TECHNICAL REPORT CERC-87-16
US Army Corps COMBINED REFLECTION AND DIFFRACTION of Engineers BY A VERTICAL WEDGE by H. S. Chen
Coastal Engineering Research Center
DEPARTMENT OF THE ARMY
Waterways Experiment Station, Corps of Engineers PO Box 631, Vicksburg, Mississippi 39180-0631
8a,
December 1987 Final Report
\ NY Lif, iI Hl eC ns. Le \ Approved For Public Release, Distribution Unlimited it | \
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Prepared for DEPARTMENT OF THE ARMY US Army Corps of Engineers Washington, DC 20314-1000
Under Waves at Entrances Work Unit 31673
(.S. Rees Bos ea ee ieee. Kea CEES.
TECHNICAL REPORT CERC-87-16
US Army Corps COMBINED REFLECTION AND DIFFRACTION of Engineers BY A VERTICAL WEDGE
by H. S. Chen
Coastal Engineering Research Center
DEPARTMENT OF THE ARMY Waterways Experiment Station, Corps of Engineers PO Box 631, Vicksburg, Mississippi 39180-0631
Be: I 1. 1 ; fe b; i : \ _ Woods Hole Qeeanosra sits } Instit
December 1987
EA Oss NE Ie Wi Tf one is
; a Final Report sai) We) BREAN WATE ik o//// |i\ \ is erty | fray \ Approved For Public Release, Distribution Unlimited Zi fay yey yi a IS) | / | \ a i if |i | \ Ba / } / | | off | |
Prepared for DEPARTMENT OF THE ARMY US Army Corps of Engineers Washington, DC 20314-1000
Under Waves at Entrances Work Unit 31673
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Technical Report CERC-87-16
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PROGRAM PROJECT TASK WORK UNIT ELEMENT NO. | NO NO. ACCESSION NO Washington, DC 20314-1000 31673
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Combined Reflection and Diffraction by a Vertical Wedge
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18. SUBJECT TERMS (Continue on reverse if necessary and identify by block number) Breakwaters (LC) Me os Numerical analysis (LC) Rasa a | ee | Water waves 19. ABSTRACT (Continue on reverse if necessary and identify by block number)
This report presents an analytical solution for the combined wave reflection and diffraction by a vertical wedge, of arbitrary wedge angle, subject to the excitation of a plane simple harmonic wave train coming from a far field. Results of two special cases are calculated: one a thin semi-infinite breakwater and the other a wedge of 90-deg angle. The results are presented in amplification factor diagrams. A subroutine WEDGE is also documented in the appendix. The subroutine can be used to calculate the case of arbitrary wedge angle.
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PREFACE
The work in this report was authorized by the Office, Chief of Engineers (OCE), Coastal Engineering Functional Area of Civil Works Research and Development, under Waves at Entrances Work Unit 31673, Harbor Entrances and Coastal Channels Program, at the Coastal Engineering Research Center (CERC) of the US Army Engineer Waterways Experiment Station (WES). Messrs. John H. Lockhart, Jr., and John G. Housley were OCE Technical Monitors. Dr. Charles L. Vincent is CERC Program Manager.
This report was prepared by Dr. H. S. Chen, Coastal Oceanography Branch (CR-O), Research Division (CR). Work was performed under direct supervision of Dr. Edward F. Thompson, Chief, CR-O, and Mr. H. Lee Butler, Chief, CR; and under general supervision of Dr. James R. Houston and Mr. Charles C.
Calhoun, Jr., Chief and Assistant Chief, CERC, respectively.
This study was initiated after a discussion by Dr. Vincent and the au- thor on the possibility of implementing a scheme to redistribute wave energy behind islands in numerical models. The author acknowledges and appreciates the review and comments provided by Drs. Edward F. Thompson and Norman W. Scheffner. This report was edited by Ms. Shirley A. J. Hanshaw, Information Products Division, Information Technology Laboratory, WES.
Commander and Director of WES is COL Dwayne G. Lee, CE, and Technical
Director is Dr. Robert W. Whalin.
CONTENTS
IINIMVNGIG 50 0000000000000000000000000000000000000000000000000000000000000
ARS Telas TON EROWDIUG ILO 5 6 9000000000000000000000000000009000000000000000 PART II: BOUNDARY VALUE PROBLEM... .cccccccccccrcccrcrcccccccsccsscosees
MEVMaMAeICaL IO rMMLEVELOMNs o4660000000000000000000000000000000000000 AMAISAEHOAIL SOIMEROMs 690000000000000000000000000000000000000000000 IK Speeiwad, CASG@Scaodccood0d0 00 DDD DDO CODD DOODDD DODD ODDO DDDDD0000
PART III: CALCULATION AND RESULTS... 2... ccccccccvcccceccervccsvcccsees
Vertical Wedge of O-Deg Wedge Angle.....ccccccccccscccrcecrccccrcecs Vertical Wedge of 90-Deg Wedge Angle......cccesccccvcccccccrcccce
PART IV: (GOMGMWISIMONS 6 G50 000000000000000000000000000000000000000000000 INIDIDIDRVAMGINS 6 9600000000006000000000000000000000000000000000060000006000000 JNO (AQ SW RNOWALION, MINE 6 6 0000000000000000000000000000000000050000 JNPIEANDIDG 138 INOWMEMON 5 6900000000000000000000000000000000000000000000000
COMBINED REFLECTION AND DIFFRACTION BY A VERTICAL WEDGE
PART I: INTRODUCTION
1. The boundary value problem of linear wave reflection and diffraction by a vertical wedge of arbitrary wedge angle has been well formulated and presented by Stoker (1957) among many other investigators. The technique to obtain an analytical solution for the problem is also depicted in the cited book. However, analytical solutions are not available for the problem, except for the special case of wave diffraction by a thin semi-infinite breakwater, that is, a wedge with wedge angle equal to zero.
2. The solution of the thin semi-infinite breakwater was presented in the dimensionless diffraction diagrams by Wiegel (1962). The diagrams have been especially useful in preliminary engineering design and have been in- cluded in the Shore Protection Manual (SPM) (1984). Although equally useful, the combined reflection and diffraction diagrams are not available, perhaps because of the complexity of the diagrams which makes them difficult to create without using modern high-speed computers for computation and graphing.
3. The objectives of the present study are (a) to obtain an analytical solution for the combined wave reflection and diffraction by a vertical wedge of arbitrary wedge angle subject to excitation of a plane simple harmonic wave train coming from infinity and (b) to provide the combined reflection and dif- fraction diagrams. The diagrams included in this report have two cases: one for a thin semi-infinite breakwater and the other for a 90-deg vertical wedge. Subroutine WEDGE for computing the combined reflection and diffraction by a vertical wedge of arbitrary wedge angle is also documented in the report (Appendix A).
PART II: BOUNDARY VALUE PROBLEM Mathematical Formulation
4. In this study our primary interest is the wave reflection and diffraction by a vertical wedge of arbitrary wedge angle in a constant water depth h * subject to the excitation of monochromatic incident waves of infi- nitesimal amplitude coming from infinity. Let (r,0,z) be cylindrical coordi- nates, with z= 0 representing the undisturbed water free surface and upward direction representing the positive z-axis. The tip of the wedge is chosen to be the origin of the coordinates and two rigid walls of the wedge to coincide with 6=0 and 6= 85 » respectively, as illustrated in Figure 1. Carte- sian coordinates (x,y,z), corresponding to the cylindrical coordinates, are
also occasionally used and shown in the same figure. Therefore, the wedge
zZ
Figure 1. A vertical wedge of arbitrary wedge angle
* For convenience, symbols and abbreviations are listed in the notation (Appendix B).
angle is 27 - on » and the water region is defined by oF 2620 and 0222-h.
5. The velocity field for the wave reflection and diffraction in an ideal fluid can be represented by the velocity potential function 4(r,®,z,t) which must satisfy the Laplace equation, where t is the temporal coordinate. We assume that the waves are sinusoidal in time with radian frequency w . Water depth is constant, and the bottom is rigid and impermeable. Therefore, the vertical and temporal components of the velocity potential function, which follow from separation of variables, can be factored out and the velocity po-
tential written as
mn cosh k(z + h) iwt O(r,8,z,t) = Ay cosh kh o(r,8)e (1) where Ay = -iga,/w i= yal g = gravitational acceleration aS incident wave amplitude
k = wave number
¢ = horizontal component of the velocity potential function
6. Substituting Equation 1 into the Laplace equation and using both the kinematic and dynamic boundary conditions at the free surface, the Laplace equation is then reduced to the Helmholtz equation which is written in polar
coordinates as follows:
ES aot gine ai gh kro =0 (2)
where k must be a real number and satisfy the dispersion relationship 2 W = gk tanh kh (3)
7. The free surface displacement NN from the mean water level z= 0 can be obtained from linear wave theory and is represented as 1 ao
= Oe iwt n(r,8,t) = a 8G aj o(r, Oe (4)
8. Thus only the horizontal part of the velocity potential function 9 is needed to be determined as a solution of Equation 2 in the water region on > 620 , with the following boundary conditions at the rigid and imperme-
able walls of the wedge:
ad _ 0
ona ae @ = 0 ernal 85 (5)
9. A condition at infinity is also required to ensure a unique solu- tion. The classic approach is to use the Sommerfeld radiation condition at infinity which states that the scattered wave 0 must behave like a cylin-
drical outgoing progressing wave at infinity such that '
a lim vr (ste + sn4,) = 0 (6)
The total wave represented by 9 is the linear superposition of an incident wave %; » a reflected wave from the the ®=0 wall of the wedge o >» and
the scattered wave Os from the tip of the wedge.
9= 6,4 6,4 9, (7) Equation 6 can be satisfied if eo tkr od ~ at ro (8) = ise
10. The incident wave coming from a large distance from the tip of the wedge is assumed to be a plane progressive wave of amplitude ay and incident angle a to the x-axis as given by
=f etkr cos (6-a)
(9)
Consequently, the perfectly reflected wave from the y = 0 wall of the wedge
is
a etkr cos (6+0) (10)
Thus the boundary value problem (in which the governing equation is Equa- tion 2, the boundary condition is Equation 5, and the radiation condition is
Equation 6) is completely formulated.
Analytical Solution
11. Analytical solution to the problem formulated in the preceding section is obtained by following the solution technique by Stoker (1957). To obtain the solution, the water region is divided into three subregions--I, II, and III--by the incident wave ray passing through the tip of the wedge and the reflected wave ray reflected away from the tip of the wedge, as shown in Fig- ure 2. Obviously, the total wave in subregion I is the sum of the incident, reflected, and scattered waves; the total wave in subregion II, where the reflected wave does not exist, is the sum of the incident and scattered waves; and the total wave in subregion III, where the incident and reflected waves
have been shaded out, is only the scattered wave. For certain combinations of
y
Tt
Figure 2. Three subregions and the wedge
the wedge angle and incident wave angle, subregions II and III may not exist
at all. In general, the solution function can be written as
$= ¢(r,8) + 6 (r,8) (11) where 5 + om T-a>O>0 o (F298) = o; Tta>O6@>7T-a (12) 0 oe > @ > ta ©
The equation reveals that oF is the sum of the incident and reflected waves 5 and $. and is a known function. The scattered wave oe is the only un- known function to be determined in the problem. Nevertheless, the total wave ¢ instead of the scattered wave 0. is the desired solution to be obtained in this study.
12. The solution for the total wave $ is pursued. The finite cosine
transform of 9 , denoted by $ » is introduced by the formula
VT
o(kr,n) = i o(kr,®) cos ut dé (13)
where n= 0,1,2,... are integers, and v is related to the wedge angle as
defined by
@ = vt (14)
Applying the finite cosine transform and using the boundary condition in Equa-
tion 5, Equation 2 becomes
etree +r = + (IED - (2) ¢=0 (15)
Equation 15 is a form of the Bessel equation for which general solutions are Ja sy 6k) and Y/vike) ;
respectively. Since Ya/vik®) are singular at the origin, the solution is
the Bessel functions of the first and second kinds,
chosen to be
o(kr,n) = a Jy 6k) (16)
where a, are constants to be determined. 13. Taking the finite cosine transform of Equation 11 and using Equa-
tion 16, we have
VT VOT
a Jy (kX) - i 6 cos = dé (17) 0 0
~o- n Q ie} n Is Q @D i]
or
(kr) - co (18)
2 |
Son ss
Then applying the operation lim... vr(9/dr + ik) to both sides of Equa-
tion 18, and using the Sommerfeld radiation condition (Equation 6) we have
VT F 0 , ng “ Lim (Fe is ix) a. Tes) © i 6, cos * ae| = 0 (19) (
14. Equation 19 can be asymptotically evaluated to determine ale Firstly, the first term involving the Bessel function is evaluated. The func- tion J py Sk) at reo behaves asymptotically (Abramowitz and Stegun 1964)
as follows:
Jay (ke) ~ Vo cos (ex - = - z) (20)
Hence, we have
Lim "(Fe a, th) py : E: ei (kr-nn/2vtn/4) (21)
Secondly, the second term involving the integral of oF is evaluated. The asymptotic behavior of the integral over 9% = (0,v™) and at large distance r>© can be found by the method of stationary phase. The integral, after sub-
stituting oP from Equations 9, 10, and 12, can be written as
vt TO. e ie ° 6 i d cos ng dé -f E: Sees=e)) + epliee cos(842) | Coste fo) v av) 0 0 TO. i C= () + gue eerie) cos = do (22) T-a
In the integrals, there are three points of stationary phase at 9 = 4% and 6=7 +a. If the same argument as that of Stoker (1957) is followed, of the three contributions only the first one 9 =a furnishes a nonvanishing con- tribution for r*@ when the operator Y“r(9/dr + ik) is applied to it. The physical significance of this statement is that only the incident wave is
effective in determining the cosine coefficients of the solution. Therefore,
VT lin “e(% + ax) f 9, cos ne d6 ~ 2V2Tk cos = sare) (23) 0
Substituting Equations 21 and 23 into Equation 19, we obtain the unknown co-
efficients a
na eint/2v
anes ZECOS Ra (24) 15. Since the solution ¢ in the cosine series expression is ee ee 2 = nO 6(r,8) = = B(r,0) += JY F(xsn) cos & (25) n=1
10
the solution is obtained by substituting Equations 16 and 24 into Equation 25
as follows:
inn/2v a. ng
o(r,9) = = JQ (kr) + 2 » e Jay ke) cos - cos —— (26)
n=1
Equation 26 is the solution for the combined wave reflection and diffraction by a vertical wedge of arbitrary wedge angle and is considered to be extended from the solution by Stoker (1957) who only solved the problem of a thin semi- infinite breakwater. The solution in Equation 26 and the one by Stoker are not only in nonclosed form but also in terms of Bessel functions. It seems that the calculations of the solutions are very difficult without using a mod- ern high-speed computer. This is probably the reason why Stoker arrived at his solution expressed in the same cosine series but did not use it to calcu- late the result. Instead, he further transformed the expression into a very complex integral form for further approximation in calculating the result.
16. Notably, the solution at the origin point is obtained by simply
substituting r= 0 into Equation 26 to arrive at 2 $(0,8) == (27)
Therefore, wave response at the origin point depends only on the wedge angle
and does not depend on the incident wave angle.
Two Special Cases
17. The solutions for two special cases are used to verify Equation 26: one for the case of a thin semi-infinite breakwater and the other for the case of an infinite wall extending from x = -~ tom,
18. The vertical wedge should reduce to a thin semi-infinite breakwater as the wedge angle reduces to 0 deg. Therefore, solution of the combined wave reflection and diffraction by a thin semi-infinite breakwater is obtained by substituting v=2 (that is, oR = 27) into Equation 26 which then becomes
11
OGe,@)) = JQ (kr) + 2 » einm/4 5 (kr) cos = cos x (28)
n=l
n/2
Equation 28 is precisely the same one obtained by Stoker (1957).
19. The vertical wedge should also become an infinite wall extending from x = -© to © with the water occupying only the half plane of y20 as the wedge angle increases to 180 deg. In this situation the scattered wave is absent from the solution, and the total wave is only the sum of the incident
and reflected waves as follows:
GeO) = eikr cos (6-0) + eikr cos (8+0) ag
After expansion of the exponential functions in terms of Bessel functions
(Abramowitz and Stegun 1964), Equation 29 becomes
foe)
$(r,8) = 2 Jo (kr) ap 2 » i"J (kr) cos n@ cos n@ (30)
n=1
Equation 30 is the same equation reduced from Equation 26 by substituting
v= 1 into it.
12
PART III: CALCULATION AND RESULTS
20. Results of the combined reflection and diffraction by a wedge of arbitrary wedge angle can be calculated from Equation 26. Since the solution is not only in terms of Bessel functions but also in a nonclosed form, the computer program WEDGE is therefore written to calculate the solution.
21. In the program the subroutine BESJ for calculating Bessel function of fractional or integer order was used. The subroutine was originally writ- ten by Amos, Daniel, and Weston in 1975 (Morris 1984) and is collected in the Naval Surface Weapons Center Library of Mathematics Subroutines (Morris 1984).
22. In the calculation the summation of the infinite terms in Equa- tion 26 was carried out to the term which is preceded by eight successive terms of the absolute value of the Bessel function, all equal to or less than Oman The solution has a truncation error less than Omen and it is of the order of one.
23. In this study, results of the combined wave reflection and diffrac- tion for the wedge are calculated for two cases: one for a vertical wedge of
O-deg wedge angle and the other for a vertical wedge of 90-deg wedge angle.
Vertical Wedge of 0-Deg Wedge Angle
24. When the wedge angle is equal to zero, the wedge is actually a thin semi-infinite breakwater extending from x =0 to ». Figure 3 shows the thin semi-infinite breakwater along with the polar coordinates. In this case the diffraction results for various incident wave angles in the water region from 6 =m to 2m and r/dA < 10 , where A is the incident wave length, have already been presented by Wiegel (1962) and are shown in the SPM, Vol- ume I (1984). The present results combine reflection and diffraction effects and cover the water region from 6 =0 to 2m and r/dA < 10. Therefore, the present results for this particular case can be considered to be a comple- mentary and extended version to the ones in the SPM.
25. In this study wave response was calculated at 1,460 grid points intersected by r/\ = 0.5,(0.5),10 , which means that the values of r/) are from 0.5 to 10.0 with each value increment being 0.5. Hereafter, all similar
expressions are to be interpreted in the same way (e.g., 96 = 0,(7/36),2m for
13
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eee IS oy egy a
Figure 3. Thin semi-infinite breakwater and polar coordinates
the incident wave angle a = 0,(mn/12),7) . The wave response at the origin
point is obtained substituting v = 2 into Equation 27, as follows:
o(0,6) = 1 (31)
Those calculated values were used to interpret the value for each non-
overlapping pixel of size O.lr/A by O.lr/X in the area within the 10r/A
radius from the origin. A diagram was then constructed by patching those
The wave response diagrams for each incident
pixels over the entire area. Notably, the values in the dia-
wave angle are shown in Figures 4 through 15.
grams constitute the amplification factor which is defined as the ratio of the
total wave height to the incident wave height. Therefore, in subregions II
14
89 16 .28 46 68 .88 86 .16 38
= = = © ®@® 8& 8&8 &Y &
Figure 4. Amplification factor diagram for the thin semi-infinite breakwater for incident wave angle = 0 deg
and III (as defined in Figure 2) where the reflected wave is absent, the am- plification factor is essentially the diffraction coefficient as defined in the SPM.
26. Figures 4 through 15 reveal that the amplification factors in sub- region I change very rapidly between 0 and 2.35 over the subregion, and the diagram patterns become very complex because of the interesting superposition of the incident, reflected, and scattered waves. (In the legend of Figures 4 through 15, the width of the pixel is one incident wave length, and the values are amplification factors.) Such patterns would be very difficult to con- struct without using a high-speed computer and computer graphics. In sub- regions II and III, the amplification factors change smoothly from 1.15
roughly along the reflected wave ray reflected from the origin point to nearly
15
hh eet SS BS S
Amplification factor diagram for the thin semi-infinite breakwater for incident wave angle
- 88 20 35 -85 -15 -45 #5 -85 .35
Mh = se ee SF SS &
Amplification factor diagram for the thin semi-infinite breakwater for incident wave angle
-68 25 35 -85 5 UG: -45 7S -8S ooh
Figure 7.
Amplification factor diagram for the thin semi-infinite breakwater for incident wave angle = 45 deg
18
i ee ee
-88 -29 33 -83 -15 -45 #5 -85 -35
woe hw =— se = FSF FS SF
Amplification factor diagram for the thin semi-infinite breakwater for incident wave angle
a ):) 25 35 85 1S 4S 7S BS 35
Mh == =a = FG FS &
Amplification factor diagram for the thin semi-infinite breakwater for incident wave angle
68 8 £35 85 -15 45 #5 -85 “35
ww = = = ©& SF FS ®&
Amplification factor diagram for the thin semi-infinite breakwater for incident wave angle
-68 .29 -39 -85 -15 45 75 -65 35
me = = = fF fF = @&
Amplification factor diagram for the thin semi-infinite breakwater for incident wave angle
66 28 9 85 15 45 orf 65 .35
Ny hes = =| @ O&O SF @&
Amplification factor diagram for the thin semi-infinite breakwater for incident wave angle
68 29 39 -65 15 -45 HS -65 .35
Figure 13. Amplification factor diagram for the thin semi-infinite breakwater for incident wave angle = 135 deg
24
Ne = = = ®SF @ SF @&
Amplification factor diagram for the thin semi-infinite breakwater for incident wave angle
-68 29 33 85 15 45 75 05 35
RN" - - & & * &
Amplification factor diagram for the thin semi-infinite breakwater for incident wave angle
- 68 29 33 -85 -1S 45 5 -85 -35
0.00 at the back wall of the wedge in the shadow zone. The diagram patterns are relatively smooth and simple.
27. Notably, the diagrams do not include phase information of the wave response which is usually unimportant in most engineering practice. Should the phase of the wave response need to be known, one can always use the com— puter program WEDGE to calculate it.
28. The contour diagrams of the amplification factor, similar to the ones presented by Wiegel (1962) and shown in the SPM (1984), were also plotted as typically shown in Figure 16. Examination of those contour diagrams indi- cates that, in subregion III, the results are identical to Wiegel's results. But in subregion II the contour patterns for the amplification factor (or the diffraction coefficient K" used in the SPM (1984)) equal to 1.0; thus the present results are far more complicated than Wiegel's. The author believes that Wiegel's results may lose accuracy because of insufficient resolution of the computational tools during the late fifties and early sixties. Neverthe- less, such inaccuracies are usually either tolerable or immaterial in most engineering practice.
29. The contour patterns in subregion I are very complex, and it is difficult to track specific contours. Therefore, for clarity only the patched
diagrams are presented, and the contour diagrams are omitted in this report. Vertical Wedge of 90-Deg Wedge Angle
30. When the wedge angle is equal to 7/2(6, = 31/2) , the vertical wedge occupies the entire fourth quadrant of the space as shown in Figure 17. Wave response was calculated at 1,100 grid points intersected at r/d = 0.5,(0.5),10.0 and 6 = 0,(1/36),31/2 for the incident wave angle a = 0,(1/12),7 . The wave response at the origin is obtained by substituting
v = 1.5 into Equation 27 as follows:
6(0,0) =< (32)
Those calculated results were used to construct the amplification factor dia- grams by following the same procedures described for the case of the thin semi-infinite breakwater. The diagrams are shown in Figures 18 through 27. Because of symmetry of the results, the diagrams for the incident wave angle
a > 31/4 can be obtained from those for an incident wave angle of tT-a.
2H,
Ne ee er i eee en ee eens —_
Figure 16. Amplification factor contour diagram for the thin
semi-infinite breakwater for incident wave angle = 90 deg Therefore, the diagrams for the incident wave angles omitted in this report.
Sil,
a = 1/6,(1/12),7 are
The diagrams indicate that, for each corresponding incident wave
angle, the results in subregion I are very similar to those obtained for the
vertical wedge of O0-deg wedge angle. However, the results in subregions II
and III from both wedges are discernibly different.
28
, - . , .— ? et t ra y ' HS, t Cr usin: At ies ’ if Fo. Had ’ : Bay “3 Ds t I ? Cs Dee Uae I t Os 5 1 ! ' I Bog Re | iL aeeneny beeen een spen abs === PRCT pBOcH DOSS poeact OSS UCG| > 4 ‘ 1 | H 4 \ ‘ wee ey ee | ; | \ 5 Ys spethoOO Got See! Hts U 1 1 1 ‘ A ioe BO at? ol St tar 1 1 5 4 Zoaeo fy yao me 08 ie 1 \- a ry Ch OG, | \ Nanos \ eens OTS Geer 1 5 act 4 we Yow Pt eS a 4 = ' ‘ 4 Apts oh : et ies yee? + uA ine Oe Tae eet) 4 4 a=) : eee ’ , ‘ iF ay Me ave ot -4 = “TO ee Se at ae Pe a ' ‘ AS ay) eee My OR Secs eae 4 Noonan ‘ \. Ds ’ iM . ral 4 ’ » ‘ ef Ya ' ‘ H 4 ‘ -% 1 Yo ~ ipaeeren oe vee ‘ Popa Qo a oe eee ee Sf * Ae tS we A i ’ ats = H . _ oe ak Soe “ey, O t Desc “D. ‘ a ed x i ae a ' ty ’ r % oe Ese a aS a 4 0 oes & aa ' ee ! ees Wae> Ss Be ee aan ~~ ‘ a ~s i y i anal bs, Baad Cea ' pate Sy ra micscs 4 -<T N aN 1 ~—.) ’ . : we ' (> : e ‘ ' ~ ' : -_ a cer) u le ae Tinian H wet ~ . =e Pe s Saas ' y - Senet aac7 :
-=—--~
_-<--
A 90-deg wedge and polar coordinates 29
Figure 17.
Figure 18. Amplification factor diagram for the
90-deg wedge for incident wave angle
30
0 deg
—_~ = = © ® ® & FS S'
-08 -18 -26 48 -68 -88 -98 -18 -36
Figure 19, Amplification factor diagram 90-deg wedge for incident wave angle =
31
for the 15 deg
Nh ss & SO BS
-88 -25 “EES -85 1 -45 75 -85 -35
Nye = —- SF 2 @& ©
Amplification factor diagram for the 90-deg wedge for incident wave angle
-68 .25 -55 -85 15 -45 75 -85 35
Figure 21. Amplification factor diagram for the 90-deg wedge for incident wave angle = 45 deg
33
6.68 6.25 6.55 8.65 1.15 1.45 1.75 2.65 2.35
Figure 22. Amplification factor diagram for the
90-deg wedge for incident wave angle
34
60 deg
mm = = =—§ ©& & & @
.68 .25 55 65 -15 4S 7S -6S 35
NN = = = © FSF FS &
Amplification factor diagram for the 90-deg wedge for incident wave angle
-68 25 -55 .65 13 45 #5 65 .35
Figure 24. Amplification factor diagram for the 90-deg wedge for incident wave angle = 90 deg
36
8.68 8.25 6.55 6.85 1.15 1.45 1.75 2.05 2.35
Mme = = = & & S&S &
Amplification factor diagram for the 90-deg wedge for incident wave angle
-48 .25 -55 .85 13 -45 75 85 .35
Ne NM =e |= =| OO F&F &
Amplification factor diagram for the 90-deg wedge for incident wave angle
-68 25 -55 65 US 45 45 65 35
Nhe =|=- |= =| © FB F&F &
Amplification factor diagram for the 90-deg wedge for incident wave angle
-68 .25 -55 -65 15 4S 75 85 .35
PART IV: CONCLUSION
32. An analytical solution for the combined wave reflection and dif- fraction by a vertical wedge of arbitrary wedge angle is obtained and ex- pressed in Equation 26. The analytical solution is in terms of Bessel functions and in nonclosed form. The computer subroutine WEDGE, written for calculating the solution, is documented in Appendix A.
33. The amplification factor diagrams for a vertical wedge of O-deg wedge angle and a vertical wedge of 90-deg wedge angle are calculated and pre- sented. The calculated results indicate that the wave response in subre- gion I, where the incident, reflected, and scattered waves all exist, is in a very complex pattern with the amplification factor varying from 2.35 to 0.0 over the subregion. The wave response in subregions II and III is a rela- tively simple pattern with the amplification factor decreasing from 1.15 roughly along the reflected wave ray reflected from the origin point to nearly 0.00 at the back wall of the wedge in the shadow zone.
34. Diagrams of the special case of a vertical wedge of O-deg wedge angle can be considered complementary and extended versions to the ones pre-
sented in the SPM (1984).
40
REFERENCES
Abramowitz, M., and Stegun, I. A. 1964 (Jun). Handbook of Mathematical Func- tions, National Bureau of Standards, Applied Mathematics Series 55, pp 358-495.
Morris, A. H., Jr. 1984 (Jun). "NSWC Library of Mathematics Subroutines," NSWC TR 84-143, Strategic Systems Department, Naval Surface Weapons Center, Dahlgren, Va., pp 43-44.
Shore Protection Manual. 1984. 4th ed., 2 vols, US Army Engineer Waterways Experiment Station, Coastal Engineering Research Center, US Government Printing Office, Washington, DC.
Stoker, J. J. 1957. Water Waves, Interscience Publishers, Inc., New York, pp 109-133.
Wiegel, R. L. 1962. (Jan). "Diffraction of Waves by a Semi-infinite
Breaker,'"' Journal of the Hydraulics Division, American Society of Civil En- gineers, Vol 88, No. HY1, pp 27-44.
41
APPENDIX A: SUBROUTINE WEDGE
1. Subroutine WEDGE is used to calculate the value of 94 in Equa-
tion 26 which is generally a complex number. Its absolute value is the ampli- fication factor, and its phase is the phase indicator from the phase of the incident wave. As mentioned in the main text, 4 is a function of the Bessel function of either fractional or integer order, depending on the wedge angle, and is the aummaeien of a series of infinite terms. The subroutine BESJ, doc- umented in the Naval Surface Weapons Center (NSWC) Library of Mathematics Sub- routines (Morris 1984)* is used in the WEDGE subroutine. The programming of the WEDGE subroutine is very straightforward if a truncation term in the se- ries in Equation 26 is determined. The program is written in FORTRAN language
and is listed in this appendix.
Description
2. The following subroutine is available for computing 9 in
Equation 26: CALL WEDGE(F, FABS, FPHA,XRL, XTH, WEDGEA, WAVEA, IDX)
where the arguments are all real values except F which is a complex value.
Input arguments are as follows:
a. (XRL,XTH)=(r/A,60) where (r,9) are polar coordinates of the lo- cation where $ is to be computed, and A is the incident wave length. Therefore, XRL is the radius vector or radius distance normalized by the incident wave length. XTH is the vectorial angle in degree.
- WEDGEA = wedge angle in degree.
Io
c. WAVEA = incident wave angle in degree. d. IDX = an index (set to 0 in this subroutine).
Output arguments are as follows:
* References cited in the Appendix can be found in the References at the end of the main text.
Al
a. F= @ in Equation 26 (wave response normalized by incident wave amplitude).
b. FABS = amplification factor, the absolute value of 9.
ig ies] Las] Il
phase difference, the phase of $6.
Example and Test Run
3. To serve as an example as well as to ensure the subroutine is the correct one, the user should run the test program listed in Figure Al and make
sure the output is the same as that listed in Table Al.
FROGRAM WEDGE! (INPUT ,OUTPUT, TAPES=INPUT, TAFES=QUTPUT) TESTI CONPLEX F TEST2 PI=3. 141592654 TESTS WRITE (4,4) TEST4
4 FORMAT(//3%,' WEDG-ANG WAV-ANG LOCATION WAVE RESPONSE TESTS { ABS-VAL PHASE’/3X,’ (DEG) (DEB) XRL XTH(DEB) (NORMTESTS PALIZED) AMP-FAC (RAD) ‘/3X,35¢° -*)/) TEST?
WEDGEA=90, TEST WAVEA=135. TESTS XRL=2.0 TESTIO ¥TH=30. TEST11 IDY=0 TESTI2 CALL WEDGE (F,FABS,FPHA, XRL,XTH,WEDGEA,WAVEA, IDX) TESTI WRITE(6,40) WEDGEA,WAVEA,XRL,XTH,F,FARS,FPHA TESTIA 40 FORMAT(1X,8F9. 2) TESTIS STOP TESTI4 END TESTI7
Figure Al. Computer program list 1
Table Al
Sample Output of the Test Program
WEDG-ANG WAV-ANG LOCATION WAVE RESPONSE ABS-VAL FHASE (DEG) (DEG) RRL XTH(DEG) (NGRMALIZED) AMP-FAC iRAD) 90.00 135.00 2.00 40.66 a5 Ail 268 a fe Bo Mil
A2
4. The input arguments in the test programs are as follows: a. WEDGEA = 90; the wedge angle is 90 deg. b. WAVEA = 135; the incident wave angle is 135 deg.
c. XRL = 2.0 and XTH = 30; the location of the wave response $4 to be computed is at radial vector of two incident wave length dis-— tances and vectorial angle of 30 deg in polar coordinates.
d. IDX = 0; the index IDX is set to 0. This case is shown in Figure A2. The outputs are given in Table Al which is self-explanatory.
5. If the location of the wave response $ to be computed is a very large distance, for example XRL greater than 18 or so, the output might print the message that the number of terms is insufficient for summation in comput- ing $ . In this situation, the user must replace the integer 200 in PARAMETER(NN=200), listed in Card WEDGE15 in the SUBROUTINE WEDGE list, by a
larger integer to ensure the accuracy.
Figure A2. Example problem
A3
Subroutine WEDGE Listing
6. Subroutine WEDGE, which is listed in this section, calls subroutine BESJ which, in turn, calls subroutine JAIRY and function GAMLN. Subroutines BESJ and JAIRY and function GAMLN are borrowed from the NSWC Library of Mathe- matics Subroutines (Morris 1984). Including these borrowed subroutines here in the list is only for the purpose of allowing subroutine WEDGE to be self- contained and complete. The computer program of subroutine WEDGE is listed in
Figure A3.
A4
SUBROUTINE WEDGE(F,FABS,FPHA,XRL,XTH,WEDGEA,WAVEA, IDX)
THIS COMPUTER PROGRAM WAS WRITTEN BY H.S. CHEN OF CERC IN 1986 UNDER CORP’ CIVIL WORK R&D PROGRAM. NEITHER ANY OF AGENCIES NOR ANY INDIVIDUAL ASSUMES ANY LEGAL LIABILITY OR RESPONSIBILITY FOR THE ACCURACY OF THE PROGRAM.
Es I 2
WAVE REFLECTION AND DIFFRACTION BY A VERTICAL WEDGE.
INPUT : XRL = R/L, (RADIUS VECTOR) / (WAVE LENGTH) XTH = VECTORIAL ANGLE IN DEGREE WEDGEA = WEDGE ANGLE IN DEGREE WAVEA = INCIDENT WAVE ANGLE IN DEGREE OUTPUT : F = FHI, WAVE RESPONSE NORMALIZED BY INCIDENT WAVE AMPLITUDE FABS = ABSOLUTE VALUE OF PHI, THE AMPLIFICATION FACTOR FPHA = PHASE OF PHI IN RADIAN.
PARAMETER (NN=200)
DIMENSION BJ (NN) ,W(NN) , XM(NN)
COMPLEX F,1M
DATA TOLR/1.E-8/,1TER/8/
PI=3.141592654
CPI=PI/180.
XKR=XRL#2. O#P I
TH=XTH#CPI
WA=WAVEA*CPI
XNU=(360.-WEDGEA) /180.
XM(1)=1.0
IF(IDX.NE.0) GOTO 14
CALL BESJ(XKR,0.0,1,W,NZ)
Bd(1)=W(1)
ICOUNT=0
DO 10 N=1,NN
NL=N+1
ICOUNT=ICOUNT+1
IF(ICOUNT.LE.ITER) GOTO 8
NNN=N
GOTO 14 8 XM(N1)=FLOAT(N) /XNU
M=INT(XM(NL) )
ALPHA=XM(NI) -M
ML=M+1
CALL BESJ(XKR,ALPHA,M1,W,NZ)
IF(N1.EQ.NN) WRITE(4,9) XKR,ALPHA,ML,W(ML) ,NZ 9 FORMAT(/'’ ##** NO. OF TERMS FOR SUMMATION IS INSUFFICIENT *#*%',
1 /’ XKR,ALPHA,M1,W(ML),NZ =',2F10.4,15,E15.6,15/)
BJ (N41) =W(ML)
Figure A3. Computer program list 2 (Sheet 1 of 25)
A5
WEGDEL
Re HE EF € RE KR HE KE KK KR RR KR K€ EK KK HR HE KR KE KE RE & KF KF & E¥NOTICEL]
*¥NOTICE2 *NOTICES *NOTICE4 *¥NOTICES
%¥ # FF HK He EH EK HK KF RH HE KH HF HH HH HK KR HR HK HK HH HE HE KR KF RNOTICES
-WEDGE2 WEDGES WEDGE4 WEDGES WEDGES WEDGE7 WEDGES WEDGE? WEDGE10 WEDGE1L1 WEDGE12 WEDGELS
-WEDGE14 WEDGE1S WEDGE14 WEDGE17 WEDGE18 WEDGE19 WEDGE20 WEDGE21 WEDGE22 WEDGE23 WEDGE24 WEDGE25 WEDGEZ4 WEDGE27 WEDGE28 WEDGE29 WEDGES WEDGES1 WEDGES2 WEDGE3S3 WEDGES4 WEDGESS WEDGE364 WEDGES7 WEDGESS8 WEDGE39 WEDGE40 WEDGE41 WEDGE42 WEDGE43 WEDGE44
oa
AIAIAIYIAIGIDYIYVGYIYVWIMVWYW IIIA AIAAIIIBWIHIOIGN IHS
IF (ABS(BJ(N1)).GT.TOLR) ICOUNT=0 10 CONTINUE 14 CONTINUE F=BJ(1)/2. DO 20 N=1,NNN N1=N+1 XMN=XM (NL) TM= (0.0, 1.0) ##XMN#BJ (NI) #COS (XMN#WA) #COS (XMN*TH) F=F+TM 20 CONTINUE F=4./XNU*F FR=REAL (F) FI=AIMAG(F) FABS=SORT (FR¥FRtFI¥#FI) IF (WA.LE.1.E-8) FABS=FABS/2. IF(FABS.LT.TOLR) GOTO 30 FPHA=ATAN2(FI,FR) RETURN 30 FPHA=0.0 RETURN END SUBROUTINE BESJ(X,ALPHA,N,Y,NZ) WRITTEN BY D.E. AMOS, S.L. DANIEL AND M.K. WESTON, JANUARY, 1975. REFERENCE SAND-75-0147
ABSTRACT BESJ COMPUTES AN N MEMBER SEQUENCE OF J BESSEL FUNCTIONS J/SUB(ALPHAtK-1)/(X), K=1,...,N FOR NON-NEGATIVE ALPHA AND Xx. A COMBINATION OF THE POWER SERIES, THE ASYMPTOTIC EXPANSION FOR X TO INFINITY AND THE UNIFORM ASYMPTOTIC EXPANSION FOR NU TG INFINITY ARE APPLIED OVER SUBDIVISIONS OF THE (NU,X) PLANE. FOR VALUES OF (NU,X) NOT COVERED BY ONE GF THESE FORMULAE, THE GRDER IS INCREMENTED OR DECREMENTED BY INTEGER VALUES INTO A REGION WHERE ONE OF THE FORMULAE AFPLY. BACKWARD RECURSION IS APPLIED TO REDUCE ORDERS BY INTEGER VALUES EXCEPT WHERE THE ENTIRE SEQUENCE LIES IN THE OSCILLATORY REGION. IN THIS CASE FORWARD RECURSION IS STABLE AND VALUES FROM THE ASYMPTOTIC EXPANSION FOR X TO INFINITY START THE RECURSION WHEN IT IS EFFICIENT TO DO SO. LEADING TERMS OF THE SERIES AND UNIFORM EXPANSION ARE TESTED FOR UNDERFLOW. IF A SEQUENCE IS REQUESTED AND THE LAST MEMBER WOULD UNDERFLOW, THE RESULT IS SET TO ZERO AND THE NEXT LOWER ORDER TRIED, ETC., UNTIL A
MEMBER COMES ON SCALE OR ALL MEMRERS ARE SET TO ZERO. OVERFLOW CANNOT OCCUR. BESJ1 CALLS SUBROUTINE JAIRY AND FUNCTION GAHMLN. DESCRIPTION OF ARGUMENTS INPUT X | Hollen @
ALPHA - ORDER OF FIRST MEMBER OF THE SEQUENCE, ALPHA.GE.90
N - NUMBER OF MEMBERS IN THE SEQUENCE, N.GE.1 OUTPUT Y - A VECTOR WHOSE FIRST N COMPONENTS CONTAIN
VALUES FOR J/SUB(ALPHAtK-1)/(X), K=1,...,N
Figure A3. (Sheet 2 of 25)
A6
WEDGE45 WEDGE46 WEDGE47 WEDGE48 WEDGE49 WEDGES WEDGES1L WEDGES2 WEDGESS WEDGES4 WEDGESS WEDGES6 WEDGES7 WEDGES8 WEDGES9 WEDGE6O WEDGES1 WEDGEG2 WEDGES3S WEDGE64 WEDGESS BESJ101 BESJ102 BESJ103 BESJ104 BESJ105 BESJ106 BESJ107 BESJ108 BESJ109 BESJ110 BESJi11 BESJ112 BESJ113 BESJ114 BESJ1I15 BESJ116 BESJ117 BESJ118 BESJ119 BESJ120 BESJ1i21 BESJ1i22 BESJ1253 BESJ124 BESJ125 BESJ126 BESJ127 BESJ128 BESJ129 BESJ130 BESJi31 BESJ132 BESJ133 BESJ134
OHOonrw i ~Iianrnraianain
ise)
ia]
On
1
1
5 “
1
NZ - ERROR NZ=0 NZ=-1 NZ=-2 NZ=-3 NZ.GT.
ERROR CONDITIONS
IMPROPER INPUT AR UNDERFLOW
DOUBLE PRECISION DX,T DIMENSION Y(N)
DIMENSION Ciii,10) DIMENSION C1(88) DIMENSION A1(952), DIMENSION GAMA(26) ,TE DIMENSION FNULIM(2) ,P DIMENSION CR(10) ,DR(1
EQUIVALENCE EQUIVALENCE EQUIVALENCE EQUIVALENCE EQUIVALENCE EQUIVALENCE EQUIVALENCE
ROG) sc (11,9) ,€ (ALFA(L,1 (BETA(1,1 (BETA(1,5 DATA ELIM1,ELIM2, TOL
DATA PP(1)/8.72909153 PP(3)/1.24578576
TOLS=LN(1.E-3) DATA TOLS
DATA UPOL(1) ,CON1,CON 6, 6666666666667E- ol, 1.0416656566667E-01/
DATA RTWO,PDF,RTTP,PI 7. 8539816339745E-01,
DATA FNULIMN(1)/100./,
- A NON-FATAL ERROR
vALFA(26,4) “(C2 (22 A2(52)
(ALFA(1,3),
(BETA(1,3)
INDICATOR NORMAL RETURN - COMPUTATION COMPLETED X 15 LESS THAN 0.0 ALPHA IS LESS THAN 0.0 N 1S LESS THAN 1
0 LAST NZ COMPONENTS OF Y SET T0 0.9 BECAUSE OF UNDERFLOW
GUMENTS - A FATAL ERROR (NZ.6T.0)
RX ,DTM, DEN BETA (26,5)
BL (52) ,B2(52) ,B3(26)
MP (3) ,KMAX(5) ,AR(8) ,BR(10) P(4)
0)
wUPOL (10)
1(1)) 2(1)) ),A1(1)) PAN) BHA) »B21)) ),B3(1))
i Obds ; 44, f 1.E-15 PP(2)/2.6569393226503E-01/,
Q35SSE+00/, E PP(4)/7.70133574743039E-04/
3555 B6559E-O1/,
/-6.9077552799821E+00/
2,CONS ,CONS48 / L OOTOOOODODODOE SO: SB, 355333333333 3E-01, 1.4142155623731E+00, DT / 1.3483997249265E+00,
7.9788456080286E-01, 1,.5707963267949E+00/
FNULIM(2)/60./
CE=-ALOG(TOL) , TCE=-0.75*ALOG(TOL) DAA CE nee / J.45387 763749 LE+O1, 2.5904082296185E+01/ DATA INLIN / 150 /
DATA AR(1)/8.35503472
Figure A3.
AR(2)/1
282265745563 5E-
(Sheet 3 of 25)
22222E-02/,
A7
~.
BESJ135 BESJ136 BESJ137 BESJ138 BESJ1359 BESJ140 BESJ141 BESJ142 BESJ143 BESJ144 BESJ145 BESJ146 BESJ147 BESJ148 BESJ149 BESJ150 BESJ151 BESJ1S2 BESJ153 BESJ154 BESJ1595 BESJ1564 BESJ157 BESJ158 BESJ159 BESJ160 BESJ161 BESJ162 BESJ165 BESJ164 BESJ1465 BESJ146
(BESJ167
BESJ168 BESJ169 BESJ170 BESJ1i71 BESJ172 BESJi73 BESJ174 BESJ1735 BESJ176 BESJ177 BESJ178 BESJ179 BESJ180 BESJ181 BESJ192 BESJ183 BESJ184 BESJ185 BESJ186 BESJ187 RBESJ188 BESJ199A
ted Poe
Gl hoe
oOo oon bein OOO OF Sted be
con Pid he OOO OS ed he
oN Oo} Om et hoe
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AR(3)/3. AR (7) /
BRL) BR(3) BR(S) BR(7) BR(9)
Citi) Ci(3) G7)
7 /
/
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Ci(st)/
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C1(S1)/
C1(55)/ 0 CCRT) P= 7/0 C1(59)/-6. EWG) (2s
C1(63)/
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9194576623194E+03/, C1(68)/ 8.0617221817373E+03/, 3586550006434E+04/, C1(70)/ 1.165539353546864E+04/, 30564697B6134E+03/, C1(72)/ 1.2009029132164E+03/, 0809091978B840E+02/, C1(74)/ 1.7277275025845E+00/, Oly ELETO)E O.005 CLLPTIE D.O%,
»O204291330966E+04/, C1(79)/-9. 6980598 S88638E+04/, Ci(8o0)/ 1.
254700123253E+05/, C1(81)/-2.0340017728042E+05/,
»2220046498S502E+05/, C1 (835) /-4.1192654968898E+04/, C1(84)/ 7. C1(86)/ 4.
1095143024894E+03/, C€1(85)/-4.9391530477309E+02/, 0740420012735E+00/, C1(87)/ 0.0/, C1(88)/ 0.0/
1. S11763561466350E+06/, 3.7632712976564E+06/ , 1,.2683652733216E+04/, 4, 2
SS
o218768981263E+047, 43805296995 56E+01/,
970GB1911B4S2E+07/, C214) / 5.09526024926465E+07/, ALOS1L4B21153535E+07/, C2(16)/ 6.6344512274729E+07/, 7567176660763E+07/, C2(18)/ 1.35288767166422E+07/,
Figure A3. (Sheet 4 of 25)
A8
BESJ189B BESJ189C€ BESJ189D BESJ190
ESEJ1914A BSEJ191B BSEJI91C BSEJ191D BSEJI91E BESJ192
BESJ193A BESJ193B BESJ193C BESJ193D BESJ193E BESJ195F BESJ1936 BESJ193H BESJ1931 BESJ193J BESJ193K BESJ193L BESJ193M BESJ193N BESJ1930 BESJ1935P BESJ9310 BESJ1935R BESJ193S BESJ194
BESJ195A BESJ195B BESJ195C BESJ195D BESJ195E BESJ199F BESJ1956 BESJ195H BESJ1951 BESJ195d BESJ 195K BESJ19SL BESJ1935M BESJ195N BESJ1950 BESJ194
BESJ197A BESJ197B BESJ197C BESJ197D BESJL97E BESJ197F BESJ1976 BESJ197H BESJ1971
DATA
oOoOnyA oon Pol ro K— OOO OW SP he
DATA
oon eS Ww Pe
DATA
Oommen Str OO CON OO eed Pe
DATA
cd Poe
AOU) a Al (3) / ALCS) / AL(7) / ALC?) # AL(11)/
AL(L7)/ 1
A1l(37
AlL(39)/-7. A1(41)/-5. AL(43)/-4., AL1(435)/-3. AL(47)/-2.
A1(49)/-2
AL(S1)/-1. AZ(L) /- A2(3) / A2(5) / A2(7) / A2(9) 7 A2(11)/ 7
A2(19)/ 3
A2(2 A2(25
A2(3 A2(37
AZ(39)/-1. A2(41)/ 2. A2(43)/ 4. A2(45)/ 5.
» 7856181 280864E+06/, »SBB6089753717E+04/,
-4444444444444E-05/, .8489288489288E-05/, 466913572741 79E-04/, .6182429706150E-
04/7, 32721040083235E-64/,
.0073885876275E- 047, AL(13)/ 1. ANUS) We
73060773591 788E- s027477415091E-
o4/, 04/,
»5166881654592E-04/, AL(19)/ i. AL(21)/ ft. AL(23)/ 9.3 AL(25)/ 8. AL(27)/ 6. AL(29)/-1. AL(S1)/-1. AL(33)/-1. ALCS) 7-1. VID
1640527147474E-047, 0377241042299E-64/, 212051724950E-035/, 4296310571570E-05/, 93735541355 459E-047, 4198627255669E-05/, SOBNSSSGOS305E-047, 4680975564647E-04/, 1974497568425E-047, S7699S54989119E-05/7,
2937434815522E-05/7, 6961156600937E-035/, 4813486800988E-05/, 5540053297204E-05/, BS99679S90418E-05/,
»2845929716472E-05/,
8481621752767E-05/,
S4A21197145774E-04/,
O446550359494E-05/, »6747110669971E-04/, »J650142760860E-04/, »1488669202982E-04/, » 644984192 A2Z(13)/ 4, A2(15)/ 2. AZ(17)/ 1.
5090E-05/, 7439429929051E-05/, 6993971497922E-05/, 3005667479396E-05/,
~J925748581935E-06/, AZ(21)/-2, a) / 2c Wier ke A2(27)/ 3. A2(29)/-6. A2(31)/-3. A2(33)/-2. cD) Hie iee )/-6.2
6539676969794E-06/, 7273929609125E-06/, S130471509356E-067, 7819419920177E-04/7, 3793850631886E-05/7, 1091625602736E-04/, 78950273791 352E-04/, 7TSE2Z4528034085E-047, 298287282021E-035/7,
72335430236696E-05/, S139544314829E-05/, 53800612449019E-05/, 6839620854582E-05/7,
Figure A3.
C2 / C /
0) 2) AL(2)
Al(6) / AL(8) /
AL(12)/ 1
AL(20)7
AL(40)/-6.
A1(42)/
AL(48) /-
AL(52)/-
(N30) YS
A2(6) / A2Z(8) /
A2Z(10)/ 9. A2(12)/ 4. A2(14)/ 3S. A2(16)/ 1. A2(18)/ 7. A2(20)/ 1. aye A2(24)/-8.1 A2(25)/-1. A2(28)/ 2. A2(30)/-2. A2(32)/-3. A2(34)/-2. A2(36)/-1.25 A2(38)/-4.
A2(22
A2(40)/ SS. A2(42)/ 3. 5. 245952949591 1E-
A2(44)/
A2(46)/ 3.
920 222)
2.6599558954626E- 2.4873 AL(10)/ 2.1 ~8626763663754E-04/, AL(14)/ 1. A1(16)/ 1. EEO Ihe 1.0979829837271E-04/7, CAN A1L(24)/ 8. Al(26)/ 8. AL(28)/ 2. AL(30)/-1. OU CS2) Joos AL(34)/-1. AL(36)/-1. A1(38)/-8.
9 (YA) ho
-0818640461266E+05/, »1001714062
6925E+02/
07792 6592768783
207792E-04/, 245E-04/7, 04/, O43734466E-04/, 6342485712346E-04/,
4109170592902E-04/, 4O50349739127E-047, 2366744559825E-04/,
82526078369356E-035/, 8571085247871E-05/, 0249754B40779E-05/, 2224174518292E-04/, 1644493147205E-04/, 12192491B810E-04/, 3381550396749E-04/, 0618431920797E-04/, 2692304558819E-05/
4404
255772102E-05/,
-5.0473104430356E-05/, A1(44)/-3. Ai(46)/-3. 2.0495 A1(50)/-2. 1.6651933002139E-05/
9868872771760E-05/, 1741425660902E-05/, 2272063487E-05/, 0535275310648E-05/,
eo ice eee sees Uae (HME Ma 1 1
Seen ee -04/, 0222587683 a 04/,
Gisroionae vas,
4586909303469E-05/, O0757033496520E-05/, 6275751200554E-05/, 9321093824794E-05/, 8262086574450E-06/, 4404004981425E-07/7, 9134686709849E-06/, 1726937967866E-06/, 0201141879802E- 05/, 0247195276182E-04/, SBS59B230603501E-04/, 1368011524758E-04/, 2856408261914E-04/, S4406306069E-04/7, 6286073058812E-05/
606904B239460E 62642745985679E
-06/, -05/, 05/, 9A349B2039310E-05/,
(Sheet 5 of 25)
A9
BESJ197J BESJ197K BESJ198
BESJ199A RESJ199B BESJ199C BESJ199D BESJ199E BESJ199F BESJ1996 BESJ199H BESJ1991 BESJ199d BESJ199K. BESJ199L BESJ199M BESJ199N BESJ1990 BESJ199P BESJ199@ BESJ199R BESJ199S BESJ200
BESJ201A BESJ201B BESJ201C BESJ201D BESJ201E BESJ201F BESJ2016 BESJ202
BESJ203A BESJ203B BESJ203C BESJ203D BESJ203E BESJ2035F BESJ 2036 BESJ203H BESJ2051 BESJ203) BESJ 203K BESJ203L BESJ 203M BESJ205N BESJ2030 BESJ203P BESJ2030 BESJ203R BESJ2035 BESJ204
BESJ205A BESJ205B BESJ205C BESJ205D
con eS
Om NO Of eS ed ro OO NO OP ol he
ce on fs Gy ho oe
te} PO ~O CO NS & Of & Oy PO
oonoowS
boo
DATA
DATA
DATA
AZ(47)/ &. A2Z(49)/ &. A2(S1)/ 3.7
Bl (1) Bi(3) BL (3) Bi(7) Bi (9) Bi(ii)/ Bi (13)/
SSS Ss SS PI td fe SI or bo
Bi(2
si (27) #1 B1L(29)/
0647852757842E-035/, O157789453946E-035/, FABLES HI VEVESORY 5
Ger
»7998872141355E-02/, ~BBS014022351135E-03/, 22475 »1443042172729E-04/, »694310076064BE- ~S481888931850E-04/7, sd2211615 Jeol (OS) fhe BNA YY the BL CHS) / the BUEG2119) ele SY We El Cza) 7 sn
311058920E-03/, 04/,
S4957E-04/, 9754183803305E-04/, 5931689966182E-04/, 3144806811996E-04/, 1044914450460E-04/, 4199822420424E-05/, 1346626214280E-05/,
4928295321 543E-037, -5.0291654957204E- Ba Saale
04/, 7946399697078E-047,
B1(33)/-3.96141953504646E-035/, B1(35)/-1.2608973578025E-057, B1(37)/ 8.0999616541427E-06/, B1L(39)/ 1.7396412547293E-035/, BiC4i)/ 2.1446526379082E-05/, B1(43)/ 2.2896778381471E-05/, B1(45)/ 2.5032197608091E-035/, BL{47)/ 2.2500588110529E-05/, B1(49)/ 2,.1641842744810E-05/, BL(S1)/ 2.0638874978217E-035/, B2(1) / 3.9221507672129E-04/, B2(3) / 2.7952063399Z202E-04/7, 245) / 6.9327110565704E-035/, B2(7) /-1.55744996354327E-035/, B2(9) /-4.18861861459669E-035/, B2(11)/-4.8766544741379E-03/, B2(13)/-4.7473562089009E-035/ B2(15)/-4.3353509649451127E- 05/, BGS 5,9482263860322E-05/, eee 3.3779330612337E-05/, 2(21)/-2.9526956173081E-057, ie 3) /-2.5800617466688E-035/, B2(25)/-2. 2582355095 1835E-05/, B2(27)/-4.74617796559976E-04/, B2(29)/-3.2039022806704E-04/, B2(21)/-4.2577810128544E-057, B2(33)/ 7.9709268407968E-05/7, B2(35)/ 1.1246677525220E-047 , B2(37)/ 1.0865163484877E-04/, B2(39)/ 9.2929839659355E-05/, WE Pea eA SNS Wai ¢ B2(43)/ 35.92596454732320E-05/,
Figure A3.
AZ (48) / A2(50)/ A2(352)/
BL(2) / B1(4) 7 Bi(é) / Bi(8) / Bicto)/ Bi(t2)/ B1(14)/ Bi(16)/ B1(18)/ B1(20)7/ Bi(22)/ B1(24)/ Bi¢26)/ B1(28)/ B1(30)/ I C2) B1i(34)/ B1(36)/ B1(38)/
B1(40)/ Bi(42)/ B1i(44)/ B1 (46) / B1 (48) / Bi(50)/ 61(52)/
BAVA) ff B2(4) / 2Co) F H2(8) 7 2(10)/ 2Ch2) 7 B2(14)/ B2(14)/ B2(18)/ Be (20) 7 B2(22)/ B2 (24) 7 B2(26) / AL CAE})) 7 B2(30) / A CRA) / B2(34)/ H2(36)/ B2¢38) /
B2(40)/ B2(42)/ 2(44)/
A10
— FI td ee FD OO
cn con c-
SOF eee pre ono on
PO rl br ho bro Po or
oo — = a 2
Rr id ore BPP Po td te
r=
cn c- co
» 08023 »89199657354470E-0357,
30785 »2823607372035E-05/7, »20981015356199E- © 1150754925622 -0116924199708E
90778844E-035/,
5280437558585E-05/
-T99S491L1L064359E-03/, -9009660676105E-
O3/,
2287887657294E-04/,
.717872817897GE-047, »93235283546292E-04/, 8895 »2228058079888E- -76836855901972E-04/, »A4S479OSOLITISE- » 20245444949350E-04/, »0182877074057E ~7413054575385E »9900226964622E-05/, «782047095 46S9E-04/, 9482 -9400855046082E-04/, »12035892907610E-05/, ©42892608575735E-07/, »53690700926215E-
214849575E-04/7,
o4a/, O4/,
-04/ 5 -05/,
2135851275E-04/,
03/
»9867297884215E-05/,
23994659923246E-05/, S8981118E-05/,
O5/, Wai! 4 -05/
793258155238E-04/, ee cert EN 04/7, 25868306999E-05/,
TAO ISES WS 05/7, 4, -8701003118674E-05/, »cSOBLSOS8LS8463E »0923501931577
6900488937914E-05/, -05/, 5E-05/, S0B5716753541E-05/,
~1588856077211E-035/, ~7oF7 891 482834E-05/, - 4130983 »1147965676891E »7786456714732E-04/, »6110501611996E »F457129429497E-05/7, »OS138236 »1310364210848E -O143795159766E-04/
5676128E-05/, =05/,
-04/,
OB27E-04/, -04/,
AO2Z9S1LSS0L609E-035/, SVE SAAZ NG /a Se — o5/, 2216930882699E-05/,
(Sheet 6 of 25)
BESJ205E BESJ205F BESJ2056 BESJ206
BESJ207A BESJ2078 BESJ2070 BESJ207D BESJ207E BESJ207F BESJ 2076 BESJ207H BESJ2071 BESJ207d BESJ207K BESJ207L BESJ207M BESJ207N BESJ2070 BESJ207P BESJ 2070 BESJ207R BESJ 2075 BESJ208
BESJ 209A BESJ209B BESJ209C BESJ209D BESJ209E BESJ209F BESJ 2096 BESJ210
BESJ211A BESJ211B BESJ2Z11C BESJ211D BESJ211E BESJ211F BESJ2116 BESJ211H BESJ2111 BESJ211J BESJ 211k BESJ211L BESU211M BESJ211N BESJ2110 BESJ211P BESJ2110 BESJ211R BESJ2115 BESJ2t2
BESUZ13A BESJ213B BESJ213C
moon
M4 poe ~O CON O&O OWS OP re ao cn & C4
4 Poe
Beir O ON oO On SS
B2(45)/ 4. 5853948516536E-05/, 82(46)/ 4.0144551389149E-05/, B2(47)/ 3, 5048173003133E-05/, B2(48)/ 3.0515799503435E-05/, B2(49)/ 2.64935611995052E-05/, B2(50)/ 2.2936363369100E-05/, B2(51)/ 1.9789305666402E-05/, B2(52)/ 1.7009198463541E-05/ DATA BS(1) / 7.3646581057258E-04/, B3(2) / 8.72790805144619E-04/, B3(3) / 6.22614862573514E-04/, B3(4) / 2.8599815419430E-04/, B3(S) / 3.8473757287937E-06/, B3(6) /-1.8790600363597E-04/, B3(7) /-2.9760364659456E-047, B3(8) /-3.4599812683267E-04/, BS3(9) /-3.53558247091604E-04/, B3(10)/-3.3571563577505E-04/, B3(11)/-3.04352112478904E-04/, B3(12)/-2,.6672272304761E-04/, BS(13)/-2.2755421412282E-04/, B3(14)/-1.8992261195456E-04/, B3(15)/-1.5505891859909E-04/, B3(16)/-1.2377824076187E-04/, B3(17)/-9.6292614771764E-05/, BS(18)/-7.2517832771442E-05/, B3(19)/-5. 2207002889563E-05/, #3(20)/-3.5034775051190E-G5/, B3(21)/-2.0848976103555E-05/, B3(22)/-8.7010609684977E-04/, B3(23)/ 1.1569868667510E-G6/, B3(24)/ 9.1642647412278E-06/, B3(25)/ 1.5647778542887E-05/, B3(26)/ 2.0822362948247E-05/ DATA GAMA(1L) /6.29960592494744E-01/, GAMA(2) /2.5198420997898E-01/, GAMA(S) /1.54790300415466E-01/, GAMA(4) /1.10713062414616E-01/, GAMA(S) /8.5730939552740E-02/, GAMA(6) /6.9716131695868E-02/, GAMA(7) /5.8608567199371E-02/, GAMA(8) /5.04698873535631E-02/, GAMAI9) /4.42600580689146E-02/, GAMA(10)/3.9372066154351E-02/, GANA(11)/3.5428319592446E-02/, GAMA(12)/3.2181885750210E-02/, GAMA(13)/2.9464624079116E-02/, GAMA (14) /2.7158167711293E-02/, GAMA(15)/2.5176827297386E-02/, GAMA (16) /2.3457075530608E-02/, GAMA(17)/2.1950839013491E-02/ GAMA (18) /2.0621082823565E-02/, GAMA(19)/1.94388240897988E-02/, GAMA(20)/1.838106S380068E-02/, GAMA(21)/1.7429321323196E- 02/, GAMA (22) /1.65685835778661E-02/, GAMA(23) /1.5796528598792E-02/, GAMNA(24) /1.5972950149410E- -02/, GAMA(25) /1.4419325083994E-02/,
GAMA(26)/1.
S8184805973534E-02/
TEST INPUT ARGUMENTS
NZ=
KT=1
IF(N-1) 92
Kale?
NN=N
IF (X)
IF (ALPHA)
POE
IF (N.EQ. 1)
Ie?
60 TO 118
Til
DOMit9) l=11).N (IN 0F
RETURN
CONTINUE
IF (ALPHA.LT.O.)
108,
109
93,110,120 91,114,116
RETURN
60 T0 91
Figure A3.
All
(Sheet 7 of 25)
BESJ213D BESJ213E BESJ213F BESJ2136 BESJ214 BESJ215A BESJ215B BESJ215C BESJ215D BESJ215E BESJ219F BESJ2156 BESJ215H BESJ2151 BESJ213d BESJ 215K BESJ215L BESJ215M BESJ2164 BESJ217A BESJ217B BESJ2170C BESJ217D BESJ217E BESJ217F BESJ2176 BESJ217H BESJ2171 BESJ217J BESJ 217K BESJ217L BESJ217M BESJ217N BESJ218 BESJ219 BESU220 BESJ221 BESJ222 BES ae BESJ2 Rie BESJ22 BESJ22 BESJ22 BESJ229 BESJ23 BESJ231 BESJ232 seer ee BESJ23 BESJ237 BESJ23 BESJ239
[os al oe a oe a oe
oO
(52) f3) (cal (se) fs)
850
840
880
DFN=DBLE (FLOAT (N) )+DBLE (ALPHA) -1.D+0 FNU=DFN
XO2=X*.5
SX02=X02*X02
DECISION TREE FOR REGION WHERE SERIES, ASYMPTOTIC EXPANSION FOR X TO INFINITY AND ASYMPTOTIC EXPANSION FOR NU TO INFINITY ARE
APPLIED.
IF(SXO02.LE. (FNUt+1.)) GO TO 850 TA=AMAX1 (20. ,FNU) IF(X.GT.TA) GO TO 880 IF(X.GT.12.) GO TO 840 XO2L=ALOG (X02) NS=SX02-FNU
60 TO 852
FN=FNU
FNPI=FN+1.
XO2L=AL0G (X02)
IS=KT
IF(X.LE.90.5) GO TO 134 NS=0 DFN=DFN+DBLE (FLOAT (NS) ) FN=DFN
FNPL=FN+1.
IS=KT
IF(N-1+NS.G6T.0) I5=3
60 TO 134 NS=AMAX1(36.-FNU,O.) DFN=DFN+DBLE (FLOAT (NS) ) FN=DFN
IS=KT
IF(N-1+NS.G6T.0) I5=3
60 TO 130
CONTINUE
RTX=SQRT(X) TAU=RTWO#RTX TA=TAUtFNULIM (KT) IF(FNU.LE.TA) GO TO 500 FN=FNU
IS=KT
UNIFORM ASYMPTOTIC EXPANSION FOR NU TO INFINITY
CONTINUE
XX=X/FN
W2=1.-XX#XX
ABW2=ABS (W2)
RA=SQRT (ABW2) IF(ABW2.67.9.2775) GO TO 200
CASES NEAR X=FN, ABS(1.-(X/FN)¥*#2).LE.0.2775 COEFFICIENTS OF ASYMPTOTIC EXPANSION BY SERIES
Figure A3. (Sheet 8 of 25)
Al2
PESJ240 BESJ241 BESJ242 BESJ243 BESJ244 BESJ245 BESJ246 BESJ247 BESJ248 BESJ249 BESJ290 BESJ251 BESJ252
BESJ258 BESJ209 BESJ260 BESJ261 BESJ262 BESJ263 BESJ264 BESJ265 BESJ256 BESJ267 BESJ268 BESJ269 BESJ270 BESJ271 BESJ272 BESJ2753 BESJ274 HESJ275 BESJ276 BESJ277 BESJ278 BESJ279 BESJ280 BESJ2981 BESJ282 BESJ283 BESJ284 BESJ285 BESJ286 BESJ287 BESJ288 BESJ289 BESJ270 BESJ291 BESJ292 BESJ293 BESJ294
Iona
ba bh
ioe) noonn
ra
ZETA AND TRUNCATION FOR A(ZETA) AND B(ZETA) SERIES BESJ295
BESJG296 KMAX IS TRUNCATION INDEX FOR A(ZETA) AND B(ZETA) SERIES=NAX(2,SA) BESJ297 BESJ298 SA=0. BESJ299 IF (ABW2.EQ@.G.) GO TO 21 BESJ300 SA=TOLS/ALOG (ABW2) BESJS301 SB=SA BESJ302 DO 22 1=1,5 BESJ303 KMAX (IT) =AMAX1(SA,2.) BESJ304 SA=SA+SB BESJS05 CONTINUE BESJ306 KB=KMAX (9) BESJ307 KLAST=KB-1 BESJ308 SA=GAMA (KB) BESJ309 DO 24 K=1,KLAST BESJ310 KB=KB-i BESJS11 SA=SA#W2+GAMNA (KB) BESJ312 CONTINUE BESJ313 Z=W2*SA BESJ314 AZ=ABS(Z) BESJS15 RTZ=SQRT (AZ) BESJ3516 FNIZ=FN¥*CON2 BESJ317 RTARY=RTZ#FN13 BESJ318 ARY=-RTARY*#RTARY BESJ319 AZS2=AZ#RTZ*CON1 BESJS2 ACZ=FN*¥AZ32 BESJ321 MEN oLE55)) (0) Wo) 227/ BESJ522 BESJ323 TEST FOR UNDERFLOW, 1.E-2B0=EXP(-444.), ONE WORD LENGTH BESJ324 UP FROM UNDERFLOW LIMIT OF CDC 46600 BESJ325 BESJ326 IF(ACZ.GT.ELIM2) GO TO 180 BESJ327 ARY=-ARY BESJS2 PHI=SQRT(SQRT (SA+SA+SA+SA) ) BESJ32 BESJ330 B(ZETA) FOR S=0 BESJ331 BESJ332 KB=KMAX (3) BESJ333 KLAST=KB-1 BESJ334 SB=BETA(KB, 1) BESJS39 DO 23 K=1,KLAST BESJ334 KB=KB-1 BESJ337 SB=SB#W2+KETA(KB, 1) BESJ358 CONTINUE BESJ339 KSP1=1 BESJ340 FN2=FN#FN BESJ341 RFEN2=1./FN2 BESJ342 RDEN=1. BESJ343 ASUM=1. BESJ 344 RELB=TOL*ABS (SB) BESJ345 BSUM=SB BESJ346 DO 25 KS=1,4 BESJ347 KSPI=KSP1+1 BESJ348 RDEN=RDEN#RFN2 BESJ349
Figure A3. (Sheet 9 of 25)
Al3
oa
ioe)
ann
onan
A(ZETA) AND B(ZETA) FOR S=1,2,35,4
KB=KMAX (5-KS) KLAST=KB-1 SA=ALFA(KB,KS) SB=BETA(KB,KSP1)
DO 26 K=1,KLAST KB=KB-1 SA=SA#W2+ALFA(KB,KS) SB=SB#W2+BETA(KB,KSP1) CONTINUE
TA=SA#RDEN
TB=SB*#RDEN ASUM=ASUM+TA BSUM=BSUM+TB IF(ABS(TA).LE.TOL.AND.ABS(TB).LE.RELB) GO TO 152 CONTINUE
CONTINUE
BSUM=BSUM/ (FN*#FN1S)
60 TO 400
CONTINUE TAU=1./RA T2=1./W2 IF(W2.GE.0.) 60 TO 30
CASES FOR (X/FN).GT.SQRT(1.27735)
AZ32=ABS (RA-ATAN(RA) ) ACZ=AZ32#FN
CZ=-ACZ
Z732=1,.5*AZ32 RTZ=Z52**CON2 FNIS=FN#*CON2 RTARY=RTZ#FN1S ARY=-RTARY#RTARY
60 T0 150
CONTINUE
CASES FOR (X/FN).LT.SQRT (0.7225) AZS2=ABS(ALOG((1.+RA)/XX) -RA)
TEST FOR UNDERFLOW, 1.E-280 = EXP(-644.), ONE WORD LENGTH UP FROM UNDERFLOW LIMIT OF CDC 4600
ACZ=AZ32*FN
CZ=ACZ
IF(ACZ.GT.ELIMN2) GO TO i980 Z32=1.5#AZ32
RTZ=Z232**CON2 FNIS=FN**CON2 RTARY=RTZ#FN15 ARY=RTARY*RTARY
Figure A3. (Sheet 10 of 25)
Al4
BESJ3550 BESJ351 BESJ352 BESJ353 BESJ354 BESJ355 BESJ356 BESJ397 BESJ398 BESJ359 BESJ340 BESJ341 BESJ342 BESJ563 BESJ364 BESJ365 BESUS466 BESJ367 BESJ548 BESJ349 BESJS70 BESJ371 BESJ372 BESJ373 BESJ3574 BESJ375 BESJ376 BESJ377 BESJ378 BESJ379 BESJ380 BESJ381 BESJ382 BESJ383 BESJ3584 BESJ385 BESJ386 BESJ387 BESJ388 BESJS89 BESJS90 BESJ391 BESJ392 BESJ393 BESJ394 BESJ399 BESJ396 BESJ397 BESJS98 BESJ399 BESJ460 BESJ401 BESJ402 BESJ403 BESJ404
oO
150
101
155 145
400
402
CONTINUE PHI=SQRT((RTZ+RTZ) *TAU) TB=1.
ASUM=1.
TFN=TAU/FN UPOL(2)=(C(1,1)*T24+C(2,1)) #TFN RCZ=CONI/C2Z CRZ32=CONS48*RCZ BSUM=UPOL (2) +CRZ32 RELB=TOL¥ABS (BSUM) AP=TFN
KS=0
KP1=2
RZDEN=RCZ
DO 155 LR=2,8,2
COMPUTE TWO U POLYNOMIALS FOR NEXT A(ZETA) AND B(ZETA)
LRPI=LR+1 DO 101 K=LR,LRP1
KS=KS+1
KPL=KP1+1
S1=C(1,K)
DO 102 J=2,KP1 S1=S1*T2+C(J,K)
CONTINUE
AP=AP*TFN
UPOL(KP1)=AP*S1
CR(KS) =BR(KS) #RZDEN RZDEN=RZDEN*RCZ DR(KS)=AR(KS) #RZDEN CONTINUE
SUMA=UPOL (LRP1) SUMB=UPOL (LR+2) +UPOL (LRP1) #CRZ32 JU=LRP1
DO 151 JR=1,LR
JU=JU-1 SUMA=SUMA+CR (JR) #UPOL (JU) SUMB=SUNB+DR(JR) #UPOL (JU) CONTINUE
TB=-TB
IF(W2.GT.0.) TB=ABS(TB) ASUM=ASUM+SUMA*TB BSUM=BSUM+SUMB#TB
IF (ABS(SUMA).LE. TOL. AND. ABS(SUMNB).LE.RELB) GO TO 165 CONTINUE
TB=RTARY
IF(W2.6T.0.) TB=-TB BSUM=BSUN/TH
CONTINUE CALL JAIRY(ARY,RTARY,ACZ,AI,DAI) TEMP (1S)=PHI* (AI *ASUM+DAI*BSUM) /FNIS GO TO (401,202,650), IS
TEMP (1) =TEMP (3)
Figure A3. (Sheet 11 of 25)
Al5
BESJ405 BESJ406 BESJ407 BESJ408 BESJ409 BESJ410 BESJ411 BESJ412 BESJ413 BESJ414 BESJ4195 BESJ416 BESJ417 BESJ418 BESJ419 BESJ 420 BESJ421 BESJ422 BESJ423 BESJ424 BESJ425 BESJ426 BESJ427 BESJ428 BESJ429 BESJ430 BESJ431 BESJ432 BESJ433 BESJ434 BESJ4355 BESJ434 BESJ437 BESJ438 BESJ439 BESJ440 BESJ441 BESJ442 BESJ443 BESJ444 BESJ445 BESJ446 BESJ447 BESJ448 BESJ449 BESJ450 BESJ451 BESJ452 BESJ4535 BESJ454 BESJ455 BESJ456 BESJ4397 BESJ458 BESJ499
oO
oa
ET=1 BESJ440
401 I8=2 BESJ441 DFN=DFN-1.D+0 BESJ462 FN=DEN BESJ443 60 TO 130 BESI444
BESJ445 SERIES FOR (X/2)#**2.LE.NU+1 BESJ446 BESJ447
134 CONTINUE BESJ448 GLN=GAMLN (FNP1) BESJ449 ARG=FN*XO2L-GLN BESJ470 IF(ARG.LT.-ELIM1) GO TO 123 BESJ471 EARG=EXP (ARG) BESJ472
300 CONTINUE BESJ473 §sil. BESJ474 AK=3. BESJ475 eh. BESJ474 Tain BESJ477 S1=FN BESJ478 MO 123 Reilly BESJ479 $2=T2+51 BESJ480 T=-T#5X02/52 BESJ481 S=S+T BESJ482 IF(ABS(T).LT.TOL) GO TO 127 BESJ495 T2=T2+AK BESJ484 AK=AK+2. BESJ485 S1=S1+FN BESJ486
125 CONTINUE BESJ487
7 CONTINUE BESJ488 TEMP (1S) =S*EARG BESJ499 60 TO (301,202,600), IS BESJ490
301 EARG=EARG*FN/ X02 BESJ491 DFN=DFN-1.D+0 BESJ492 FN=DFN BESJ493 1S=2 BESJ494 GO TO 360 BESJ495
BESJ496 SET UNDERFLOW VALUE AND UPDATE PARAMETERS BESJ497 BESJ498
180 Y(NN)=0, BESJ499 NN=NN-1 BESJ500 DFN=DFN-1.D+0 BESJSO1 FN=DFN BESJ502 IF (NN-1) 170,171,130 BESJ 503
17 KT=2 BESJ504 15=2 BESJ505 GO TO 130 BESJ504
123 Y(NN)=0. BESJS07 NN=NN-1 BESJSO8 FNPL=FN BESJ509 DFN=DFN-1.D+0 BESJS10 FN=DFN BESJ511 IF(NN-1) 170,172,173 BESJS12
IW2MKeT=2 BESJ5S13 1S=2 BESJS14
Figure A3. (Sheet 12 of 25)
Al6
oa
aInadnn
170
ioe) P=) i)
300
ott 512
IF(SX02.LE.FNP1) GO TO i33 60 TO 130 ARG=ARG-XO2L+ALOG (FNPI1) IF(ARG.LT.-ELIM1) GO TO 123 60 TO 134
NZ=N-NN
RETURN
BACKWARD RECURSION SECTION
CONTINUE
NZ=N-NN
IF(KT.EQ@.2) GO TO 250 CONTINUE
BACKWARD RECUR FROM INDEX ALPHA+NN-1 TO ALPHA ¥Y<(NN) =TEMP (1)
Y(NN-1) =TEMP (2) IF(NN.EQ@.2) RETURN DX=X
TRX=2.D+0/DX DTIM=DFN#TRX
TM=DTM
K=NN+1
DO 230 I[=3,NN
K=K-1 ¥(K=-2) =TM#Y (K-21) -Y¥ CK) DTM=DTM-TRX
TM=DTM
CONTINUE
RETURN
Y(1)=TEMP (2)
RETURN
ASYMPTOTIC EXPANSION FOR X TO INFINITY WITH FORWARD RECURSION IN OSCILLATORY REGION X.GT.MAX(29, NU), PROVIDED THE LAST MEMBER
OF THE SEQUENCE IS ALSO IN THE REGION.
CONTINUE IN=ALPHA-TAU+2. IF(IN.LE.O) GO TO 562
INPI=IN+1 DALPHA=ALPHA-FLOAT(INP1) KT=1
GO TO 5i1
DALPHA=ALPHA
IN=06
IS=KT
ARG=X-PIDT#DALPHA-PDF SA=SIN(ARG)
SB=COS (ARG) RA=RTTP/RTX
ETX=8. #X
DX=DALPHA
DX=DX+DX
DTM=DX*DX
Figure A3. (Sheet 13 of 25)
Al7
BESJS15 BESJS16 BESJS17 BESJ518 BESJS19 BESJ320 BESJ921 BESJG22 BESJS23 BESJS2
BESJS29 BESJ526 BESJ927 BESJ528 BESJ529 BESJS3
BESJSS1 BESJ552 BESJ933 BESJ534 BESJS35 BESJ536 BESJ537 BESJSS
BESJ539 BESJS540 BESJ541 BESJ542 BESJ343 BESJ544 BESJS45 BESJS46 BESJ347 BESJS548 BESJ549 BESJ590 BESJS51 BESJ592 BESJS53 BESJS54 BESJS95 BESJ956 BESJ597 BESJ598 BESJ399 BESJ560 BESJS61 BESJ562 BESJS563 BESJ964 BESJ969 BESJ566 BESJ567 BESJ568 BESJ569
ioe)
ao
on oa ~~
T2=DTM-1.D+0 T2=T2/ETX $2=T2 RELB=TOL*ABS(T2) TL=ETX
S1=1.
FN=1,
AK=8.
DO 504 K=1,13 TI=TI+ETX FN=FN+AK DX=FN TRX=DTM-DX AP=TRX T2=-T2*AP/T1 S1=S1+T2 TLSTI+ETX AK=AK+8. FN=FN+AK DX=FN TRX=DTH-DX AP=TRX
T2= T2*AP/T1 §2=52+T2 IF(ABS(T2).LE.RELB) GO TO 505 AK=AK+8.
4 CONTINUE 3 TEMP(1S)=RA*(S1*SB-S2*SA)
60 TO (506,507) ,1S DALPHA=DALPHA+1. 15=2
TB=SA
SA=-SB
SB=TB
60 TO 303
FORWARD RECURSION SECTION
IF(KT.E8.2) GO TO 250 S1=TEMP (1)
S2=TEMP (2)
TX=2./X
TM=DALPHA*TX IF(IN.E@.0) GO TO 520
FORWARD RECUR TO INDEX ALPHA
DO S10 I=1,1N
=52
S2=TM*52-S1
TM=TM+TX
S$1=5
CONTINUE
IF(NN.EQ@.i1) GO TO 535
$=82
Figure A3.
(Sheet 14 of 25)
A18
BESJ570 BESJS71 BESJ572 BESJS73 BESJS74 BESJS75 BESJS7
BESJS77 BESJS78 BESJS79 BESJ580 BESJ581 BESJ582 BESJS83 BESJ584 BESJS85 BESJS86 BESJS87 BESJS88 BESJS89 BESJS90 BESJS91 BESJS92 BESJS953 BESJ594 BESJS95 BESJ596 BESJS97 BESJ598 BESJS99 BESJ400 BESJ401 BESJ4602 BESJ603 BESJ604 BESJ605 BESJ4606 BESJ407 BESJG608 BESJ609 BESJ410 BESJ411 BESJé12 BESJé13 BESJ414 BESJ415 BESJ616 BESJ417 BESJ4618 BESJ419 BESJ620 BESJ421 BESJ622 BESJ623 BESJ624
aInnon
920
600
6950
677
603
S2=TH#S2-S1 TM=TH+TX 51=5 CONTINUE
FORWARD RECUR FROM INDEX ALPHA TO ALPHA+N-1
Y(t) =S1
¥(2)=52
IF(NN.E@.2) RETURN DO 530 I=3,NN Y¥(T)=TM*¥Y (1-1) -¥ (1-2) TM=TM+TX
CONTINUE
RETURN
Y¥(1)=S2
RETURN
BACKWARD RECURSION WITH NORMALIZATION BY ASYMPTOTIC EXPANSION FOR NU TO INFINITY OR POWER SERIES.
CONTINUE
COMPUTATION OF LAST ORDER FOR SERIES NORMALIZATION KM=AMAX1(S.-FN,O.)
TFN=FN+FLOAT (KM)
TA=X02L-TA
TB=-(1.-1.5/TFN)/TFN IN=CE/(-TA+SQRT (TAX TA-CE*¥TH))+1.5 IN=IN+KN
GO TO 603
CONTINUE
COMPUTATION OF LAST ORDER FOR ASYMPTOTIC EXPANSION NORMALIZATION
GLN=AZ32+RA
IF(ARY.GT.30.) GO TO 475 RDEN=(PF(4) #ARY+PP(3)) *ARY+1. RZDEN=PP (1) +PP (2) #ARY TA=RZDEN/RDEN
IF(W2.LT.G.16) GO TO 451 TB=GLN/RTARY
60 TG 677
TB=(1. 259992104940. 16798947 30#W2) /FNLS GO 10 677
CONTINUE
TA=CONL*¥TCE/ACZ
TA=((0, O4935827160#TA-O. LLLLLLLL11) #TAtG. 6666666667) €TAXARY IF(W2.LT.0.10) GO TO 451 TB=GLN/RTARY
IN=TA/TB+1.5
IFC IN. GT.INLIM) GO TO 4062 DX=FLOATC(IN)
DTM=DFN+DX
DX=X
TRX=2.D+0/Dk
DTM=DTM#TRX
Figure A3. (Sheet 15 of 25)
Al19
BESJ625 BESJ626 BESJ627 BESJ628 BESJ429 BESJ630 BESJ631 BESJ652 BESJ433 BESJ634 BESJ6355 BESJ436 BESJ637 BESJ438 BESJ639 BESJ4640 BESJ641 BESJ642 BESJ643 BESJ644 BESJ649 BESJ4646 BESJ647 BESJ648 BESJ649 BESJ4650 BESJ651 BESJ652 BESJ695 BESJ654 BESJ655 BESJ696 BESJ4&57 RESJ658 BESJ659 BESJ460 BESJ661 BESJ642 BESJ6635 BESJ664 BESJ6465 BESJ466 BESJ667 BESJ468 BESJ469 BESJ670 BESJ471 BESJ6/72 BESJ473 BESJ4674 BESJ4735 BESJ676 BESJ677 BESJ478 BESJ479
aod
a
605
601
604 615
oy
TM=DTM TA=0. TB=TOL KK=1 CONTINUE
BACKWARD RECUR UNINDEXED
DO 601 I=1,1N
S=TH
TB=TM*TE-TA
TA=S
DTM=DTM-TRX
TM=DTM
CONTINUE NORMALIZATION IF(KE.NE.1) GO TO 4604 TA=(TA/TB) #TEMP (3) TB=TEMP (3)
KK=2
IN=NS
IF(NS.NE.0) GO TO 605 Y (NN) =TB
NZ=N-NN IF(NN.E@.1) RETURN 5=TB
TB=TM#TB-TA
TA=S
DTM=DTM-TRX
TM=DTM
K=NN-1
Y(K)=TB
TF (NN.EQ@.2) RETURN KM=K-1
BACKWARD RECUR INDEXED
DO 602 I=1,KM Y(K-1) =TM#Y (K) -¥ (K#1) DTM=DTM-TRX
TM=DTM
K=K-1
CONTINUE
RETURN
CONTINUE NZ=-2 RETURN CONTINUE NZ=-3 RETURN CONTINUE NZ=-1
Figure A3.
(Sheet 16 of 25)
A20
BESJ4580 BESJ681 BESJ4682 BESJ483 BESJ4684 BESJS85 BESJ4686 BESJ487 BESJ4688 BESJ689 BESJ490 BESJ4691 BESJ492 BESJ693 BESJ4694 BESJ495 BESJ694 BESJ697 BESJ498 BESJ699 BESJ700 BESJ701 BESJ702 BESJ703 BESJ704 BESJ705 BESJ706 BESJ707 BESJ708 BESJ709 BESJ710 BESJ711 BESJ712 BESJ713 BESJ714 BESJ715 BESJ716 BESJ717 BESJ718 BESJ719 BESJ720 BESJ721 BESJ722 BESJ723 BESJ724 BESJ725 BESJ726 BESJ727 BESJ728 BESJ729 BESJ730 RBESJ731 BESJ732 BESJ7335 BESJ734
NOaonnrorinrwarIiononiwaig
ioe)
CDC 6600 ROUTINE
AND ITS DERIVATIVE DAI(X) FOR JBESS INPUT: X - ARGUMENT, COMPUTED BY JBESS, X UNRESTRICTED RX — RX=SORT(ABS(X)), COMPUTED BY JBESS C - C=2.*#(ABS(X)*#1.5)/3., COMPUTED BY JBESS OUTPUT: AI - VALUE OF FUNCTION AI(X) DAI - VALUE OF THE DERIVATIVE DAI(X) WRITTEN BY D.E. AMOS, S.L. DANIEL & M.K WESTON DIMENSION AKI (14) ,AK2(23) ,AKS (14) DIMENSION AJP(19) sAUN(19) ,AC15) ,B(15) DIMENSION DAK1 (14) ,DAK2(24) ,DAK3 (14) DIMENSION DAJP(19) ,DAIN(19) ,DA(15) ,DB(15) DATA Ni,N2,N5,N4/14,23,19,15/ DATA M1,M2,M3,M4/12,21, 17,13 / DATA FP112,CON1,CON2,CONS,CON4 ,CONS/
1
5 <
Cl roe
Poe 10 CO SO ON fe C4 PO NO oS
fy PQ oe
“Nc on eS
1
RETURN END
SUBROUTINE JAIRY(X,RX,C,Al,DAI)
1.5089969389958E+00, 3.8000458986729E-01,
DATA AKI(1) AK1(2) /-1. AK1(4) / 8. AK1(6) / 1. AK1(B) /-1
DATA AK2(1) AK2(2) / AK2(4) / AK2(6) / AK2(8) / AK2(10)/ AK2(12)/ AK2(14)/ AK2(16)/ AK2(18)/ AK2(20)/ AK2(22)7
DATA AK3(1) AKS (2) AKS(4) AK3 (4) AKS (8)
AK3(14)/ DATA AJP(1) AJP (2)
98762 »8729020974102E- 06/, »21135041695591E-07/,
29245470528 265E- 097, »26863505659574E-10/, ~JOL41035901371E-11/, ©§4464450473454E-12/,
»7010603055935E- -1977479916481E-14/,
144734079E-
et et oO ee lO) Rl ed oo CO I |
HON fe cal ike Laie AK3(10)/-6. AR S(L2)/-1.
fHIkq
RASC]
JAIRY COMPUTES THE AIRY FUNCTION AI(X)
/ 2. 2042309098779E- 2929024278770E-01/7, 2284415200554E-04/, 6382428017212E-05/7,
»29621999359335E-07/, AKL(190)/ 1. AKL(12)/ 2. AK1(14)/-35.
Rese see ae VASES g O32535257423563E-12/, 16169897 78508E- Maid / 2.7436615086960E-
3979096975 F690E-03/,
2924875E- o4/, 1713 1890E-05/,
13/,
/ 2.8027 7812704284438E-03/, b3524996526900E-06/, D2 29435502291 6E-09/, 2440825180026E-11/, S63752444639351E-135/, 1270513469106E-14/,
-2.46480324312435E-16/ 4 7.7895295643758E-
B43565654568GE-01/,
Figure A3.
b. SEBO ONRIO TES Ol,
WII.
de die dr din died
01/,
AKL (3) AK1 (3) AK1(7) AK1 (9) AKI (11)/-
SS
1. 20 3. 8. 3.
AKI(13)/-1.
O1/, AK2 (3) AK2(5) AK2(7) AK 2 (9)
AK2(13)/ AK2(15)/-1. AK2(17)/-9,
AK2(19)/-4.
AK2(21)/-4, AK2(23)/-3. OLA AK3(3) / AK3(5) / ARSC) if AK3(9) / AK3(11)/ AK3(13)/
02/, AIP(S) /
fio (No JI 5 7-4. AK2(11)/-3. -2,0390244716791E-
— cO cn cn ~O
9.03515471619678E+00, 8. 6602540378444E-01/
OSBBLL6335919E-02/, 34614345891 23E-04/
0690258957319E-07/, 2290815882567E-09/, 3916546561568E-11/, 1067954609788E-14/,
5733922062119E-03/, 1212491739992E-04/, 3680422537055E-06/, 7589279396229E-07/, 0924537427061E-08/, 09/~" 3670476763957E-10/, 3139829654843E- 12/, 4384026199096E-13/, 5076010450328E-14/, 1907704086507E-15/
© OS542257962900E-05/, i pivomiaeaneanes 08/, »4713840457655E-10/7, » 60477904117 3128576196625
AEST BA o E-14/, 2267976598135E-15/,
~O1412605214617E-02/7,
(Sheet 17 of 25)
A21
BESJ7395 BESJ736 AIRY1 AIRY2 AIRYS AIRY4 AIRYS AIRY6 AIRY7 AIRY8 AIRY? AIRY10 AIRY11 AIRY12 AIRY13 AIRY14 AIRY195 AIRY16 AITRY17 AIRY18 AIRY19 AIRY 20 AIRY21 AIRY22 AIRY25 AIRY 24 AIRY29 AIRY26 AIRY27 AIRY28 AIRY29 AIRY30 AIRY31 AIRY32 AIRY33 AIRY 34 AIRY35 AIRY36 AIRY37 AIRY38 AIRY39 AITRY40 AIRY41 AIRY42 AIRY43 AITRY44 AIRY45 AIRY46 AIRY 47 AIRY 48 AIRY49 AIRYSO AIRYS1 AIRYS2 AIRYS3
oOo ON Oo OF & hd by
Oo On OO Pei pe
“SM & OF f hd Pore
—“SoCe On B YI Po re
(Ea) (23) C>
Noo Se
m= 10 CO NO CN Fs ty Por
AJP(4) / 3.0534272427761E-02/, AJP(S) /-4.9542470251308E-03/, AJP (4) /-1.7274955256395E-03/, AIP(7) / 2.4313763783919E-04/, AJP(8) / 5.0456477751708E-05/, AJP(9) /-6.16316582469521E-06/, AJP (10) /-9.0398574551077E-07/, AIP(11)/ 9.7024377835588E-08/, AJP(12)/ 1.0963945330520E-08/, AJP(13)/-1.0471633058877E-09/, AJP (14) /-9.6035944134465E-11/, AJP(15)/ §.2535878945413E-12/, AJP (16)/ 6.3612343901877E-13/, AIP(17)/-4.9662961411602E- 14/, AJP (18) /-3.2981028892962E-15/, AJP(19)/ 2.3579825203110E-16/ DATA AJN(1) / 3.8049788761724E-02/, AJN(2) /-2.4531954184555E-01/, AJN(3) / 1.6582062370270E-01/, AIN(4) / 7.4933004581879E-02/, AJN(S) /-2.6347628810664E-02/, AIN(5) /-5.9253559730498E-03/, AIN(7) / 1.4474440958980E-03/, AIN(8) / 2.1831183132222 -04/, AIN(9) /-4.1065207748030E-05/, AIN(10) /-4.6687499417177E-06/, AIN(L1)/ 7.1521880727714E-07/, AIN(12)/ 6.5296477085463E-08/, AIN(13)/-8.4428402756595E-09/, AJN(14) /-6.4419615897698E-10/, AIN(15)/ 7.20802 28650528E-11/ AIN(14)/ 4.7246543171785E-12/, AJN(17) /-4.6602263254704E- 137, AIN(18) /-2.6776271038919E-14/, AJN(19)/ 2, 3616131657002E-15 DATA A(1) / 4.9027542474279E- D1/, A(2) / 1 .5764727794620E-03/, A(3) /-9.6619596314031E-05/, A(4) / 1.3591608028882E-07/, A(5) / 2.9815734265486E-07/, Ald) /-1.868247675599B8E-08/, A(7) /-1.0368573766714E-09/, A(B) / 3.28660818434335E-10/, A(9) /-2.570914106327BE-11/, A(10)/-2.32357655300468E-12/, A(11)/ 9.5752327904826E-13/, A(12)/-1.2034082864972E-13/, A(13) /-2.9090771677072E-15/, A(14)/ 4.5565445458015E-15/, A(15) /-9.9900387481026E-16/ DATA B(1) / 2.7859355280308E-01/, Bi2) /-3. Seiteey ieee 53E-03 B(3) /-2.3114967738499E-05/, B(4) / 4.7131784226356E- nee B(S) /-1.1241590793133E-07/, B(6) /-2. NOLOOSDL AREAS, B(7) / 2.6094807530219E-09/, B(8) /-3.5509813610122E-11/, B(9) /-3.508499784238B8E-11/, B(10)/ 5.8300718795420E-12/, B(11)/-2.0464482875333E-13/, H(12)/-1.1052917947674E-13/, B(13)/ 2.9772477803878E-14/, B(14)/-2.8820511100994E-15/, B(15)/-3.3265631169617E-16/ DATA NID,N2D,N3D,N4D/14,24,19,15/ DATA M1D,M2D,M3D,M4D/12,22,17,13/ DATA DAK1(1) / 2.0456784230789E-01/, DAKi(2) /-6.6132273990566E-02/, DAKI(3) /-8.4984590098929E-03/, DAKL(4) / 3.1218349155629E-03/, DAKL(S) /-2.7001648982943E-04/, DAK1(5) /-6.3563629867939E- 08/4 DAKI(7) / 3.0239771240951E-06/, DAK1(8) /-2.1831119533009E-07/, DAKI(9) /-5.3619428933293E-10/, DAKIi(10)/ 1.1309803562231E- ae DAK1 (11) /-7.4302383462907E-11/, DAK1(12)/ 4.2890417082689E-13/, DAKL{13)/ 2.2381092575454E-13/, DAKI (14) /-1, 3914013564118E- 14/ DATA DAK2(1) / 2.9333234388323E-01/, DAK2(2) /-8.0619878474311E-03 3/, DAK2(3) / 2.42540172333514E-03/, DAK2(4) /-.8229754885024E-04/, DAK2(5) / ines 8642775118E-04/, DAK2(6) /-4.9745744768406E-05/, DAK2(7) / 1.3209068123950E-05/, DAK2(8) /-3.4952824044494E-06/, DAK2(9) / 9.2436245107894E-07/, DAK2(10)/-2.4473257152187E-07/, DAK2(11)/ 6.4930783764891E-08/, DAK 2 (12) /-1.7271762150154E-09/, DAK2(13)/ 4.6072576360466E-09/, DAK2 (14) /-1.2324905529155E-09/, DAK2(15)/ 3.3062040949810E-10/, DAK2(15)/-8.8925209977240E-11/, DAK2(17)/ 2.3977351987930E-11/, DAK2 (19) /-S.4801392115345E-12/, DAK2(19)/ 1.7551013202373E-12/, DAK2(20) /-4. 7630382983344E-13/, DAK2(21)/ i. 29498241 10081E-13/,
Figure A3.
(Sheet 18 of 25)
A22
AIRYS4 AIRYS9 AIRYS6 AIRYS7 AIRYS8 AIRYS9 AIRY60 AIRY61 AIRY62 AIRY6$ AIRY 64 AIRY45 AIRY46 AIRY67 AIRY48 AIRY49 AITRY70 AIRY71 AIRY72 AIRY73 AIRY 74 AIRY735 AIRY76 AIRY77 AIRY78 AIRY79 ATRY80 AIRY81 ATRY82 AIRY83 AIRY84 AIRY85 AITRY86 AIRYS7 AIRY8E& AIRY89 AITRY99 AIRY91 AIRY92 AIRY93 AIRY?4 AIRY9S AIRY96 AIRY97 AIRY98 AIRY99 AIRY100 AIRY101 AIRY162 AIRY103 AIRY104 AIRY105 AIRY106 AIRY107 AIRY108
Ld bo
oe to a Su ee — ee
Oo wo SO OF S fy Po ee
oon oO Pw Pe
SWOOP
“1 oo~ CN & te Poor
DAK2 (22) /-3.5267 -2.62786914342 / 2.8467582881135E-01/7,
DAK2(24)/ DATA DAKS(1 DAKS(2) / DAKS(4) 7 DAKS(6) / DAKS(8) / DAKS(10)/ DAK3S (12)/ DAKS(14)/ DATA DAJP(1 DAJP(2) /- DAJP(4) / DAJP(6) /- DAJP(8) / DAJP (10) /- DAJP(12)/
DAJP(14)/-3.,
DAJP(16)/ DAJP(18)/ DATA DAUN(1 DAJN(2) / DAJN(4) /- DAIN(4) / DAJN(8) /- DAJN(1G)/ DAJN(12)/- DAJN(14)/ DAJN(16) /- DAIN(18) / DATA DAI)
DA(3)
221043E-14/, 29E-15/
»9507307261908E-03/, »84907283944634E-06/, »0592563445715E-69/, ~ 563557688851 354E-11/, »8857435378444E-13/, »1737476261721E-14/, »0957477309706E-16/ ) ff 1.20262933468882E-01/, 1.6794842923050E-02/, 8.4556029509887E-04/, 2.2982786094548E-05/, 3.76343991134692E-07/, 4, 29611335 200301E-09/7, S7245881346190E-11/, »2612065309577E-13/,
ep
boo co on
2 -1.1260437448512E-15/, )
7 1.0859453963297E-02 8.553513197485709E-02/, 8.7842072529426E-02/, 9,4167406050324E-63/, 4. 1115734315683E-047, 9,8763368220840E-06/, 1.507985001S147E-07/, 1.5959991741922E-09/, 1. 24105S55050030E-11/, 7.35940097115574E-14/,
2514076235408E
4.9162732110460E-017,
DAK2(23)/ 9.
DAK3S (3) DAKS(5) DAKS(7) /-5. DAK3(9) 7-3. DAK S(11)/-8. DAK3(13)/-1.
i-4, POM a
6.952191 35151 146E-027,
DAIP(3) DAJP (3S) DAJP (7) DAJP (9) DAJP(11)/ 3. DAJP(13)/-3. DAJP(15)/ 2. DAJP(17)/-1. DAJF(19)/ 7. Af 6
DAIJN(3) /-3. DAJN(S) / 5. DAJN(7) /-3.
DAJN(9) / 1.
DAJN(11)/-1. DAJN(135)/ 2. DAJN(15)/-2. DAJN(17)/ 1.
DAJN(19)/-7. DA(2) / 3. DA(4) 7-4.
Wg foie Yo
[oH
/ / DAS) / DA(7) /- DIAG DALI) /- DA(13) /- DA(15) DATA DB(1) DB(S) DB(7) DBC?) /-
Ps Lode) [F(G. 61.5.) LF (X.6T. 1.2) T=(X+X-1, Way ey J=N1 FL=AK1 (J) F2=
60 60
2
i
i=
DB(3) /-8.4 i 8
YOR
4
-05/,
»1SL5888055463E-087, SOIT Tldor 226097,
S57 2404342042E-11/,
3. 2408911983032E-13/, »4G75524797406E-14/, »1790078647740E-16/
HAY NS
252852
SO944235E-O1/, 219009E-05/7,
»4238972021762E-07/, »3637783584458E ~40995S691G5B19E-12/, DIB, aii. DB CUS) (ot DB(15)/-7.1
-09/,
S48 7401513 702E-i13/, 7725354306791 1E-147, 179333575 79S53E-16/
TO 206
60 TO 150 2) *CON4
Figure A3.
A23
DA(&) / DA(8) /- DA(10)/- DA(12)/ DA(14)/
DB(2) / DB(4) # DE(6) /- DB(8) / DB(10)/- DB(i2)/- DB(14)/
oe ike Sie Ly
les
fi =e 6. 5. 4, ie
6200315158592E-15/,
8348115033798E-05/, 0141849117858E-07/, BS32529140058E-10/, 9088909477950E-12/, 6858825645219E-14/, 68523514651092E-15/,
»7801023526382E-03/,
97146140182135E-03/, 4288962070198E-05/,
2.2906787091599E-G6/,
456639353555956E-08/, S8673569121499E-10/, 72696091 06654E-12/, S876S205235850E-14/, SCS TAS 17 7
527706811 306E-01/, 325190697605E-02/, 3218702601900E-03/, 129732689135E-04/, 9731296981239E-06/, 3268766952539E- OB/, 076659226683B8E-10/, 39631765351 04E-12/, 3288747562750E-15/ 11164930827496-03/, 617697761721 4E- 9729580465652E-08/, 4475282664204E-10/, 9965506584722E-i2/, 6209895256874E-13/, 5938481128449E-16/,
1 sa] 0
06/,
4421 2835341992E-05/, SBO40S51841871E-04/, 2428689470978E-09/, »1699104265667E-10/, 1867422109358E-12/, 9OL9057660871E-14/, 5595061044266E-15/,
(Sheet 19 of 25)
AIRY109 AIRY116 AIRY11i1 AIRY1L12 AIRY113 AIRY114 AIRY115 AIRY116 AIRY117 AIRY118 AIRY119 AIRY120 AIRY121 AIRY122 AIRY123 AIRY124 AIRY125 AIRY126 AIRY127 AIRY128 AIRY129 AIRY130 AIRY131 BEJS132 AIRY133 AIRY134 AIRY135 AIRY1S6 AITRY137 AIRY137 AIRY139 AIRY140 AIRY141 AIRY142 AIRY143 AIRY144 AIRY145 AIRY146 ATRY147 AIRY148 AIRY149 AIRY150 AIRY151 AIRY152 AIRY1935 AIRY154 AIRY1595 AIRY196 AIRY157 AIRY158 AIRY1959 AIRY140 AIRY141 AIRY162 AIRY16$
1035
106
DO 105 1=1,h1 J=J-1
TEMPL=F 1 FI=TT¥F1-F2+AK1 (J) F2=TEMP1
CONTINUE AL=T#F1-F2+AK1 (1)
J=N1D
F1=DAK1 (J) F2=0.
DO 106 I=1,M1D J=J-1
TEMP1=F1
FL=TT¥F1-F2+DAK1 (J)
F2=TEMP1 CONTINUE
DAI=-(T#F1-F2+DAK1 (1) )
RETURN
CONTINUE T=(X+X-CON2) ¥CONS We Sul kee L
J=N2
FL=AK2(J)
F2=0.
pO 155 I=1,M2 J=d-1
TEMP1=Fl FL=TT#FI-F2+AK2 (3) F2=TEMP 1
CONTINUE RTRX=SORT (RX) EC=EXP(-C)
AI=EC¥(T#FI-F2+AK2
J=N2D FL=DAK2(d) F2=0.
DO 156 1=1,M2D J=J-1
TEMP1=F 1
(1))/RTRX
FI=TT¥FI-F24DAK2 (0)
F2=TEMP 1 CONTINUE
DAL=-EC#(T#F1-F2+DAK2(1)) #RTRX
RETURN
CONTINUE T=10. /C-1. TT=T+T
J=N1 FI=AK3(J) F2=0.
DO 205 I=1,M1 J=J-1
Figure A3.
(Sheet 20 of 25)
A24
AIRY164 AIRY165 AIRY166 AIRY147 AIRY168 AIRY149 AIRY179 AIRY171 AIRY172 AIRY173 AIRY174 AIRY175 AIRY176 AIRY177 AIRY178 AIRY179 AIRY180 AIRY181 AIRY182 AIRY183 AIRY184 AIRY185 AIRY186 AIRY187 AIRY188 AIRY189 AIRY190 AIRY191 AIRY192 AIRY193S AIRY194 AIRY195 AIRY196 AIRY197 AIRYi98 AIRY199 AIRY200 AIRY201 AIRY202 AIRY203 AIRY204 AIRY205 ALRY2Z06 AIRY 207 AIRY208 AIRY209 AITRY210 AIRY211 AIRY212 AIRY213 AIRY214 AIRY215 AIRY216 AIRY217 AIRY218
ba oS cn
300
309
306
TEMP1=F 1
FL=TT#F1-F2+AK3 (J)
F2=TEMP1
CONTINUE
RTRX=SQRT (RX)
EC=EXP(-C) AI=EC#(THFI-F2+AKS(1))/RTRK J=N1D
FL=DAKS(J)
F2=0,
DO 206 I=1,MiD
J=J-1
TEMP1=F1 FL=TT#F1-F2+DAKS (J) F2=TEMPI
CONTINUE DAI=-RTRX#EC#(T#FI-F2+DAKS(1)) RETURN
CONTINUE IF(C.G67T.5.) 60 TO 350 T=.44#C-1.
TT=T+T
J=N3
FL=AJP (J) EL=AUN(J)
F2=0.
E2=0.
DO 305 I=1,M5 J=d-1
TEMP1=F 1
TEMP2=E1 FLSTT#Fi-F2+AUP (J) EL=TT#E1-E2+AUN (J) F2=TEMPI
E2=TEMP2
CONTINUE
AL=(T#EL-E2+AIN(L))-X¥(T#FL-F2+ANP (1) )
J=N3D
FL=DAUP (J) EL=DAIN(J)
F2=0.
E2=0.
DO 306 I=1,M3D
J=J-1
TEMP1=F 1
TEMP2=E1
Fl = TT¥FL-F2+DAJP (J) El= TT#E1-E2+DAIN(J) F2=TEMP1
E2=TEMP2
CONTINUE
DAI=X#X*¥(T#FI-F2+DAJP(1))+(T#E1-E2+DAIN(1))
RETURN
Figure A3.
(Sheet 21 of 25)
A25
AIRY219 AIRY22 AIRY221
LD ms a) ~
Le)
oo
AIRY2351 AIRY232 AIRY233 AIRY234 AIRY235 AIRY23 AIRY237 AIRY23 AIRY239 AITRY240 AIRY241 AIRY242 AIRY243 AIRY244 AIRY245 AIRY246 AIRY 247 AIRY248 AIRY249 ALRY250 AIRY251 AIRY252 AIRY253 AIRY254 AIRY255 AIRY256 AIRY257 AIRY258 AIRY259 AIRY260 AIRY261 AIRY262 AIRY263 AIRY264 AIRY269 AIRY266 AIRY267 AIRY268 AIRY269 AIRY270 AIRY271 AIRY272 AIRY275
oO
(sa) (52) (se) fee) isel Ge) Ga
350
CONTINUE T=10./C-1.
TT=T+T
J=N4
FL=A(d)
E1=B(J)
F2=0,
E2=0.
DO 310 I=1,H4 J=d-1
TEMP1=F 1
TEMP2=E1 FL=TT#F1-F2+A(d) E1=TT¥E1-E2+B(J) F2=TEMP1
E2=TEMP2
CONTINUE TEMP1=T#F1-F2+A (1) TEMP2=T¥EL-E2+B (1) RTRX=SORT (RX) CV=C-FPI12 CCV=COS(CV) SCV=SIN(CV) Al=(TEMP1*¥CCV-TEMP2*SCV) /RTRX
J=N4D
FLi=DA(J) EL=DB(J)
F2=0,
E2=0.
DO 3ii 1=1,M4D J=J-1
TEMP1=F 1 TEMP2=E1
FI=TT#F1-F2+DA(J) EL=TT#E1-E2+DB(d) F2=TEMF1
E2=TEMP2
CONTINUE TEMPL=T#F1-F2+DA(1) TEMP2=T#E1-E2+DB (i) EL=COV*CONS+. S¥SCV E2=SCV*CONS-.S*CCV DAI=(TEMPL*E1-TEMP2*E2) *RTRX RETURN
END
FUNCTION GAMLN(X)
WRITTEN BY D. E. AMOS, SEPTEMBER, 1977. REFERENCES * SAND-77-1518
* COMPUTER APPROXIMATIONS BY J.F.HART, ET.AL., SIAM SERIES IN APPLIED MATHEMATICS, WILEY, 19468, P.135-134.
* NBS HANDBOOK OF MATHEMATICAL FUNCTIONS, AMS S55, BY M. ABRAMOWITZ AND I.A. STEGUN, DECEMBER. 1955, P.257
ABSTRACT Figure A3. (Sheet 22 of 25)
A26
AIRY274 ATRY2735 AIRY276 AIRY277 ALRY27 AIRY279 ATRY280 AIRY281 AIRY282 ATRY283 AIRY284 AIRY285 AIRY286 ATRY287 AIRY288 AIRY289 AIRY290 ATRY291 AITRY292 AIRY293 AIRY294 AIRY295 AIRY296 AIRY297 AITRY298 ATRY299 AIRY300 AIRY3O01 AIRY302 AIRY303 AIRY304 ATRY305 AIRY306 AIRY307 AIRY308 AITRY309 ATRY210 ATRYS11 AIRYS12 ATRY313 AIRY314 AIRYS15 AIRYS14 AIRYS17 AIRY318 GLNI GLN2 GLNS GLN4 GLNS GLN6 GLN7 GLNS GLN9 GLN1LO
NIOOIAINIAAMAAWAAAABAAHANHM
GAMLN COMPUTES THE NATURAL LOG OF THE GAMMA FUNCTION FOR
X.GT.G. A RATIONAL CHEBYSHEY APPROXIMATION IS USED ON B.LT.X.LT.1000., THE ASYMPTOTIC EXPANSION FOR X.SE.1000. A RATIONAL CHEBYSHEV APPROXIMATION ON 2.LT.X.LT.3. FOR O.LT.X.LT.8. AND X NON-INTEGRAL, FORWARD OR BACKWARD RECURSION FILLS IN THE INTERVALS 0.LT.X.LT.2 AND S.LT.X.LT.8. FOR X=1.,2 3100., GAMLN IS SET TO NATURAL LOGS OF FACTORIALS. DESCRIPTION OF ARGUMENTS INPUT X SeXGh 0 OUTPUT GAMLN - NATURAL LOG OF THE GAMMA FUNCTION AT xX DIMENSION GLN(100) ,P(S) ,Q(2) ,PCOE(9) ,Q@COE(4) DATA XLIM1,XLIM2,RTWPIL/ EG 9 aoe » 97.189385322047E- DATA P(1)/7.663451880000E-04/, P(2)/-5.940956105200E-04/, CSS) / To PESTS NOSE TSER O47, PCA) /-2.7777777546577E-03/,
i
2 DATA DATA
PW he
ted Poo
oon oon Pine Oona one
DATA
Cn ee tel Pa oe
RCo ies Q(1)/- PCOE(1)/2.
PCOE(3)/1. PCOE(S)/2, PCOE(7)/2. PCOE(9) /6.
OCOE(1)/1. QCOE(3)/9.
GLN(1) /0. GLN(4) /1., GLN(4) /4, GLN(8) /8.52 GLN(10)/1. GENCI2) 7107 GLN(14)/2.2 GLN(16)/2, GLN(18)/3.3 GLN(20)/3., 2)/4,.5 GLN(24)/5. GLN(26)/5. WAG 4 GLN(30)/7. GLN(32)/7., GLN(34)/8. GLN(34)/9, Nao Fe GLN(40)/1.,
GLN(2
GLN(28
SLN(38
BLN(42)/1.
SLN(44)/1.
GLN(45)/1.
GLN(48)/1., "445657439463E+02/,
(Sheet 23 of 25)
BLN(S0) /1
edrdend
ITIIIIIIT
-02/
7777777777786 - O3/,
973786644810E-03/ O931159546710E-01/ 159943128451E+00/ 0782472535179E+01/ 2003583800713E+01/
DODDDDD00NDOE+00/,
Q(2)/8. 3333535355333 3E- 2581943959028E- 980671310204E-01/7, 338067999387E+00/, 503677233002E+01/,
»PCOE(2)/9. ,PCOE(4)/3. »PCOE(4) /6. »PCOE(8)/3.
QCOE(2
02/ 03/,
)/-8.906016659498E+00/,
822521104714E+00/ ,QCOE (4) 76. 20038380071 3E+O1/
O/7, GLN(2) /0.0/, TILTS94G9228E+00/, TE7AIIT4A27E2E+O0/, es ere OLB2748008E+01/, O2307B84587E+01/, 216385512E+01/, 9927138384E+01/, SSOS0734S014E+01/, FS359BB41B8720E+01/, 38013889848E+01/, 160667554776E+01/, BOG3IE0S22298E+01/, JI7SSB62701E+01/, 12570289671 7E+01/, B09222555332E+01/, JOS44S701758E+01/, 213617560369E+01/, P33051245479E+017, 066317502506E+02/ 14G342117815E+02/, 219330815 1S4E+02/, 29 ee eae
3480272243 73E+
ee ee ies Ie coc Gy
Figure A3.
GLN (3) GLN(3)
GLN(7) /6. GLN(9) /1. GLN(I1)/1. GLN(13)/1. GLN(15)/2. GLN(17)/3.,
GLN(19)/
3.63590445
GLN(21) 7/4, GLN(23)/4.,
GLN (25)
GLN(27)/6. GLN(29)/6, GLN(31)/7.
GLN(33)/8. GLN(35)/8. GLN(37)/9,
GLN{39)/1. GLN(41)/1,
GLN(43)/1. GLN(45)/1.
GLN(47)/1. GLN(49)/1. GLNCS1)/1.
A27
75.931471805599E-01/, /3.178053830348E+00/,
S79251212010E+00/, 050460290274E+01/, S10441257308E+01/, 9987214495466EtO1/, S19122118274E+01/, DS71BS01060BE+91/, Z20B03E+01/, 255561646075E+01/, B471181351B84E+0i/,
/S.A7TB4A7T2Z9S9BLIE+O1/,
261701761 00E+01/, 7TEB974AZLSTLISE+Oi/, Fees Oe e eS Eun.
LSS7ISPASSLZE+O1/, 898082754220E+01/, S71G69454214Et+01/, O29681996145E+02/ LOSZ06397148E+02/, 177718813997E+92/, 205172711 494E+02/, S295Z5750356E+02/, 40673923648 2E+02/, 48477766951 8E+02/,
AND
o1/
GLNi1 GLN12 GLNIS GLN14 GLNIS GLN14 GLN17 GLN18 GLNI9 GLN20 GLN21 GLN22 GLN23 GLN24 GLN25 GLN26 GLN27 5LN28 GLN29 GLNSO GLN3S1 GLNS2 GLN33 GLN34 GLN3S GLNS6 GLN37 GLN38 GLN39 GLN40 GLN41 GLN42 GLN43 GLN44 GLN45 GLN46 GLN47 SLN48 GLN49 GLNSO GLNGi GLNS2 BLNSS GLNS4 6GLNSS GLNS6 SLNS7 GLNS8 GLNO? GLNSO GLN61 GLNé2 GLN6S GLN64 GLNGS
a
Lap)
(val (sa) (o2)
an
10
15
— ~0 CO “I cr
L4 poe OO coONM co Cn & 4 Po or
oon or cn
DATA
DATA
GLN(S2)/1. GLN(S4)/1. BLN(S4) /1 GLN(S8)/1. GLN(60)/1, GLN(61)/1. GLN(S6S)/1. GLN(65)/2. GLN(67)/2. GLN(69) 7/2. GLNI71)/2. GLN(73)/2. GLN(75)/2. GLN(77)/2. GLN(79)/2. GLN(81)/2. GLN(83)/2. GLN(85)/2. GLN(S7)/3., GLN(89)/3. GLN(91)/3. GLN(93)/3, GLN(95)/3. GLN(97)/3. GLN(99)/3.
S24095925945E+02/, 6035511 2B215SEt+02/,
~68S274454484E+62/,
7639584840 70E+02/, 8453358288614 E+G2/ 8862917542357E+02/, 968661814572Z9E+02/, O51681994826E+02/, LS5322414945E+02/, 219564418191 E+92/, SO439043546S58E+02/, SB97B3895618E+02/, 4§73729140962E+02/, 562211355500E+02/, 649216497985E+02/, 736735124285 7E+02/, B24742926876Et+02/, FL S2Z39500943E+027, OO2Z2074B6470E+02/, U91L641935802E+02/, 18152463596202E+02/, 271852877038E+02/, So2511819792E+02/, 4337940706235E+02/, J45359085S5194E+02/,
NDX=X T=X-F POT
LOAT(NDX) £@.6.0)
DX=XLIMI-X%
IF (DX.
RATIONAL CHEBYSHEY APPROXIMATION ON 2.
LT.0.0)
NXM=NDX-2 PX=PCOE (1)
DO 10 1=2,9 PX=T#PX+PCOE (1) QX=OCDE(1)
DO 15 I=2,4 QX=T#OX+QCOE (1)
DGAM=
IF (NX IF (NX
BACKWARD RECURSION
PX/QX M.GT.9)
MEG. 0)
60 TO 2 60 T0 2
60 TO 51
GO TO 40
DGAM=DGAN/(1.+T)
IF (NX
DGAM=
M.EQ.-1) DGAN/T
60 T
Ac aud
GAMLN=ALOG (DGARM) RETURN
FORWARD RECURSION FOR 3.LT.X.LT.9
Figure A3.
A28
PON Wolbiokalbiok
GLN
GLN(62)/1.
GLN(54)/2
GLN(646)/2. GLN(48)/2. GLN(70)/2,
GLN(72)/2.
GLN(74) 7/2. GLN(76)/2. GLN(78)/2. GLN(S0)/2.
GLN(82)/2. GLN(84)/2. GLN(84)/2,
SLN(88)/3. GLN(90)/3. GLN(92)/3. GLN(94)/3.
GLN(96)/3. GLN(98)/3. GLN(100)/3.
Cag) fin GLN(S55)/1. GLN(S7)/1. GLN(S9)/1.
Va Wolbia
t=)
le
Yh ts of
ied id od
ca
cn cn po ce a- ha om
hI ~J = cO
92739047 287B8E+02/, OLOO9S163993E+02/, 0934258675 25E+02/, L77Sh9S41140E+02/, 2619054983235 7E+02/, BATOL7234428E+02/, AS26884 900S0E+92/ : S18904022097E+02/, SOS649F09719E+027, 9291097 651L0E+02/, 7B0675734404E+02/, B48951552954E+02/, 9576660135508E+02/, O46868SS74657E+02/, 136528299499E+02/, 26534991 267E+02/, 5171788719695 +02 bf 40B8150588708E+02/, 499541180408E+02/ 5913542053695E+02/
3 FOR GAMMA(X)
(Sheet 24 of 25)
GLN66 GLN67 GLN468 GLN69 GLN7O GLN7i GLN72 GLN73 GLN74 GLN75 GLN76 6LN77 65LN78 GLN79 GLN8O GBLN81 GLN82 GLN83S GLNB4 GLNS85 GLN86 GLN87 GLN8B8 GLN89 GLNIO GLNI1 GLN92 GLN93 GLN94 GLNIS GLN9S GLN97 GLN9S GLN99 GLNLOG GLNIO1 GLNI62 GLNIOS GLNIO4 GLN1IOS GLN1OS GLN107 GLN168 GLN109 GLN110 GLNi1i GLNI12 GLNi13 BLNI14 GLN115 GLN116 SLN117 GLN118 GLN119 GLNi20
on]
lao)
oO
40
af
XX=2.4T
DO 24 1=1,NkR DGAMN=DGAN*X Xx XX=KX#1. GANLN=ALOGS(DGAN) RETURN
X.GT.xXLIM1
RX=1./%
RXX=RX#RX
IF((X-XLIM2).LT.0.) GO TO 41 PX=O(1) #RXX+O(2) GAMLN=PX#RX+(X-.5) #ALOG(X) -K+RTWPIL RETURN
X.LT.XLIM2
PX=P (1) SUM=(X-.5) #ALOG (X) -X DO 42 1=2,5 PX=PX#RXX+P (1) CONTINUE GAMLN=PX#RX+SUM+RTWEIL RETURN
TABLE LOOK UP FOR INTEGER ARGUMENTS LESS THAN OR EQUAL 100.
IF(NDX.GT.100) GO TO 40 GAMLN=GLN(NDX)
RETURN
END
Figure A3. (Sheet 25 of 25)
A29
GLNL21 GLNL22 GLNi23 GLNL24 GLNI2S GLNI2 GLNi27 GLN128 GLNi29 GLN1 20 BLN13i GLN1S2 GLNI33 GSLN134 GLNISS GLH136 GLN137 GLN138 GLNI29 GLN140 SLN141 GLNI42 GLN143 GLNi44 GLNI45 GLN146 GLN147 GLN148 GLNi49 GLNLSO GLNIS1 GLNIS2
deh oa ; oe wig ih aa rT: e 2a pare me
a ant iaa ta ee
O50, Kal i ; racy i", a O: Lh al ft +“ ae
oar cfhis a ¥
tet “Ate 4 EF mney nnbhiok unite o) tants
2 “ oe eigen
eee a ua en ee 16 ‘ 7? ‘ AR
| Acre oe
28 - P| a 5 : :
a us : Shih) shyb3' #e nay’ np avemniin wpe dot Ue 08 sate es . - 5 ,
; = Ww y ae 4 a mite es reLeolorie Le — ee fut 7. a3. aaa, cana etd =
PRONE I Pe
a | teen ae es ae ee _ Risto sgtetienn . ae 4) ub j ; 7 ; : aie — : a : - Fr : Fonsio| ' Ee
7 - grits”
7 et et we - wig ~ cas LF acwie HAS Ty.
- Fie ©
: ity |
ie BHR ie
a) we
ire, i a io >.
APPENDIX B: NOTATION
Incident wave amplitude Softy
Gravitational acceleration
Water depth
v=1
Bessel function of the first kind
Wave number
Non-negative integers
Polar coordinate
Temporal coordinate
Horizontal coordinate
Horizontal coordinate
Bessel function of the second kind Vertical coordinate
Incident wave angle
Free surface displacement
Polar coordinate
Angle related to wedge angle
Value related to wedge angle
Velocity potential function
Horizontal component of the velocity potential function Defined in Equation 12
Incident wave velocity potential function Reflected wave velocity potential function
Scattered wave velocity potential function
Finite cosine transform of ¢
Wave radian frequency
Bl