High-frequency scattering by a convex smooth object

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HIGH-FREQUENCY SCATTERING

BY A CONVEX SMOOTH OBJECT

HIGH-FREQUENCY SCATTERING

BY A CONVEX SMOOTH OBJECT

PROEFSCHRIFT

TER VERKRIJGING VAN DE GRAAD VAN DOCTOR IN DE TECHNISCHE WETENSCHAPPEN AAN DE TECHNISCHE HOGESCHOOL DELFT, OP GEZAG VAN DE RECTOR MAGNIFICUS DR. IR. C. J. D. M. VERHAGEN, HOOGLERAAR IN DE AFDELING DER TECHNISCHE NATUURKUNDE, VOOR EEN COMMISSIE UIT DE SENAAT TE VERDEDIGEN

OP WOENSDAG 9 OKTOBER 1968 TE 16 UUR DOOR

ADRIANUS JURIANUS HERMANS

wiskundig ingenieur

geboren te Leiden

%^ ^irt' 'V'

UITGEVERIJ WALTMAN - DELFT

DIT PROEFSCHRIFT IS GOEDGEKEURD DOOR DE PROMOTOR PROF. DR. R. TIMMAN.

Aan mijn ouders en Rientje

C O N T E N T S

Introduction 9 CHAPTER I SCATTERING OF A PLANE WAVE BY A SPHERE

1 Formulation 11 2 Asymptotic expansion of the diffracted wave 14

3 Boundary layer expansion near the sphere and the shadow

boundary ~ 18 4 Asymptotic expansion near the sphere at finite distance from

the shadow boundary 22 5 Asymptotic expansion at finite distance to the sphere . . . . 25

6 Asymptotic expansion near the axis of symmetry 25 7 Final asymptotic solution in the shadow region 29

8 Asymptotic behaviour in the lit region 34 CHAPTER II SCATTERING OF A SPHERICAL WAVE BY A SPHERE

9 Asymptotic expansion in the shadow region 35 10 Asymptotic expansion in the lit region 45 CHAPTER III DIFFRACTION OF WAVES BY AN ARBITRARY SMOOTH

CONVEX OBJECT

11 Derivation of the asymptotic equation 46

12 The incident wave is plane 50 13 The incident wave is spherical 54 CHAPTER IV EXAMPLES

14 Circular cylinder with plane incident wave 57 15 Circular cylinder with spherical incident wave 60

Appendix 65 References . . . . - 69

C O N T E N T S

Introduction 9 CHAPTER I SCATTERING OF A PLANE WAVE BY A SPHERE

1 Formulation 11 2 Asymptotic expansion of the diffracted wave 14

3 Boundary layer expansion near the sphere and the shadow

boundary ' 18 4 Asymptotic expansion near the sphere at finite distance from

the shadow boundary 22 5 Asymptotic expansion at finite distance to the sphere . . . . 25

6 Asymptotic expansion near the axis of symmetry 25 7 Final asymptotic solution in the shadow region 29

8 Asymptotic behaviour in the lit region 34 CHAPTER II SCATTERING OF A SPHERICAL WAVE BY A SPHERE

9 Asymptotic expansion in the shadow region 35 10 Asymptotic expansion in the lit region 45 CHAPTER III DIFFRACTION OF WAVES BY AN ARBITRARY SMOOTH

CONVEX OBJECT

11 Derivation of the asymptotic equation 46

12 The incident wave is plane 50 13 The incident wave is spherical 54 CHAPTER IV EXAMPLES

14 Circular cylinder with plane incident wave 57 15 Circular cylinder with spherical incident wave 60

Appendix 65 References 69

I N T R O D U C T I O N

HISTORICAL

The problem to be treated is the three-dimensional scattering of a scalar wave, by a totally reflecting smooth convex object, at high frequencies.

In this thesis we apply the ray method of J. B. KELLER [6-11] to solve this problem. This method, based on the ray-concept, is combined with the method of boundary-layer expansions. The reasoning being essentially physical in nature makes these methods more advantageous in comparison with other methods. Although exact proofs are not available, the correctness of this theory is not seriously questioned and it is believed that it yields asymptotic representations, valid for short wavelength, of exact solutions of the Helmholtz equation. The evidence for this comes from exact solutions, valid for a few simple geometrical configurations, e.g. the two-dimensional circular cylinder and the sphere, for which short-wave expansions can be found rigorously. A disadvantage of the treatment given by J. B. KELLER [6-11] and E. ZAUDERER [12] a.o. is that in order to find asymptotic solutions these authors always use results which are obtained by other methods, such as results of the treatment of the exact solution by means of the Watson transformation. The aim of the author is to derive systematically asymptotic solutions of the Helmholtz equation, by means of the ray method and boundary layer expansions, without using any result obtained by other methods. Asymptotic results are derived by solving asymptotic equations, which have different forms in different regions, with boundary conditions, on the object and at infinity and matching conditions which follow from results, obtained in neighbouring regions.

V. A. FocK [16] in his two-dimensional treatment of short-wave diffraction by a convex cylinder used the ray method and boundary-layer expansions as well.

A very extensive treatment of the asymptotic solution of the Helmholtz equation has recently been given by H. M. NUSSENZVEIG [1]. He worked out the problem of diffraction of a plane wave by a sphere, for large values of frequency, starting with the known exact solution. NUSSENZVEIG reformulated Watson's transformation and gave expansions for high frequencies in the whole space.

An imitation of this treatment has been developed by D. LUDWIG [3], which yields the asymptotic solution in the space around a smooth convex object of arbitrary shape. However, it is a purely mathematical treatment without physical backgrounds and in the author's opinion the ray theory is preferable.

Recently a method of tackling the asymptotic solution of diffraction problems has been proposed by D. S. JONES [4, 5]. He considered the problem of the diffraction of a wave at high frequencies by a circular disc. To solve this problem he derived an integral equation and found some asymptotic solutions.

His method is applicable, in principle, to the diffraction problems which we will consider here. However, it is not easy to solve the integral equation which arises in the three-dimensional scattering problem. We therefore apply the ray theory.

INTRODUCTION TO THE PROBLEM

As mentioned before, the problem of scattering of a scalar wave by a totally reflecting smooth object at high frequencies will be considered. Hence we are interested in solutions of the Helmholtz equation

A(p + k^ip = 0

with {ka) > 1 where a is a measure for the radius of curvature of the object. In the subsequent sections it seems that we even should take {ka)^ > 1. The boundary con-dition on the object is ^ = 0.

The wave function ip can be interpreted either as the velocity potential of sound waves corresponding to an acoustically soft object, or as the Schrödinger wave function, in nonrelativistic quantum mechanics [18], in which case it corresponds to a hard-core potential. The theory of material waves leads unambiguously to the wave equation

i f i ^ = - - - — ^ ^ + U { x , y , z ) ^

with U(x,y,z) as the potential energy of the particle in the external field. We consider the stationary case and with E = p^jlm the equation becomes

-j^Mp + {E-U{x,y,z)](p = 0.

Hence for the movement of a free particle we arrive at the Helmholtz equation

Mp + k^ip = 0

where k^ = E-(2m/fi^).

The object we are dealing with is impenetrable, hence we have ^ = 0 as a boundary condition.

There is no difficulty in extending the treatment to a vector wave field, so as to represent electromagnetic scattering from a perfectly conducting object in a homogen-eous medium.

CHAPTER I

SCATTERING OF A PLANE WAVE BY A SPHERE

1 FORMULATION

The scattering of a scalar plane wave by a totally reflecting smooth object in a homo-geneous medium will be treated here. We are interested in the asymptotic behaviour of the wave function for large values of the frequency. The scalar function (p has to be a solution of the three-dimensional wave equation

A(p-l-/c^(^ = 0 (1,1) especially with large values of k{ka > 1).

This function cp must be equal to zero on the surface of the object and satisfy the radiation condition at infinity. This condition will be considered in the following section.

In the first place we will consider the diffraction of a plane wave by a sphere with unit radius and its centre in the origin of the xyz space. The plane wave has the form

If the constant k is large (Ar » 1), we can use the 'ray' theory to find an asymptotic solution of the problem. Therefore we have to define rays first.

Rays are introduced by means of the following substitution

(p{x,k) = (?)(x,/c)e''''*"'> (1,2)

in which x = (x,y,z).

If we insert (1,2) in (1,1), we obtain

A(p + ik(2V(p-VS+(pAS) + k^ll-(yS)'-']f>= 0. (1,3)

If (p and 5 do not have large gradients, then for large values of k the k^ term is the leading one. Hence we put its coefficient equal to zero and we get

(ysf = 1. (1,4)

In geometrical optics this equation is the well-known eiconal equation. Its

^refl \4>id

Fig. la

If we introduce the arclength ^ along these characteristics and choose ^ positively in the direction of increasing S, then we get along these lines

S = S„ + é (1,5)

where SQ is an integration constant. We call these characteristics rays.

In geometrical optics rays are often introduced by means of the Fermat principle. In general we can easily verify [19] that in a nonhomogeneous medium the charac-teristics of the equation

(ysy = n\x)

in which «(x) is the refractive index, (we arrive at this eiconal equation, if we put in equation (1,1) k^n^{\) instead of ^^) are the curves between two fixed points Xj and Xj along which the optical path length

j n ( x ) d S c

is stationary with respect to small variations in the integration path C.

The case in question deals with a homogeneous medium with «(x) = 1, so the rays are straight lines.

The incoming rays of our problem are rays parallel to the x-axis with unit ampli-tude. If we put the sphere in this field, we find two different regions, viz. a lit region and a shadow region. In the lit region every point is reached by two rays, a direct incoming ray and a ray reflected by the sphere. In this way no rays will come into the shadow region. Hence we are dealing with a solution of (1,1) which is discon-tinuous along the shadow boundary and the assumption that the gradients are mod-erate, is violated, so the k^ term is not the leading one anymore. If we still require solutions of the form (1,2), we must remove the discontinuity.

We therefore define creeping rays [14].

Creeping rays are rays generated by that part of the incoming rays which meets the surface tangentially and follows the geodesies of the object in a direction which is the same as the incoming rays. Each point of these geodesies will generate a ray into free space tangentially to the geodesic line with a certain amplitude.

In the case discussed here the geodesies are circles in flat planes through the X-axis. As we see the geodesies of the sphere are closed curves and considering a geodesic in a certain plane, we observe that a point P of that plane is reached by an infinite set of rays, because the creeping rays circle round the geodesic many times. We also see that each geodesic is a line of creeping rays generated by two incident rays (Fig. lb). Hence a point P is reached by two infinite sets of rays and if P lies on the X-axis a cone of rays will reach this point, therefore special attention must be given to the x-axis.

These geometrical considerations yield the solution

in the lit region and

<P = (Pd (1,7)

in the shadow region, where (pi^^ and ^^efi are the incident and the reflected ray, respectively, ip^ and (p^ are the diffracted rays in the lit and the shadow region, re-spectively. These diffracted rays are caused by the creeping rays. We will see that for large values of k the influence of the diffracted rays in the lit region far from the shadow boundary is asymptotically small with respect to the incident and the re-flected ray.

2 ASYMPTOTIC EXPANSION OF THE DIFFRACTED WAVE

In this section we only consider that particular part of the field corresponding to the rays which are generated by the creeping rays. We call these rays diffracted rays.

To determine the wave function (p^ in the shadow region, we introduce new vari-ables ^, t] and V as shown in Fig. la, b, where

^—r] is the arclength along the ray after leaving the sphere; r\—\n is the arclength along the creeping ray;

V is the polar angle.

As a point is reached by two sets of different rays, we should consider that the co-ordinates are not single-valued. The rays at a point are determined by

where ^mi = <:, + 27tm, nmi = tli + 2nm, V, = v + nl 0 < rjiS In, m = 1,2,3,..., 0 < V ^ 71, / = 1,2.

If we summarize the contributions of all these rays, we get the total wave point {x,y,z)

It is evi

00 00

^ ( X , J ; , Z ) = X <P(<^ml,'/ml,Vl) + X 'P(fm2,>7m2,V2)-m=0 'P(fm2,>7m2,V2)-m=0

dent that in the new coordinates the line element ds becomes di" = d 4 ' + ((^ - r\fAr\^ + {sin t]+ {^-r{)co%t]Y dv' and equation (1,3), with S = £, — £,0 where £,Q is a constant, will be

where and 1 " (f „^^^P \ 1 ^ 1 d<p Ji d(p (C-f?)' f dri [8^ ^-r] ƒ ^ j f(i,ri) = sin>;-|-((^-»/)cosf/ (p = (joexp{i/c(^-^o)}-(2,1) function in a (2,2) (2,3) (2,4)

The boundary condition on the sphere will be

(p = 0, if •? = f/

and at infinity the solution must be of the form

cp = e'^'+giOf^ (2,4a)

where ^(0) is a finite amplitude function.

Before we give an asymptotic solution of equation (2,4), we will extend the class of solutions of equation (1,1). Instead of solutions of the form (1,2), we assume that

(p can be written in the form

(p (x, k) = (p (x, k) exp {ikS(x) ~ / c ^ x ) } . (2,5)

Inserting (2,5) in (1,1) yields

kXl-{VSf'\f-2ik'^'(pVS-V\l/ + ik[2V<p-YS + (pASl +

(2,6)

+ k^'(p{Vil/y~k'[2y(p-Yil/ + (pAi{>'] + A(p = 0.

We now have to determine the functions (p, S and i/^ and the constant i for large va-lues ofk. The term with the highest power of ^ in equation (2,6) depends on the vava-lues of the constant s. If k is large, we find an asymptotic solution of the wave function by making this term equal to zero.

From s > 1 follows that

(Wij/f = 0

and if ij/ has a real solution, we see that i/' is a constant and form (2,5) differs only a constant from form (1,2), so no more information is obtained.

Therefore s will be restricted to the range 0 < s < 1. In this case the first term of (2,6) contains the highest power of k. By making this term equal to zero, we get

1_(VS)'' = 0. (2,7) This equation is the same, when .y = 0 (1,4). After suppressing this term from

equa-tion (2,6), we notice that the second term contains the highest power of k. This leads to the equation

VS-Vi// = 0 (2,8)

The next term with the highest power of k will, in the case of ^ < j < 1, give

(y^f = 0

and as before this is not applicable for our purpose. So we stay within the range 0 < ^ ^ i . As we will see later, it is not necessary to consider s = j .

We now return to the newly introduced ray-coordinates {i,ri) and a solution of (2,7) will be

s = i-io (2,9)

where ^o is a constant, and from (2,8) follows that \p is a function of the variable r] only.

In the next sections we will try to find a solution of the form

(p(^,r,) = <p(^,,/)exp{ife({-^o)-fcV('?)}

with 0 < J < i and

n = 0

where KQ and p are yet to be determined.

If we write (pi^,ri) in this form, we see that (po(^,ri) must be a solution of

2 ^ ° + - ^ + ^ ^ 0 = 0 (2.10)

and we find recurrence relations between the other (p„, if we take each power of k equal to zero. These recurrence relations depend on p and s, so we have to find these constants first. A solution of (2,10) has the form

©0 = r = ;• (2,11)

{{^-rOKLrOf [(^-»/){sin^ + (^-^)cosf;}]*

So we expect ip to be asymptotically of the form

~ _ A(rj)k"'cxp{ikii-io)-k'f(r,)}^ ^2,i2) [_iè-ri){smr] + (^-ti)cosri}'\^

There are some constants and unknown functions in (2,12) to be determined, viz.

This is not possible with the boundary condition which requires cp = 0 on the sphere, where ^ = ?/. As we see cp^^ is singular at this line. We also have a condition at infinity where the radiation condition has to be satisfied. This cannot be done, because near the shadow boundary (/j = ^n) and large values of ^—rj

^Mexp{ik(i-io)}

and off the shadow boundary

N{ri)exp{ikii-io)}

(Pa

^ - n

Therefore (2,12) cannot be a solution of our problem, but it leads us to the method which yields the right solution. We will give an asymptotic solution by means of boundary layer expansions in the neighbourhood of the sphere and the shadow boundary.

3 BOUNDARY LAYER EXPANSION NEAR THE SPHERE AND THE SHADOW BOUNDARY

We will try to find a solution in the region ^—ri x 0 and r] x^n. To find a solution of the Helmholtz equation for large values of k in this region, we introduce new variables a and jS as follows

(3,1)

« = r U - 2 ) >

where 1 is a positive constant and )8 < a < 00,

0 < ;8 < 00.

We first consider the shadow region where P is positive and ip — (pj (1,7) and we expect the solution to be of the form

If we put this in the differential equation with the new coordinates (cc,P), we get the equation

iÜA(a_^)^+i^lA_^_^+fc^^^_^ +

x-pd(x 5a ix-pdPoL-p dp ƒ d(x (3,3) + J^^-^^-^+ik\2k^^-^ + k^^+licpX^0 oi-Pf dp \ da a-p ƒ J

where /(ij,f/) = sin»;-f-((^ —f/)cosf;.

We want to find a solution of equation (3,3) for large values of k and expect to get this asymptotic solution of (pi by equating the term with the highest power of k to zero. As we see we have to distinguish several cases dependent on the value of A. 1. If A < ^, the main contribution to the asymptotic expansion will be given by the

equation

2 ^Vlas , 9las ^ Q da a — P

This equation has a solution which is singular, if a—P tends to zero, however, we need a regular solution and obviously this cannot be found in this case. 2. We now consider the case A = ^. If we put this in equation (3,3) and equate the

term with the highest power of k to zero, the asymptotic solution (p^Q will be found by solving the equation

1 d 1 ö„.o ^ 2 , - | ^ + . - ^ = 0 (3,4)

a-Pdpa-P dp da a-p

with the boundary condition

(Pio = 0, if a = ^

and the radiation condition at infinity.

Strictly speaking the asymptotic solution of (3,3) has the following form

<Pi = r'f/c-"'>!„. (3,5)

If we substitute this expansion into (3,3) and equate the coefficients of each power of A: to zero, we obtain a set of recurrence relations of <pj„ and (pio must be a solution of (3,4). However, the general equation for (pi„ is rather complicated and difficult to solve, so we consider only the first term of (3,5) and assume that this will give an asymptotic expansion of tp, for large values of k. To solve (3,4) we introduce a new function / and new variables p and q as follows

•i(a-py <Pio = z e x p '

\2 '^ \ n: ,.\2

p = 2-*(a-py = ( - 1 ( ^ - ^ ) ^ (3,6)

If we put this in equation (3,4), we find that x must be a solution of the following parabolic differential equation

Pi + iT^ + PX = 0 (3.7)

op cq

with boundary condition

X = 0, if p = 0. (3,8)

However, this function is valid in the region near the sphere and the shadow-boundary, hence the radiation condition need not be satisfied by x- We therefore only require that X be zero on the sphere, where p = 0.

We suppose the solution of (3,7) is of the form

X = iFip,t)e"-dt (3,9)

c

where C is a contour which will be discussed later on.

If we put (3,9) in equation (3,7), we get the following differential equation for

F{p,t)

F,,-(t-p)F = 0 (3,10)

with the boundary condition

A solution of (3,10) which fulfils the boundary condition can be given as a combina-tion of Airy funccombina-tions

F(p,t) = Aiw.it-p)- H'2(0

Wl(l) WiO-p)>,

^ \n.

where Wi(0 = e 3 \U-tf H['^

<^(-0*\-HI^^ is a Hankel function of the first kind and order 3 and ^2(0 is the complex

con-jugate function of Wi(t). We have ^1(0 = u(t)+iv(t),

W2(t) = uit)-iv(t).

If we put these functions in (3,9), we get the solution

X = A

e'">lw,it-p)-w,(0 w,(r-p)U( (3,11)

The contour C will be taken around all singularities of the integrand. The only singularities are the zeros of the function Wi{t), which are points on the line t = ge^"' for real positive g. We define C as the contour which runs from 00 e*"' over 0 to 00. This contour can be closed at infinity in the half plane, where the imaginary part of

t is positive. The integral along this closing line is equal to zero.

Ret

The constant A follows from the condition at infinity (or finite distance to the sphere). In the nearby region we have found an asymptotic solution of (2,4)

q>, X k"Aexpi-lipi){e"''L,{t-p) - ^w,{t-p)\dt (3,12)

with p = (fj\è-rif,

,-m^-i

and on the shadow boundary with ^ = ^n: we have p = q^. At finite distance we require that on the shadow boundary

Vd = ^ino (3,13)

because this solution ip^ is a continuation of the asymptotic solution in the lit region. If we take /-j = 0 and A = /747r^, then (3,12) is the solution which meets condition (3,13) (Appendix).

If we expand (3,12) for large values of p and q, we use the method of steepest de-scent and we will see that the first part of the integral yields the incoming wave and the second part the reflected wave, which is zero on the shadow boundary [16].

In this way we have found near the object and the shadow boundary the asymp-totic solution of the Helmholtz equation of the form

^'' ~ j-^^^py'^U ~^) - ^ip^

e"'{w,{t-p)-^w,(t-p)}dt. (3,14) J<,)...r. „^ *^2(0.

By means of this solution we will construct an asymptotic solution in the whole shadow region.

4 ASYMPTOTIC EXPANSION NEAR THE SPHERE AT FINITE DISTANCE FRO\t THE SHADOV/ BOUNDARY

It is obvious that in this region we must find a solution which is an asymptotic solution of (1,1) and a continuation of (3,14). We therefore evaluate (3,14) for finite values of fj—^7r and small values of ^ — rj or in the coordinates/? and q : p—q^ = 0(1).

In this region we expand the integral of (3,14) as a sum of residues because the zeros /j of Wi(t) are the only singularities of the integrand.

We have

ts = Q^e^

with J = 0 , 1 , . . .

These zeros are simple zeros, so we get the following result

2 ( V 2 / j5=o W i ( g

and with the relation

M';(Ow2(o-wi(owi(o= - 2 i

which is the Wronski determinant of w^^t) and Wjit), we arrive at the folloyving expansion

<p,x- ,-7r*exp|,-fcf^ - ^ ) - , | P 4 £ e " - f 4 7 ^ (4-1)

bearing in mind that ^1(^5) = 0.

If we expand this solution at finite distance to the sphere, which corresponds to large values otp, we get with the help of the asymptotic expansion of the Airy function

w . ( r . - p ) « l e x p | / ( | p * - / . p * + 1 ^ + ^ ) 1 (4,2)

the following result

This solution is only valid at finite distance to the sphere and rj—^n large enough for the series to be rapidly convergent.

At large distance to the sphere we notice that this solution is not the right asymp-totic expansion, because it does not satisfy the radiation condition. This will be worked out later on. We expect (4,3) to be an asymptotic solution near the sphere at finite distance from the shadow boundary, as well.

To show this we suppose that at finite distance from the shadow boundary near the object the function ip^ is a continuation of (4,3) and for briefness' sake we consider one term t/, of the series

<?« = Z^exp U^^tsri + m\ t/, (^, ri • k) (4,4) and suppose that each term of the series should be a solution of the Helmholtz

equation.

We follow the same procedure as before and it is clear that we have to stretch the coordinates by

and

n = 0

The first term of the series will yield us the asymptotic expansion

Us ~ k^'UoAy,ri)

and after substitution in the Helmholtz equation and equating the coefficient of the highest power of k to zero, we find the equation for f/ps ^^'^ the boundary condition on the sphere

V 4 ^ + '-^-|-i-?!i- + 2,Uu„.fe-4 + 4 = 0 (4,5)

y' dy^ dy I 7' r ' J l y ' y ' y\

with Uos = 0, if y = 0.

Introducing the variable p = 2~*y^, used previously, we find

Uos = exp{i(t,y-iy^)}w,(«,-p)G(>;,s) (4,6)

where the function G{ri,s) is the integration constant which should be determined by the requirement that this solution is a continuation of the solution in the corner of the sphere and the shadow boundary, so, if we put (4,6) in (4,4), we have to match this one with (4,1).

This matching is easily carried out when we transform the asymptotic solution

ip, « k''tcxpL,(^% + ik^ + i(tj-^y')\w^{t,-p)G(n,s)

on the same coordinates used in form (4,1). We find

The function g{t]) and the exponent /-j must still be determined. From the matching condition follows that

lim g(ri) = 1, r2 = 0. (4,8)

In the following section we will see that this function g(ri) can be determined by matching (4,7) and the solution at finite distance to the sphere.

5 ASYMPTOTIC EXPANSION AT FINITE DISTANCE TO THE SPHERE

In section 2 we discussed the asymptotic expansion of the field in the region where the distance to the sphere was finite (2,12). We compare this solution with (4,3) and the expansion for large values ofp of (4,7) and match these solutions by pointing out that in the region where ^—rj is small but finite (2,12) and (4,3) are valid. Hence at finite distance to the sphere we assume the solution to be

[(^ - r]) {sm n+i^-ri) cos ^}]* A J {w\iO}^

This solution is a continuation of (4,3), valid in the region rj x \n, which we con-tinuate in the region where r\~\n = 0(1) and it corresponds to the form required by (2,12). Matching (5,1) and (4,7), we find

gin) = 1. (5,2)

Remembering that t^, the zeros of ^1(0, are points on the line / = ge^"' in the complex /-plane, we notice that the series of (5,1) is convergent, if rj—^n > 0.

In the region ri x ^n the series converges slowly and therefore we take, instead of this series representation, the solution in integral form analogous to (3,14). This is done in section 7, where we give the final solution.

6 ASYMPTOTIC EXPANSION NEAR THE AXIS OF SYMMETRY

Solution (5,1) is infinite on the axis of symmetry which is the line sin rj + (i — rj) cos rj = 0

and obviously the assumption of section 1, that the gradients are moderate is violated, so we have to look for an other solution in the neighbourhood of that line. In the

gen-eral three-dimensional case we will not find a line of singularities, but a surface, which we call the caustic surface. In our problem this surface is degenerated to the axis of symmetry.

In the neigbourhood of this axis of symmetry we have sin>/ + {^ — r])cosr] x 0

and we try to find an asymptotic solution by introducing the new variable <5 and C as follows

sin;j -I- (^ —^/)cos^/ = k~''ö

(6,1)

ï-r, = C

where p is a positive constant which will be determined by the stretching condition. From the form of the solution in the outside region we expect the solution of the form

ip, = f cxpU^ + i('^\2 VM,ri;k) (6,2)

where U^ is asymptotically

00

UM,r}\k) = k''Y.U„s{^,ri)k- (6,3)

In the new variables we get the equation for U^{^,t];k) as a function of C and ö

i{''^'^^4^T^{i''^^^^

dU^ dU, _do + _dC

+

+ vi -'{ •^] h+ k''Csmri-^ + C V 2 / ' ' 'dó dC

If - A V ,r ,u, • SU, dU,

+

+

k''cosr], ' K'^COS»?

dU, dU,\ k''sinn + ^^\+ „. X

do dC

_Co

(6.4)

-i(-J r,l/, + n s m ; ? ^ + ^ y +

.. 1^1 ,u dU^ dU,\ I/, fc"cosn,, ,

The stretching condition implies that 1 + p = 2p, from which follows that p If we put (6,3) in equation (6,4) and equate the term with the highest power

= 1. of A; to zero, we get the first term of the approximation as a solution of the following equation

^\os^lJUp.^ dUp^^lcosr, dö^ 5 dö dö ó

w i t h PO*" M — i n H p n p n H p n t n t n

W i l l i ^KJa fl ' l l l U C l J d l U w l l l UI u.

(i+c¥

The solution of (6,5) can be obtained by means of the substitution Uos = exp{ — iöcostj)F

where F is a solution of the Bessel equation

Fsö + IFS + cos^nF = 0.

In the region ^ > 0 we take the solution which meets the radiation condition

Ups = H,{C)expi-iöcosn)H''o^\öcosri)

and if we put it in equation (6,2), we get the asymptotic solution for ip,

ip, X /c^^^exp<^i/c^ + i( ]t^r,- iöcosr,}HXOHi'\öcosri).

(6,5)

(6,6)

(6,7)

(6,8)

The unknown constant r-^ and the function HJX) will be determined by matching this solution and the expansion near the sphere or the far field.

For large values of ö we obtain from the asymptotic expression for Ho^\ö cosri) the solution in ^, rj coordinates

iPd ~ fc'

I

_7t/c{sin>/ -1- (i —r])cosri}cosr]

X iexpji/c^ + j U j t^n-'^lnxi-n)

Comparing these two solutions we find r^ = 7.

Fisii-i) = -r: , ,, V.2 ^ (s'"'? cosf?)*.

v ^ - ^ {w;(g}

Hence we arrived at the solution near the axis of symmetry of the form ^/c\* n [., (, Tt^

(pd - e x p < i k l ^ I — i7c[sin>/ + (^ — f/)cos?;] cos>; V (sin»; cos>7)*x

2/ y/Q-n [ \ 2/ J (6,9)

f.AV

f A]

f/[,'^{A:[sin>; + (<^->;)cos»7]cosj/} \ e ' ' P i ' l 2 ) * 4 ' ' ~ 2

{w;(g}^

This solution is still incomplete. Moreover it is singular on the line sin»; + (ij—/;) cos»; = 0.

We obtain the total solution by adding the contribution of the ray generated by the tangent ray on the other side of the sphere. As we will see a combination of the form

exp (— iö cos»;) HÓ' \ö cos rj) + exp (iö cos t]) H^p^\ö cos rj) will be obtained instead of

exp (— (5 cos»;) ƒƒ Ó' '(<5 cos»;)

which yields a finite result, if ö tends to zero.

Solution (6,9) is singular on the sphere i = rj also. Hence we determine a solution on the axis of symmetry near the sphere. This solution must be a continuation of the one near the sphere and it is obvious that an asymptotic expansion in this region will be obtained by stretching ^—ri also.

However, we will compare the two solutions and in this way arrive at the solution near the sphere and the axis of symmetry.

If we use the p,q coordinates defined in section 3 and compare (6,9) with (5,1) and (4,1) we get

ip, X ncxplikU - I ) - ik[sinri + (^-»;)cos»;]cos»; - i^p^ ~ ^{ ^

(6,10) X H''p\k[sinri -|-(,J-»;)cos»;]cos»;}(sin»; cos»;)^ ^ ^ " ' " ^ ^ ^ — T '

7 FINAL ASYMPTOTIC SOLUTION IN THE SHADOW REGION

As we already mentioned the final solution in the shadow region will be obtained by putting the contributions of all possible rays together. However, it is clear that the parts which are the results from the first round of the creeping rays, are the main contribution to the field. If possible we add up all rays, otherwise we deal with the main part. We consider five regions (Fig. 3.).

Fig. 3

/ The region where

f/l > T , ^1 > ' ? ! .

'?2 > 2 ' ^2 > '?2

and sin>;,,2+('^,,2-'7i.2)cos»;,,2 = 0(1).

This latter condition states that the distance from the axis is finite. In section 5 we dealt with this matter. The final solution will be built up by solutions of the form (5,1), where we take

^ = ^i+2nm,

r\ = »;, + 27r»n

and

1 — il2+2nm, (m = O, 1, 2, ...)

respectively, and sum over m which is quite simple,

(Pd. - j n^cxplikUi-^j-^U'm^rii {({i-'/i)[sin>7i-F({i->/i)cos»;i]}* e x p ^ , ^ - ^ Uyr,,--s=0

{w\iQr

1 — exp < 2ikn + 2( (;r) t^n>

₊

(7,1) - j ; t * e x p j i / c K 2 - ^ j - ^ [ s i n * » ; 2 s = 0

{('^2-'72)[sin»;2 -I- (iJ2-'72)cos»;2]}*

1 — exp<2(

exp^i^y r . L - ^

kn + 2i[-\ t,n

This solution we can use in the region where the sum of the residues is finite.

II The region where

f/i « 2 ' ^1 ^ ' ' i '

^/2 > 2 ' ^2

»ni-The distance to the axis is finite. »ni-The solution again consists of two parts as in (7,1) and again we take

(^ = c,i+2nm,

ri = riy+2nm

and

^ = i2 + 2nm,

respectively. However, only w = 0 will give here an important contribution, because, as we explained before, the first part (rj = »;,) of the solution tends to ^j„<. and the other parts are exponentially small because of the behaviour of t^. But for complete-ness' sake we will give the total solution. The contribution of»; = »/i, will be given in the integral representation and the remainder as a sum of residues and sum over m. So we get (Pd.wt = . e x p j j / c K i - - l - / j p , * ^ s i n * » ; 471* {sin»;i-(?,-»;,)cos»;i}*

X j e""|w2(f-p,) - M ) . w , ( ( - p . ) jdf +

/ c \ - '

n^expliki Q^ + 3n\ ni] . siWrji

{(4'i-';i)[sin»7i +(i^i-»;i)cos»/i]}* (7,2)

exp^/I^YJ»,, + ^'^

s = 0

Kits)}'

1 — exp<J2i/c7r + 2(( - ) t^n

₊

-j 7 c * e x p j i 7 c U 2 - ^ j - ^ U i n * » ; 2

{({2-'/2)[sin>;2 + iii-n 2) COST] 2]}^

s = 0

(^^Pi'[2)'s['l2 2

{<its)y

1 — exp\2ikn + 2i( - ) t^n

with Pi = ( J (^1-»/,)',

«1 = u ) ( ^ i - ' '

This solution is valid for large values of ii—ri^, hence p , and qi are very large and we may expand the integral part of the solution by means of the method of steepest descent. (Appendix).

Ill The region where

and

^1 « '?i , ^2 « 12

sin'7i,2 + ('^i,2-';i,2)';os»;, 2 = 0(1).

This is the region near the sphere in the neighbourhoud of the shadow boundary and at finite distance to this region. However, this region is bounded in the direction of the axis, because sin»; + (i^ —»;)cos»; must be finite.

Near the sphere we use solution (4,7) with the function gitf) given in (5,2). But this solution is only used in the region where the series are rapidly convergent, i.e. at finite distance from the shadow boundary. We use the same variable as in section 4.

A* i^-r])\

^-m-i

and we consider the following values of ^ and »;

S, = ^i + 2nm, r] = rji +2nm

and

c, = c,2 + 2nm, r] = r]2 + 2nin.

If we add up all possible modes, respectively, we get the following solution in the region of convergence of the series

<?d,toi = - ' 7 r * e x p j ( 7 c U i - ^ j - i f p f ^ X .iM,>Vi(t,-Pi) s = 0 _{<its)y i7t*exp|(feK2-^j-'fP2[ X Wlih-P2) 1 — exp<^2ikn + 2i( - | t^n

kV

₊

(7,3)

x l ^ '

wts)r

1 -exp<j2i/c7r+ 2(( - j t,n

Near the shadow boundary where »;i x \n we have the solution

'^''•- = 47i'''p|'K^'^~^)~'*^*

|.'-|w2(.-p,)-^fw,(.-p.)}d.

m * e x p j ( / c U i + y j - i f p f | x

+

x Z e ' s = 0 , . M . ' * ^ l ( ' s - / '

^'['

{w;(g}M ' ^2 1 — exp<2(k7r + 2/( - I t^n

+

- i 7 t * e x p j i / c K 2 - | j - i f p f y x with ^ y^.M.Wl(t.-P2) - 0 {wUO}^ k V / . 37r 2 ^ ' ^ ^ 1 — exp<(2(7c7r + 2/(

-*'•"}]"'

ƒ K TTze region near the axis of symmetry we discussed in section 6 We notice that

^. « ^2 = C

and

ni ~i2 = n and use the notation

Ö = fc{sin»; + (^ —»;)cos»;}.

We add up all possible modes and arrive at the solution <?d,.ot = U ) / / _ e x p j i f c ( c - | j U s i n » ; c o s » ; ) ' x

X (exp( —;5cos»;)//!,'*(<5cos>;)-H exp(i5cos»;)7fó^\(5cos»7)} x

1

s=0

exp<;iU) tiri

{^iQY

1 — cxp\2ikn + 2i

\sn]Y.

(7,4)

For small values of c—»; we have region V. With

m^-\

we get the solution

«Pd.tot = T^exp j (7cK - ^ j - ilp*Msin»; cos»;)* x

X {exp(-i(5cos»;)WÓ («^cos»;) + exp(i^cos»;)//ó ((3cos»;)} x (7,6)

X y g . M ^ * ' i ( « s - / ' )

{<its)Y

1 -exp<j2/k7r-l-2(( - j t,n

It can be made clear that (7,6) is an asymptotic solution of the Helmholtz equation, though we have derived (7,6) from the matching principle.

8 ASYMPTOTIC BEHAVIOUR IN THE LIT REGION

In the preceding sections we discussed the asymptotic solution of the Helmholtz equation in the shadow region of the sphere. We will now investigate the behaviour of ip in the lit region.

In the region far from the shadow boundary, in an arbitrary point P three kinds of rays will contribute to the asymptotic solution of the Helmholtz equation.

First of all we have the incoming and the reflected rays. The incoming ray is given and the reflected ray can be determined by laws of geometrical optics and is known [16]. A contribution is also given by the creeping waves which radiate the diffracted rays. Hence we get the following solution

<?tot = ^ I n c + ^refl + ^rf.tot ( 8 , 1 )

where i^j ,„( is given in (7,1).

We will meet a complication, if we take the point P near the axis of symmetry. It is obvious that instead of (7,1) we must take (7,5) or (7,6) as ^d,,„t.

CHAPTER II

S C A T T E R I N G OF A S P H E R I C A L WAVE

BY A S P H E R E

9 ASYMPTOTIC EXPANSION IN THE SHADOW REGION

We will now discuss the field of a spherical wave scattered by a sphere. As in the preceding sections we are particularly interested in the asymptotic behaviour of the wave function for large values of the frequency. The solution method to be used is the same as before, in the case of scattering of the plane wave.

We consider a sphere with unit radius and three-dimensional source at finite distance IFQI to its centre. Therefore the incoming wave is of the form

_ _ exp{i/c|r-ro|} '^'"' |r-roi

where r represents the position vector of the observation point.

We introduce the same coordinates ^,»; as in section 2 and assume that the in-coming rays are tangent to the surface of the sphere in »; = »;o = arccos {|ro|~^}, where the amplitude is equal to (tg r]p)^^.

The boundary condition implies that the wave function ^ is equal to zero. At infinity we require ip to satisfy the radiation condition. On the shadow boundary we require a continuous solution and therefore the solution of the lit region and the one in the shadow region must be equal. In the lit region we have a solution of the form

9 = (Pinc + <Pr^{l +

(Pd-On the shadow boundary we have ^^en = 0 and ip, asymptotically zero, so the solu-tion of the shadow region ip, becomes equal ipi„^ on the shadow boundary at finite distance from the sphere.

We determine the solution in the shadow region in the same way as we did in the 'plane' case. We start in the region near the sphere and near the shadow boundary, where i^—rj) x 0 and rj x rjp.

We introduce a = k^i^-rip), p = 0iti-np) with (9,1) P < a < CO, 0 < p < CO.

The power of k in (9,1) is determined in the same way as in section 3. We also in-troduce

00 n

ip, = exp {iki^-^p)]k'^ Z k~^^(Pn- (9,2)

n = 0

Only the first term of the series will be determined and must be a solution of the equation

^ « # « — . % + 2 , - ^ + i ^ = 0. (9,3)

a — P dp a-p vp da a-P

We follow the same procedure as in the previous chapter and introduce

\i{a-pf X = (po exp

p = 2-Ho^-pf = (^J(^-'?)^

q = 2-^a =n\\^-r,p) (9,4)

where the function x has to be a solution of the equation

^ + i ^ + PX = 0 (9,5) dp dq

with the boundary condition

X = 0, if ;7 = 0

and 'shadow' condition like we discussed earlier in this section. In the same way as before these considerations lead to the solution

<Po = e x p ( - i | p * ) j^e'"'Fit)Lit-p) --^w,it-p)\dt (9,6) where

Wi(0 = uit)+lvit)

so v{t) is the imaginary part of Wi{t) for real /. The contour C is the same as before. This solution satisfies the boundary condition and is valid near the sphere. The unknown function Fit) still has to be determined. This is done by the condition

Vd = <Pinc if '7 = '/o and ^ — r] being finite and the solution symmetric with respect to

the source and the observation point, if they are both at finite distance from the sphere. We therefore express the coordinates of the source in p and q coordinates

iPo,(lo)-If the distance to the sphere is finite, then p and q tend to infinity, so we must ob-tain the solution for large values of the coordinates p and q first. This solution results from (2,12) and has the form

<Pd

sxp{ik(i-io)}k''(po

{tg»; + (^-»;)}* (9,7)

Fig. 4

If»; tends to rjp we notice that the denominator is equal to |r—FQI* which is symmetric in r and FQ (Fig. 4).

If we choose F(t) equal to A cxp[—ltqp—i^pp^]w^(t—po), then in the limit case we get a symmetric function and we can make ip, equal t„ ip;„^ by the proper choice of

A and r^.

The asymptotic solution which we have to expand for large values ofp, q, pp, qp follows from (9,6) and (9,7)

^ ^ r ^ e x p { / / c ( ^ - ^ o ) - / | ( p ^ + pg)} ^, (DA '>-' ; — - — — ~ - X {tg»; + (^-»;)}* le"'^-^°'w,{t-pp)\vit-p)-^w,{l-p)\dt. c ( ^ïV) J (9,8)

finite distance to the sphere and we get by means of the method of steepest descent (Appendix) and putting »; = »;o

9d «

2Ak'^{fj *7t* exp |</c(c - ^o) + f

(Pin exp{ikii-ip)} tg'7o + '^-'?o' So we find 2 "e-j-'•5 = h A 271*

In this way we constructed a solution in the region near the sphere and the shadow boundary. We only need formula (9,8) in the region where p and q are of the same order as pp and qp. In the region near the sphere the order is different and we can expand (9,8) for large values ofpp, qp with p, q being finite.

We use the asymptotic expansion of Wi{t—pp) for large values of po which states

w,(^-Po) = P o - * e x p | / ( ^ | p o ^ - r p o n ^ + ^ ) | . (9.9)

Taking into consideration that

Po = [2) tg^'^o and pp > p,

we get the following expansion of the field

cxp{ikii-ip)-iip^} ^ <Pd

x

2[7rtg»;o{tg»; + (^-»;)}]*

f'"f^'-^^-J)^'^'-4''

This result is not only a solution in the region near the shadow boundary. We also use this one as a solution at finite distance from this boundary. This can be derived

in the same way as we did in section 4 and it is clear that we can take the sum of residues as a solution in this region

,5 ~ -7iiexp{("/c((^-Co)-'IP*} ^ [7itg»;o{tg»; + (^-f;)}]* X E s = 0 it,q Wjjt.-p)

WAts)}

2

We used the relation between vit) and wit) which resulted from the Wronski deter-minant

w\it)vit)-w,it)v'{t) = 1

and, in the zeroes t^ of the function H',(/) this leads to

vits) =

-TTT-Solution (9,11) is valid at small distance from the sphere. At finite distance we expand (9,11) for large values ofp which yields

K2)

^^Py^^^~^o)-j-ip, X -^^ ^ ^ X [7rtg»;o(^-»;){tg»; + (^-»;)}]*

r f^Y ^ (9,12)

00 e x p j ( ( - j t,iri-t]p)

"".io {w[it,)r

Bearing in mind the values of t^ on the line ge^"' we notice that for rather large »;—»;o this solution is asymptotically equal to zero. Actually we are now able to give the complete shadow solution but for the region near the caustic which in our case is the line tg»;+(ij —»/) = 0, viz. the axis of symmetry. The derivation in this region can be carried out in the same way as in section 6 by introducing the new variable

Ö = A-{sin»; + (1^—f;)cos»;}.

At finite distance from the sphere we arrive at a solution which is a continuation of (9,12)

(Pd

ni[ -\ cosrjexp{ikii^ — Qp) — iöcosI]}

{tg'7o(^-'7)}*

X H\,'\öcosr,)Y

exp<ii( 2) ^sin-rio)

{<its)y

(9,13)

Solution (9,13) is singular on the sphere ^ = »;. Hence we determine a solution on the axis of symmetry near the sphere. This solution must be a continuation of the one near the sphere and it is obvious that an asymptotic expansion in this region will be obtained by stretching ^—ri as well.

However, we will compare the two solutions (9,13) and (9,11) and in this way arrive at the solution near the sphere and the axis of symmetry

(pd

'k\^

-7t ( - j e* cos rj exp [ikii — ^p) — i^p* — id cos»;} tg*'7o

x H < ^ > ( 5 c o s » ; ) Ê e " - f 4 ^ .

(9,14)

In all these solutions we used the coordinates

.=m,-.?.

« = 1 2I

i^-lo)-Finally we can give the complete solution as follows / 77»e region where

li »'7o, ^1 > ' 7 i , '72 > lo, i2 > '72 and sin»;i,2 +('^i,2-'7i,2)cos»;i,2 = 0(1). We find (Pd.t

M2) ^^pykiii-^o)-^

[7rtg»;o(c,->;i){tg»;,+(êi-»;.)}]* 00 exp<i!i 2I hini~no) ^k\-i I—exp<2/c7ri+2|

fjt.ni^] V

+

^^2) ^ ^ P ] ' ' ' ( ^ 2 ~ ' ^ o ) - ^ ['rtg»;o(<^2-'72){tg'72 + (^2-'72)}]* kV X Z s = 0 exp<;i| 2I hil2-*]o)

WiiQY

l-exp-^2fc7ri-|-2l-j t,ni\ .

(9,15)

We use this solution in the region where the summation over s is rapidly convergent when t]i,2~no is finite. Obviously we must come to an other solution, if the summa-tion is not rapidly convergent. This is the case in region II.

II The region where

Il ~ '7o, <?i > ' 7 i ,

'72-In this region we give the solution of the ray which corresponds with <^i,»/i in integral form and the other contribution can be totalled up as before

- _ exp{i7c(.^i-{o)-/fp,^} 2[7rtg»;o{tg»;,+(^i->;i)}]*

-l^"''{<^-P^)-^-^(^-P^)}<^^ +

[;rtg»;o({i-»;,){tg»;i+(.^i-»;,)}]* „ e x p { / ( - ) f/»;,-»;o-h27r)j. p . . .^ -)-._, ^ Z - ^ ^ ; , . . . , 2 ^ [ l - e x p | 2 / c 7 r / + 2(^^jr,7./jj {w;(g}^

-I-+

7r(|j 'expj,-

fc(^2-^0)-TTI [TT tg»7o('^2 -'72){tg'72 + (1^2 - ' 7 2 ) } ] * X E exp<!i( 2 ) tsiil2-flo)} r

{w'i(g}' 1—exp <^2/c7ri-l-2|-j t,ni

(9,16)

For large values of pi and qi the integral in the formula must be changed in an in-tegral of the form (9,8) and the first part of the solution is as follows

- j exp\ikiii-ip)-iiiPi^ + Po^)-'^

2[7r{tg»;,+(^,-»;,)}]*

x ƒ e''<''-'°>wi(/- Po) | K ' - Pi) - ^ wi(r - p i ) | d/.

This solution is valid if qp > q^ and qp being large, if q^ tends to qp, we must expand the integral by means of steepest descent (Appendix) which leaves only

^ i n c

/ / / The region where

^i ~ rii, ^2 ~ '72

and

sin»;,,2 + ({i,2-'7i,2)cos»;,,2 = 0(1).

In this region ispo ^ P i , 2 , ?o ^ Q'1,2 and we get the result

-nicxp{ikiii-^p)-iipy} in tgrio tgfji)*

, le-^'^-^^^\l-exp\2kniJ^Ytjr

_ niexp{ikii^2-^o)-'iP2^} ^ intgr]ptgr]2)^

+

X Z JM2 Wlits-Pl) l-exp<{2/c7r(-l-2( - ) t,ni «=o {w'i(0}

Near the shadow boundary rji x rjp we have the solution exp{//c(^i-^o)-i|Pi*} ^ (Pd,i 2(7ctg»;o)* vit)

^ f ' ^ " " ^ ' - ^ ' ^ - ^ " ' ^ ' - ^ ' ' ^

^ -7riexp{i/c(.;i-.;o-H27r)-/|pi^} ^ (7ttg»;otg»;i)*

x Z

g.-,,,,' w , ( f , - p i ) {w;(g}' f / A i n - 1

1-exp<2A:7r/-l-2 - (JTTIJ-

-I-_^ - Tti exp {ikii2 - io) - iiP2^} ^

intgr]ptgri2)^ ir,,2 Wlits-P2) - 0 {w\itjy X y e { / A * )"l-i l-exp<^2/c7r/-|-2 - t,ni> (9,17) (9,18)

where

q\ ={~Xi^i-i;o+2n}.

All these solutions are valid, if the distance to the axis of symmetry (caustic) is finite.

IV The region where

^ , * ^ 2 = ^ ,

'7i * '?2 = '7

and Ö = ^:{sin»; + (i^—»;)cos»;} is finite.

We now make use of solution (9,13) which yields in region { » »;

'kV

nil - 1 cos»; (Pd.io, ~ 7 - ^ ^ T m exp{(fc(^-<Jo)} X {tg'7o(^-'7)}^ X (exp(—/^cos»;)//J,''(5cos»;)-t-exp(i(5cos»;)Hj,^*(^cos»7)} x

x Z

e x P i ' ( 2 ) '"il-lo) 1 — exp •< 2/c7r j-I-2|

^)'^]/

{w'lits)V

For small values of ^-rj we come to region V and the solution

7rl - I cos»; J- ,^ <Pd.to, ~ - - ^ expii/c((^-(^o)-'IP* + x r ^ tg^'7o I 4 J X {exp( —/(5cos»;)//^''(dcos»;)-1-exp(((5cos»;)//ó^'((5cos»;)} x

Z^-''

.M vv,(f,-p)

{<its)V L

1 — exp<2/c7ri + 2| f , 7 r / i .

This last formula completes the solution in the shadow region. We only treated the case where the source is at finite distance from the sphere. It is possible to arrive at solutions if tg»;o is small. This will cause practically no alteration in the formulas and they can easily be found. Attention should be paid to the fact that the region where p and pp are of the same order is quite near the sphere, hence we must take integrals of the form (9,8) instead of (9,10) and we must supply residue expansions of (9,8).

10 ASYMPTOTIC EXPANSION IN THE LIT REGION

In the lit region we get the same kind of solution as discussed in section 8. There is a region at finite distance from the shadow boundary where the field consists of a com-position of the incident, reflected and the diffracted rays.

So we get

<?d,tot ~ Vinc +9rcfI+ <Pd ( 1 0 , 1 )

CHAPTER III

D I F F R A C T I O N OF WAVES BY AN A R B I T R A R Y

SMOOTH CONVEX OBJECT

11 DERIVATION OF THE ASYMPTOTIC EQUATION

Here we investigate the three-dimensional diffraction of waves by a smooth convex object. Two cases are considered, viz. a plane and a spherical incident wave.

Although the object is the same, we must introduce a different coordinate system in both cases. We discuss only the field caused by the creeping rays. In section 1 we defined creeping rays as geodesic lines starting on the shadow boundary in the direction of the incident rays. We call these lines u^ = constant. The arclength along these rays is defined by M' and the shadow boundary is M' = h{u^). The coordinate lines, on the surface, M' = constant, are perpendicular to the geodesies. The surface is determined by the vector

x' = x'iu"), i = 1,2,3

a = 1,2 (11,1)

and the line element on the surface by

ds^ = du'" + gdu^^ (11,2)

with

where

Sij = ^kf—i—-,; 9ii = 1, fi'i2 = 0 and gfjz = g du du'

f l if fc = /

l o if Tc # / .

The diffracted rays are tangentially to the geodesies and therefore we introduce the new coordinates K \ M^ and u^ as follows

z\u\u\u^) = x'iu\u^) + (u^-u')—- (11,3)

where z' are the Cartesian coordinates of a point in space, which lies on a tangent to one of the geodesies considered.

In the M' coordinates we derive the Laplace operator A which has the form

A = G ' 4 - ^ . - r ? . A | (11,4)

Idu'du' 'du'\

G'^ is the contravariant metric tensor and ffj the Christoffel symbol of the second

kind.

First we derive the covariant metric tensor ^ , dz'dz' ^ii — "kl : ;• du' du' We find r / 3 K 2 . d'x' d'xJ G,, =iu -u) 5y — - , ~ - p , , ou du r f 3 U2j: S'X' a V ou du du .3 ..u^a , / 3 u2c <^^x' a V G22 = g + iu'-u\)-^ + (u'-u')%^^,

du' ^ ' "du'du^ du'du^

G i 3 = G23 = 0, (11,5)

G33 = 1.

From (11,2) follows that the unit vector »»;, normal to the surface, has the form

d^x' 1

«i (11,6)

du'' giu\u^)

with g{u',u^) as the radius of curvature of the geodesic u^ = constant. The deter-minant G of the metric tensor is

^ _ ( « ^ - M ' ) H _ , , ,dg

Q'

.3 . . h

{-'"'-"•>l?}^

4(1

, aV a V / - aV a V V

ig^ du'du^ du'du^ \ 5u'^ du'du^ and with (11,2) we find

We introduce R = 0:

d^x'

a V

" a w ' ^ a M ^ a w ^

and find

G^^=^[{g^Hu^-u')'£\\iu^-uWR^],

G''=-^{u'-uyR, 22 1 iu^-u'f (11,8) G g G'^ = G " = 0, G " = 1.

In section 1 we noticed the wave function being of the form

q>, = (p,cxp{ikiSpW)}

with Sp = constant.

Introducing the same substitution here, we obtain the following form of the Helm-holtz equation dg^ \du'du' ' du") 2^<Pd I (Pd . du' u'-u' (pd _du' du') = 0. (11,9) On the object we have the condition (p, = 0, at infinity we require the radiation condition and on the shadow boundary at finite distance from the object

(pd = ^ i n c

-An asymptotic solution of (11,9) can be found by equating the term with the highest power of k to zero. This leads to an ordinary differential equation in u' with solution

(Pd Fiu\u^)

where the function F{u',u^) only depends on u' and u^. Solution (11,10) is singular on the object where «^ = M' and on the surface where

3* + ( « ^ - « ' ) ^ = 0.

du'

This surface we call a caustic surface.

In the case of the sphere the caustic is reduced to the axis of symmetry. In the last chapter we treat two problems without a caustic. In principle our method makes it possible to give asymptotic solutions near the caustic as well. In this thesis we only consider solutions valid at finite distance from the caustic.

We now apply the method of boundary layer expansions to find a solution valid near the object. We therefore stretch the coordinates near the object and the shadow boundary. The shadow boundary on the object is the given line u' = hiu^). We in-troduce the new coordinates

a = ^iu'-hiu"-)},

p = k^{u'-hiu^)}.

The power ^ of ^ follows from the same reasoning as in the case of the sphere. Putting this in equation (11,9) and equating the coefficient of the highest power of k to zero we get a differential equation for the asymptotic solution

g^ a 1 d^^2i^-^ + ^ = 0 (11,11) a-P dp a-p dp da a-p

where we take for g the value of the radius of curvature on the shadow boundary on the surface g = g{hiu'^),u^}.

Introducing the coordinates

p = 2-*g-Hoi-p)\ q — 2"*g~*a

and the new function x

X = (p,expiij;p^)

we arrive at the equation for x

~j+'^ + P^ = ^ ( l l ' i 2 ) dp dq

The solution of this equation depends on ipi„^. We therefore consider different forms of the incident wave.

12 THE INCIDENT WAVE IS PLANE

Dealing with a plane incident wave we obtain a solution of (11,12) of the form

X = A j e " 4 w 2 ( r - p ) - ^ w , ( r - p ) | d t (12,1)

where the contour C is the same as before. This function x determines the asymptotic solution near the shadow boundary and the object. To find the constant A we use the condition on the shadow boundary, which states that at finite distance from the object the asymptotic solution tends to ipi„^. Therefore we continue (12,1) in that region with the help of (11.10) and get the solution

We first determine the value of

_ gHKu')^}

' ' - dj^ •

du'

In general, for a point source all tangent rays on the shadow boundary M' = hiu^) are generated by this source. Hence we give the source x' as a point on a tangent line

fix'

x' = x'{hiu\u'}-riu')^. du

The condition that 3c' is in fact a fixed point for all rays, gives dx' = 0

and therefore differentiation with respect to u^ gives

fdx' ^ dx'\ fd^x' a V \ dx' ^ \du' du") \du'' du'duV du'

Multiplication with dx'/du^ and summing up over / leads to

r = 4

aw*

where we used the properties of the metric tensor of the surface. Therefore ^i is the distance from a point on the shadow boundary on the surface to the source.

In this case we are dealing with a plane incident wave, hence

01 = 00.

This leads to

du'

on the shadow boundary. For large values of p we get 47t*/lexp{i/c(So + u^)}

«Pine =

g'

We determine ^o in such a way that on the shadow boundary ^inc = exp{ik{So+u^)}.

This can always be done. Hence we have

^'^y

and the asymptotic solution near the shadow boundary is

- _ i ^iCxp{ikiSp + u')-iip^}

xie"^\w2it-p)-^w,it-p)yt

where

P = [2]Q~HHu'),u'}iu'-u'r,

ffg-*{hiu')y}lu'-hiu')l

Writing (12,2) as a sum of residues we find

r„ ^ • i^iexp{/fc(So + M')-(|p*} (Pd ~ —in

{...(-'-»-)|^f

X y ^ i M W i ( f s - p )

- 0 {w;(0}^ bearing in mind that Wi{tJ = 0.

At finite distance from the object we expand the Airy function for large values ofp and we get

'^) \'gHhiu'W}g^

(12,3) (Pd « F—"-^ ? r ^ T ^ exp j i/c(So + «^) + ' ^ j - X | ^ ( „ 3 _ „ i ) | ^ i + ( „ 3 _ „ . ) | ! e x p | i Y ^ Y e - * { / i ( w ' ) , u ' } f s [ « ' - / < « ' ) ] | 2 ^=0 {w'i(g} This solution is valid near the shadow boundary.

Because of the exponential behaviour of solution (12,3) we assume, that at finite distance from this shadow boundary the exponential behaviour of the solution is [12]

exp|i/c(So + u') + ( / - j t, ]^g-\u',u^)du'

we are able to derive a solution in that region. We introduce the variable y = Q~*k^iu' — u')

and the solution of the form

Again we assume that each t//^ can be expanded as an asymptotic series of the form

oQ n

ilj, = k'^Y.k~Hns- (12,5)

If we put this in equation (11,9) and equate the coefficient of the highest power of k to zero we get the ordinary differential equation for cpp^

it, tf i

1 d'll^o a^os I 1 2it,

y y y)

with the solution

lAos = exp{/(f,y-iy')}Wl(f,-p)iƒ(«^u^s) where

P = 2 - ' y ^

The asymptotic solution we have found has the form

ip, X W' Z exp<^«/c(So + «') + / U /, J g-\u',u^)du' +

+ ' ( f . y - i y ' ) [ w i ( < , - p ) H ( u \ u ' , s ) .

We continue this solution at finite distance from the object and match it with (12,3). This matching leads to Hiu',u^,s) and r^. After some calculations we find, at finite distance from the object,

(Pd^ P 7 t V ( " ' , " V ' du -cxp\ikiSp + u') + —\ X (12,6) X Z s = 0

and near the object

exp\i[^\\,''i e-\u',u')du'

h(u^

{<its)V

(Pd n^ghxpjikjSp + u')-ifp*} ^ ^,„wjit^-p)

g^Hu^-u')'-^'' du

Ze-- 0 {w\it,)y

with

P = (fjQ-Hu'-u')\

q = (^^'i^U-Hu',u')du' + g-\u'-u')^.

All these solutions are singular in the points

3* + ( u ^ - u ' ) ^ = 0.

du'^

Therefore all the derived solutions are valid at finite distance from these singularities which we call the caustic points. It is possible, in principle, to apply the same method of boundary layer expansions to construct an asymptotic solution near these caustics D. LuDWiG [2] discussed the behaviour of the field near the caustic in the general case. We therefore give no expansion by means of the boundary layer theory.

We are now able to give the complete solution in the shadow region at finite distance from the caustic. We must sum up all ray contributions. This is a geometrical problem which will not be solved here.

The solution in the lit region can be found in the same way as in the case of diffrac-tion by a sphere. At finite distance from the shadow boundary we get the incident wave, the reflected wave and the diffracted wave. The latter is generated by the creeping rays and has been discussed before.

13 THE INCIDENT WAVE IS SPHERICAL

Remembering that the geometry which we consider depends on the incident wave, it is obvious, though the object is the same as the one of section 12, that the coordinate system which we introduce here is different. However, we use the same notation.

We now take as a solution of (11,9) the form

X = ^ | e ' " « - ' " ' > W i ( ( - p o ) ( f ( « - p ) - ^ W i ( r - p ) ' ) d f

where Pp,qp are the p,q coordinates of the source. The shadow boundary on the object is called u' = hiu^) and as we mentioned in section 12

dg^ du'

is the distance from a point on the shadow boundary {hiu\u^} to the source. We take the origin of «^ in the source, so 5o = 0.

Along the diffracted ray the arclength is u'—u'. We can derive the function

u' = hiu^) by eliminating M^ from

u'=^

As a solution in the shadow region near the shadow boundary we have /l/c"exp{iku^-jf(p*-hpo*)} ipd « 7-^ , • X lau^ dg* X ƒ e c •'»(«-«o) Wiit-Po)\vit-p)-Wi(0 w,it-p)}dt (13,2)

This solution should be equal to ipi„c on the shadow boundary at finite distance from the object. Hence we expand (13,2) for large values ofpp and/» and find

>('fe"' + f )

= (Pine =

exp(iku )

from which follows

A- ^ ^ , ^ , r , - ,

Considering the case that the source is at finite distance from the object and the observation point near the object, we expand the solution for large values of pp. Because the observation point may be at finite distance from the shadow boundary we use the variables of (12,7) and get the solution

exp(i/c«^ — i f p * ) r ita\ , X K O r N( J

(pd « ~H , . X J e"^\<t-p) - - ! ^ w , ( f - p ) U f

* c (. wi(0 I

with

p = ('^)%-Hu'-ur,

1 = 2 J g-Hu'y)^u' + g-\u'-u')\. M"^)

This solution can be expressed as a sum of residues in the region where this sum is rapidly convergent and we get

~ ^, -niexp{iku - i | p ^ } y j,,qWi{t,-p) "P" ~ 1 1 :T Z e 7 , „ , . 2 -du' ^ + iu'-u') dj^ du' (13,4)

A solution at finite distance from the object will be gained by expanding w^^t^-p) for large values of p

ni

. n^g'iu',u^)cxpliku'--^

(13,5)

i^'MV

The final solution is a superposition of all ray contributions in the point of observa-tion. It is not possible to write down this solution in a general form, but we can construct it for each geometry. In the lit region there is a superposition of the in-coming, reflected and diffracted ray.

As we mentioned before special attention should be given to the behaviour in the neighbourhood of the caustic. In the next chapter we meet two problems without a caustic line or surface.

CHAPTER IV

EXAMPLES

14 CIRCULAR CYLINDER WITH PLANE INCIDENT WAVE

As an example of the general method given in Chapter III we will consider a circular cylinder in the field of a plane incident wave. The incident wave comes in with an angle v with the axis of the cylinder. If v = 0, we are dealing with the two-dimenional case. We must pay special attention to the radiation condition. The diffracted field at infinity will behave like the field of a line source; this can easily be shown by means of the theory of Green's functions and therefore the field behaves like r ~* instead of

r~' which appears in the case of diffraction around a closed finite obstacle.

We treat a circular cylinder with unit radius and a plane incoming wave with an angle of incidence equal to v. We take the axis of the cylinder along the x^-axis and the incident rays parallel to the x'ox' plane.

We introduce the coordinates u', u^