Copyright
Joseph Bishop Keller.

Asymptotic solution of some diffraction problems online

. (page 5 of 5)
Online LibraryJoseph Bishop KellerAsymptotic solution of some diffraction problems → online text (page 5 of 5)
Font size
QR-code for this ebook


1 +



sin



t. 2 s sin t cos ^ 2



sin I



- 5 sin |)



. i5(^)3(sin^ f .1). ^ ^o

2' 2^(s sin p+b cos"^ |)sin



cos



2-1



For the far field, 3 '^ R, p -^J a ar^d f ^ R - 2b^sin » (see Fig. 10) so that
(220) u„. . - ^ e ° ^Vl - i-^ y * ojTkb^)-^ .



b ik(R-2b sin ^) .

do ksin »



The result (220) agrees with the approximate solution of W. Franz and K, Dspperman [2^,



Fjcample 22 ; TUf fraction of a plane wave by a sphere (— => 0)



au



If in the preceding problem, we replace the boundary condition by ^ - 0,



Bv



we obtain (l95), (206) and (207). The first few terms in the expansion of the
field are then found to be given by



- 73 -



ikx



3b

b^ sin p iic(s - -2^ sin |)



(221) u~e-"'^+y ii >-g- e

Us(s sinp + b cos £)



i
^b~k



fi + 2 sin^ I b^ o b , _

sin fr 2 s sin *■ cos k 2 sin



-=• (—) (sin ^ -1)- -r



2^ ^ ^ 2^(s sinp + b^cos-^ |)sin | cos



For the far field, we find

b^ ik(R-2b„sin i) ( 1 + 2 sin^ i \ r- ^-,



The result (222) agrees with the approximate solution of W« Franz and K.
Depperman [2^ ,



:}■••}



10. The impedance bovmdary condition

We will now consider reflection from a surface on which an impedance (or
mixed) boxindary condition is to be satisfied. The impedance boundary condition
is
(223) 1^ - ik Z u.

In (223) Z is called the impedance of the boundary and we will assume that it is
independent of k.

Let the incident and reflected fields have the expansions



- 7U -



ii^Y U,y,z) 00 w (x,y,z)

(22U) u, -^ e ° r" -2

inc i— /., \n

n-o (ik)



(225) u 1.

The limits Z ->oo and Z -> correspond, respectively, to the boiindary conditions

u ■ and ^tr = 0,
ov



Example 23 : TUffraction of a plane wave by a parabolic cylinder (-^ - ikZu)

-ikx
A plane wave e is incident from the right (see Figure 1 ) on the para-
bolic cylinder with E^. (16 ), From Eqs. (226) - (228), we find that on the
parabolic cylinder

(229) Y - -X - - r^(0) cos ©

(230) V - (Z + cos |)(-Z + cos %)~^

r..i^ . 1 , ^^n-l 1 f O^Vl ^i"|c°^| aVn-1

(^^^^ ^n 9—7 • -9^; 0-— f °^ ? -a? P • -§r-

cos 2 " ^ °°^ 7 " ^ 1

If these initial values are inverted into the recursion formulas ( 9 ) one
finds and verifies inductively that for n > 1



-75-



(232) fj„ ■ p'°*'* ' (CCS I - Z)-(2-l) g^ a.,„(=o. |)*-2J-1 .



For j ^ and n > 2 the constants a . . satisfy
— jtn



.-1



a . r^^(t-2n-3)(Uj.2n-t.l)a..,^,.^^^.^



z[(t-3)(t-Uj-l)+(t-Uj)(t-2)-2n(2t-Uj-3)1a^_^^^_^^^^^

Z(t-2J)(UMm-2t)a..^^^_^^„_^

p(iij-t)(t-2) + (2n+2j+l-t)(2n+2j-t)~|a^_^^^_2^^_3^



. Z'(t-2j)(t-2j.l)a._^^^^^.^



>



For j » we have Instead



(23U) a . . - E



J-1



a-4. + i (3+2n-t)a. ^ . , , + Z(t-3)a. , . , ^ ,
jtn 2(2 o-l,t-U,n-l j-l,t-3,n-l



(t-2J-2n)a^_^^^_2,n-l ^ 2(2j-t)a.



i-l,t-l,n-lj) •



The complete asymptotic expansion of the field is thus given by

ik(r-2p) 00 n Un

—h 9 ^ CII

cos ^(cos 7 - Z) n=o 3^ t=o



(235) u-e-i^./f



l^-n (£j;



a.^ Jikp(cos I - Z)^- (£)^ (cos |)^-2J



Hie first few terms are



- 76-



(236) u -v^ e



-ikx



*/F



gik(r-2p)

Q Q

COS ^(cos ^ - Z)



Z + cos






9



Uipk(cos ^ - Z)



-Zcos



U



+ d-Z"^) co8^ I + Z cos'^ I + Z^ cos I -Z^ + £(-3Zco8^ I + (Z +1) cos I

)



.^^.yL. z^



a ' 2 6
cos ■= cos X



For the far field we find that



(237) u.



ref



^ "^ cos |(cos I - Z)



'Z + cos w +



Uipk(cos I - Z)^



-Z cos



U



+ (l-Z^)cos^ I ♦ Z cos^ I + Z^cos I -Z^ + i-

J [8pk(coE



1?F



15Z cos^ I



- U3Z^ cos"^ I + (UUZ-llZ^)cos^ % + (9Z^+20Z^-I6)cps^ %



+ (U0Z-8Z^)cos^ I + (32Z^-8Z^)cos^ | + l6Z^cos^ | - 2Uz'^cos | ♦ 8Z^



0[(kp)-^



The amplitude of the far field is



(238)



lAl



-y^ sec







(cos I - Z)

Q

(cos ^ + Z)



-v> 1 +



[8pk(cos I - Z)]^(cos^ I - Z^)



9 Q
15ZC08^ I



- 26Z^co8^ I + (U0Z-U0Z^)co8'^ I + (56Z^-lii)co8^ |

+ (9Z^+az^+28z)cos^ I + (78Z^-16z'*)coE^ | * (U8Z^-UZ^)cos^ |



- lOZ^ cos^ I - 20Z^ cos I + lOZ^



- 77 -



Example 2U « Diffraction of a plane wave by a circular cylinder (— - ikZu)

ikx
If a plane wave e is incident on the circular cylinder given by Eq,

(192), we find from (226)-(228) using the coordinates (s,p) that

(239) T(Sq,P) - -bpSin |



(2U0)
(2U1)



(s^,p) - (Z + sin |)(sin | - Z)"-""



o



V%'p^ "



3v



n-1



Z - sin
2co8 I



3r



3v - 2 + cos
n-1^



3v



n-1



sin |(Z-sin |) ^^ 2sin |(Z-sin |)



3s



, n > 1



The first few terms of the asymptotic expansion of the field are found



to be



(2U2) u^e



ikx V^o „. B



e



ik(s- I b^sin |)



2s "*•• ? sin I - Z



z-^i^-Fk hrrz-r - ;^



2^(sin £■ -Z)



13 sin^ I - 11 Z sin I ♦ 8 - 19 Z^
8Z+3Z^ ^ 8Z^ 8Z^ 1 ^o r ^3 B



sin I sin^ I sin^ |J 2^(sin | -Z)



- 3Zsin^ I - (l+llZ^)sin | + 9Z + 3Z^ ♦ ~



^ z3



sin t sin



rr



]



3b|



2V(sin I -Z)



Gsin^ I - (3Z^+2)siii I + 8Z + ^^



8Z



sin * sin ^



]



+ i|^ r. sin^ I - Z sin^ I + sin ^ + zl



- 78 -

References

[l] Luneburg, R.K,, a. The Mathematical Theory of Optics, Brown Univ., 1?UU
b. Propagation of Electromagnetic Waves, New York Univ., 19U8.
[2] Kline, M., An Asymptotic Solution of Maxwell's Equations, Comm, on Pure
L-and Applied Math., Vol. IV, No. 2-3, Aug. 1951, pp. 225-263
and Asymptotic Solution of Linear Hyperbolic Partial Differential
Equations, Joum, of Rational Mech. and Analysis, Vol, 3, No. 3,
May, 195it.
[3] Friedlander, F.G., Geometrical Optics and Maxwell's Equations, Proc.

Cambridge Phil, Soc. Vol. k3, Pt. 2, 19U6, pp. 281i-286,
[U] Bremmer, H., The Jumps of Discontinuous Solutions of the Wave Equation,
Comm. on Pure and Applied Math., Vol. IV, No. I4, Nov, 1951 pp.
U19-U27.

[5] Copson, E.T., The Transport of Discontinuities in an Electromagnetic Field,
Comm. on Pure and Applied Math., Vol. IV, No. k, Nov, 1951, pp.
U27-U35.

[6] Keller, J,B,, The Geometrical Theory of Diffraction, Proceedings of the
Symposium on Microwave Optics, McGill Univ., June 1953.

[7] Keller, J.D., Geometrical Acoustics I. The Theory of Weak Shocks, Joum.
Appl. Physics, Vol, 25, No. 8, pp, 938-9U7, Aug. 195U.

[8] Birkhoff, G.D., Some Remarks Concerning Schroedinger ' s Wave Equation, Proc.
Nat'l. Acad, of Sciences, Vol, 19, 1933 pp. 339-3Uh and in
Collected Math. Papers, Araer, Math. Soc, 1950, Vol. II, pp. 813-
818. Also Quantum Mechanics and Asymptotic Series, Amer, Math,
Soc, Bulletin, Vol, 39, 1933, pp. 681-700 and in Collected Math,
Papers, Vol, II, pp. 837-856.

[9] Brillouin, L., Remarques sur la Mecanique Ondulatoire, J. Phys. Radixim 7,

pp. 353-368 (1936) J Las Mecanique Ondulatoire j une methode generale
de j?esolutior por approximations successives Comptes Rendus 183,
2U, 1926.

[10] Dirac, P,A,M., Tho Principles of Quantum Mechanics, Oxford, 19U7, Third
Edition, pp. 121-123.

[11] Keller, J.B., Derivation of the Bohr-Sommerfeld Quantum Conditions From an
Asymptotic Solution of the Schroedinger Equation. Research Report
No. CX-10, July, 1953, Institute of Mathematical Sciences, New York
University, Division of Electromagnetic Research, N.Y.

[12] Lowell, S.C., The Propagation of Waves in Shallow Water, Comm. on Pure and
Applied Math. Vol. 2, No. 213, pp. 275-291, 19U9.



^ 75-



[13] Schensted, C, The Electromagnetic Transport Equation and the Luneburg-
Kline Method of Solution, Willow Run Research Center, Eng,
Research Inst,, Univ. of Michigan, Report l5-25-(50U)-3.

[lU] Friedrichs, K.O. and Keller, J.B., Geometrical Acoustics II: Diffraction,
Reflection and Refraction of a Weak Spherical or Cylindrical
Shock at a Plane Interface, to be published in Jour. Appl. Phys.

[15] Friedlander, F.G. and Keller, J.B., Asymptotic Expansions of Solutions of

2 2

(V +k )u - 0, Research Report No. EM-67, Sept. 195U, Institute

of Mathematical Sciences, New York University, Division of Elec-
tromagnetic Research, N.T.

[16] Kay, I. and Keller, J.B., Asymptotic Evaluation of the Field at a Caustic,
Jour. Appl. Phys., Vol. 25, No. 7, pp. 876-883, July, 195U.

[173 Gerjuoy, E., Total Reflection of Waves From a Point Source, Coram, on Pure
and Applied Math., Vol, VI, No. 1, pp. 73-91, 1953.

[18] Lamb, H., On Sommerfeld's Diffraction Problem and On Reflection by a Para-
bolic Mirror, Proc, London Math. Soc, 1906, pp. 190-203.

[19] Friedlander, F.G., On the Reflexion of a Spherical Sound Pulse by a Para-
bolic Mirror, Proc. Comb, Phil. Soc, Vol. 36, Pt. U, pp. 383-
393.

[20] Erdelyi, A., et. al., Higher Transcendental Functions, McGraw-Hill, New
York, 1953.

[21] Wentzel, G., Eine Verallgemeinerung der Quantenbedingungen fur die Zwecke
der Wellenmechanik, Zeits, f . Hiysik, Vol. 38, p. 518, 1926.

[22} Keller, J.B.- and Keller, H.B., Determination of Reflected and Transmitted
Fields by Geometrical Optics, Jour. Optical Soc. Am., Vol UO,
No. 1, pp. U8-52, Jan. 1953.

[23] Keller, J.B. and Preiser, S., Determination of Reflected and Transmitted
Fields by Geometrical Optics II, Inst. Math. Sciences, New York
University, Division of Electromagnetic Research, Report No, EM-20,
April, 1950.

\^li] Imai, I., Die Beugung elektroraagnetischer Wellen an einem Kreiszylinder,
Zeit. fur Physik, Bd. 137, pp. 31-U8 (195U).

[25] Depperman, K. and Franz, W., Theorie der Beugung an der Kugel unter

Bemcksichtigung der Kriechwelle, Ann. der Physik, Ser. 6, Vol. lU
pp. 253-26U, 195U,





Date Due




■Ay>S^56


MAY > ia«i




1


WUfdO'dS


lAfK^ '






"ta^W


^cB I 6J)971






MY '?W ^


^is


1 2 3 5

Online LibraryJoseph Bishop KellerAsymptotic solution of some diffraction problems → online text (page 5 of 5)