a2(m) (1/x2) + ... ] = (1/2πx)1/2 e-jx ejπ/4ejmπ/2 [ 1 -j a1(m) (1/x)- a2(m) (1/x2) + ... ] Now set x = βr = (j-1)(r/δ) so then e-jx = e-j(j-1)(r/δ) = e(1+j)(r/δ) Jm(x) ≈ (1/2πx)1/2 e(1+j)(r/δ) ejπ/4ejmπ/2 [ 1 -j a1(m) (1/x)- a2(m) (1/x2) + ... ] Now look at = = e(1+j)(r-a)/δ and there are no more terms because m is the same top and bottom. Next, look at = e(1+j)(r-a)/δ ej2π/2 = - e(1+j)(r-a)/δ When we add, we get + = e(1+j)(r-a)/δ * [ 1 - ] Now both x and xa are large so approx the ratio as ≈ 1 - ja1(m+1)(1/x) + ja1(m-1)(1/xa) Then we have + = e(1+j)(r-a)/δ * [ ja1(m+1)(1/x) - ja1(m-1)(1/xa) ] (*) Now look at the other term Jm(x) ≈ (1/2πx)1/2 e(1+j)(r/δ) ejπ/4ejmπ/2 [ 1 -j a1(m) (1/x)- a2(m) (1/x2) + ... ] = (2m/x) e(1+j)(r-a)/δ ejπ/2 and again we do our approximation on the ratio to get = (2m/x) e(1+j)(r-a)/δ ejπ/2 [ 1 - ja1(m)(1/x) + ja1(m-1)(1/xa)] = j (2m/x) e(1+j)(r-a)/δ [ 1 - ja1(m)(1/x) + ja1(m-1)(1/xa)] Now look at hm : hm ≡ [- + + ]. = - j (2m/x) e(1+j)(r-a)/δ [ 1 - ja1(m)(1/x) + ja1(m-1)(1/xa)] + e(1+j)(r-a)/δ * [ ja1(m+1)(1/x) - ja1(m-1)(1/xa) ] = e(1+j)(r-a)/δ * { - j (2m/x) [ 1 - ja1(m)(1/x) + ja1(m-1)(1/xa)] + [ja1(m+1)(1/x) - ja1(m-1)(1/xa) ] } first term last two terms = e(1+j)(r-a)/δ * { - j (2m/x) + ja1(m+1)(1/x) - ja1(m-1)(1/xa) } = e(1+j)(r-a)/δ { { j[ - 2m + a1(m+1)](1/x) - ja1(m-1)(1/xa) } = e(1+j)(r-a)/δ j{ [ - 2m + a1(m+1)](1/x) - a1(m-1)(1/xa) } This whole thing is then order 1/x so we get hm ≈ 0 *********************** Compare my limit to the M notation: Jm(x) = (1/2πx)1/2 e-jx ejπm/2 ejπ/4 [ 1 -j cm (1/x) - dm (1/x2) + ... ] Pause. Compare this to Jm(ej3π/4z) = Mm(z) ejθ(z) . (2.3.3) so we have x = ej3π/4z = (j-1)z/ e-jx = e-j(j-1)z/ = e(1+j)z/ My limit expressed in terms of z is then Jm(x) ≈ (1/2π ej3π/4z)1/2 e(1+j)z/ ejπm/2 ejπ/4 [ 1 -j cm e-j3π/4 (1/z) - e-j3π/2dm (1/z2) + ... ] |Jm(x)| ≈ (1/2π z)1/2 ez/ | 1 -j cm e-j3π/4 (1/z) - e-j3π/2dm (1/z2) + ... | ≈ (1/2π z)1/2 ez/ | 1 -j cm e-j3π/4 (1/z) | and the leading factors agree with (2.3.5). Now, | 1 -j cm e-j3π/4 (1/z) | = | 1 -j cm(-1-j) (1/z) | = | 1 +j cm(1+j) (1/z) | = | 1 + cm(j-1) (1/z) | So | 1 + cm(j-1) (1/z) |2 = (1 + cm(j-1) (1/z)) (1 + cm(-j-1) (1/z)) = 1 + cm(-j-1) (1/z) + cm(j-1) (1/z)) = 1 - 2 cm (1/z) => | 1 -j cm e-j3π/4 (1/z) | ≈ 1 - cm (1/z) = 1 - a1(m) (1/z) Now NIST says, a1(m) : 2k-1 = 1 so numerator is (4m2- 1), denom = 8, so a1(m) = (4m2- 1)/8 Then my limit becomes |Jm(x)| ≈ (1/2π z)1/2 ez/ [ 1 - a1(m) (1/z) ] = (1/2π z)1/2 ez/ [ 1 - (1/z) ] = (1/2π z)1/2 ez/ [ 1 - (1/z) ] This agrees with my (now corrected) (2.3.5) How about the phase part? Jm(x) ≈ (1/2π ej3π/4z)1/2 e(1+j)z/ ejπm/2 ejπ/4 [ 1 -j cm e-j3π/4 (1/z) - e-j3π/2dm (1/z2) + ... ] ≈ e-jπ3/8(1/2π z)1/2 e(1+j)z/ ejπm/2 ejπ/4 [ 1 -j cm e-j3π/4 (1/z) - e-j3π/2dm (1/z2) + ... ] The leading factor has this phase (ignoring the [...] for the moment): θ = -π3/8 + z/+mπ/2 + π/4 = -3π/8 + z/+mπ/2 + 2π/8 = -π/8 + z/+ mπ/2 This agrees with the leading terms of (2.3.5). Now for the bracket [ 1 -j cm e-j3π/4 (1/z)] = 1 + cm(j-1) (1/z) // from a foot above = (1 - cm (1/z) + j cm(1/z) θ = tan-1 (y/x) ≈ y/x ≈ cm(1/z) / 1 ≈ cm(1/z) ≈ (4m2- 1)/8(1/z) = (1/z) and this agrees also with (2.3.5). Fact: I have proven that my two limit formulas are the same and fixed some bugs!