Application of the isotropic road roughness assumption

Journal ofSound

and Vibration (1987) 115(l),

APPLICATION

131-144

OF THE

ROUGHNESS

ISOTROPIC

ROAD

ASSUMPTION

A. N.

HEATH

Department of Mechanical and Industrial Engineering, University of Melbourne, Parkville 3052, Australia (Received 1 May 1986, and in revised form 9 July 1986)

Results of this paper are of interest in applications of the isotropic road roughness assumption [l] to problems of vehicle vibration. The cross-spectrum between heights of parallel profiles on a homogeneous and isotropic random surface is considered. Formulae involving single integrations are derived which express the cross-spectrum in terms of the spectrum of a single track. These show that the cross-spectrum at a particular wavenumber is independent of the single-track spectrum at lower magnitude wavenumbers, and provide some useful insight into the relationship between the spectra. Closed form solutions for the cross-spectra of the common single-track road roughness models are obtained. The limit behaviour of the cross-spectrum at small and large wavenumbers is also discussed.

1. INTRODUCTION

In the analysis of vehicle vibration it is often useful to treat excitation by road roughness as a stationary, Gaussian, random function of vehicle position [2]. Given a dynamic model, vehicle speed as a function of time and the appropriate means and correlations stochastically defining the excitation, vibration averages can be calculated by a variety of techniques [3-71. For simple tyre representations [8] inputs are profile heights along the road. Then the excitation is defined by the mean and autocorrelation function for each wheel track and the cross-correlations between tracks. The correlations may be represented equivalently by Fourier transforms (auto- and cross-spectra), which are convenient functions for linear models because they are simply related to the vibration spectra. It has been suggested [ 1,9] that for some purposes cross-spectra may be generated from the single-track spectrum with sufficient accuracy by treating the vehicle tracks as parallel profiles on a homogeneous and isotropic random surface. Previous theoretical development of this isotropic road roughness assumption has centred on the relationship between the correlation functions. However the spectra are usually of greater interest in applications. In the present work the relationship between the spectra is considered, leading to some useful results and a better understanding of the assumption. Formulae involving single integrations are derived which express the cross-spectrum in terms of the single-track spectrum. These show that the cross-spectrum at a particular wavenumber is independent of the single-track spectrum at lower magnitude wavenumbers, and give some qualitative insight into the relationship between the spectra. By using the formulae closed-form solutions for the cross-spectra of the common road excitation models are obtained; previously only numerical techniques [9, lo] were applicable. The limit behaviour of the cross-spectrum at small and large wavenumbers is also discussed and some previous false assertions are corrected. 131 0022-460X/87/100131+ 14 %03.00/O

@ 1987 Academic Press Inc. (London) Limited

132

A. N. HEATH 2. PROPERTIES

OF ISOTROPIC

RANDOM

SURFACES

In this section notations are introduced and some previous results are briefly reviewed. Homogeneous and isotropic random surfaces are considered. Homogeneous random surfaces have statistical properties which are independent of co-ordinate translations. The properties of isotropic random surfaces are independent of co-ordinate rotations. Realizations are specified by the height z(x,, x,) above a horizontal plane x,x1. Heights along two parallel profiles separated by a distance c are also considered: A(x) = z(x, 0) and B(x) = z(x, c), with c > 0 as shown in Figure 1.

x

Figure

1. Specification

It

of the road surface.

Since the surface is homogeneous, A(x) and B(x) must be realizations of stationary random functions with the same mean. It will be assumed that the datum plane has been chosen so that the mean height is zero. Auto- and cross-correlation functions for the tracks are defined by RAA(YV)= E{A(x)A(x+Y)),

&B(Y)

&B(Y)

&~A(Y) = E{B(x)A(x+Y)),

= E{Ab)Wx+y)1,

where E{ *} denotes ensemble averaging. The autocorrelations are from definition equal: RAA(y) = &,(-Y)

Since the surface

is isotropic,

= E{Hx)Hx+y)),

even, and from homogeneity

= L(y)

= b(y),

the cross-correlations

R.&YV) = &(-YY)

of the surface

(2)

say.

are even and identical

= RBA(y) = h(y),

(1)

[l]:

w.

(3)

In the light of relations (2) and (3) only two correlations, R,(y) and R,(y), need be considered. From isotropy the 2-D correlation function of the surface is independent of rotations, so that the functions are related by [l] Rx(y) An auto-spectrum S,(n) transforms of the correlations

= R&y’+

c’).

and a cross-spectrum S,(n) are defined as the Fourier R,(y) and R,(y) respectively, with the transform pair

00 S,(n)

=

(4)

00 R,(Y)

I --oo

ev

(-Qvny)

dy,

R,(Y)

=

S,(n) --oc

exp (i2vny)

dy.

(5a, b)

ISOTROPIC

ROAD

133

ROUGHNESS

Since R,(y) and R,(y) are real and even, it follows from the properties of the Fourier transform [ 1l] that both S,(n) and S,(n) are real and even. Moreover, from the Bochner Theorem [12] SD(n) is non-negative. As in reference [l] a normalised, real cross-spectrum g(n) is defined by g(n) = S,(n)lS,(n). (6) The square of this quantity is the ordinary coherence function between heights on parallel tracks. From the properties of this function [13] the magnitude of g(n) is no greater than 1. Not all single-track spectra are admissible under the isotropy assumption; it is necessary that the associated 2-D spectrum for the road surface be non-negative. In reference [14] it is shown that this is satisfied when

for all n 2 0. For condition (7) to apply it is sufficient that S,(n) with InI [14]. 3. THE RELATIONSHIP

BETWEEN

S,(n)

decreases monotonically

AND

Single integration formulae expressing the cross-spectrum spectrum S,(n) are derived and discussed in this section.

S,(n)

S,(n)

in terms of the auto-

3.1. DERIVATION OF FORMULAE Taking the Fourier transform of both sides of equation (4) and substituting for I&( *) in terms of equation (5b) gives a double integration formula relating the two spectra: m 00 exp (-i2Tny) dy SD(q) exp (i2rrn,Jy2+ c’) dn,. (8) S,(n) = I -cc I -cc Reversing the order of integration (justified by convergence in a generalized function sense [15,16] of the new inner integral) and noting that with respect to n, the integrand is even in its real part and odd in its imaginary part, one may write

SdnJ drill

&(n)=2

Re {exp [i2r(n,m-

ny)]} dy,

(9)

where Re { .} denotes the real part. The inner integral is given by (see Appendix 1) 2rcn

-6(27rcJn2-nf),

-

fornZn,ZO,

(1Oa)

forn,>n20,

(lob)

m __cn J,(27rcJn:_nZ) ‘_

,

where 6( *) is the Dirac delta function [ 151 and JI( * ) is the first order Bessel function [ 171. Splitting the outer integral of equation (9) and substituting the appropriate solutions gives, for n 3 0, -8(2&n’-

S,(n)=2

n:)

dn,

cc

-2

I’ ”

(11)

134

HEATH

A. N.

The first integral simplifies to S,(n) after a change of the integration variable to u = 27&n’nf and an application of the sifting property of the delta function [ 111, with use made of the evenness of the integrand. Since the cross-spectrum is an even function the result for unrestricted n follows directly, upon replacing n by In] where necessary, to yield So(n

Sx(n)=SrJ(n)-27X Two equivalent forms of equation dn variable to n = n, - n gives

I

)n J,(2srcjn;-n’) /I g’inf- ,I

(12) are useful.

Firstly,

dn,.

a change

(12) of the integration

X S&r’+

S,(n)=S,(n)-25-c

t~‘)J,(27rcu) du.

(13)

I0 Secondly, yields

by parts (with S,( n ) assumed

integration

f&(n)=where Sb( .) denotes

differentiable

S’~(n,)J,(2~c\in;-n-)

the derivative

,

of S,(.) with respect

and bounded

at infinity)

7

dn,,

(14)

to its argument.

3.2. INTERPRETATION Equations (12)-(14) show that the cross-spectrum at a particular wavenumber is independent of the single-track spectrum at lower magnitude wavenumbers. This result may be explained by interpreting the road surface as a sum of elemental, random plane waves (for mathematical details see reference [12]). cc. (see Figure 2) generates A plane wave with wavenumber n P = l/Ap and orientation a sinusoid of lower wavenumber n = I/ A = sin (p )/A, = np sin p on track A and a similar sinusoid on track B, shifted in phase by 6 = 2anc cot CL.The auto-spectrum S,(n) represents a sum of mean-square contributions from plane waves of all orientations yielding sinusoids of wavenumber n on the vehicle tracks: i.e., plane waves of the same or higher wavenumber np = n/sin p as g is varied over the range -7r/2 to 7~12. A similar sum, weighted by the power factor cos S to account for the phase difference between the tracks, defines the cross-spectrum. Since the second-order statistics of isotropic surfaces are defined by S,(n) [l], a knowledge of S,(n) for InI 2 n, specifies the mean-square lnPl 3 n,, and therefore defines S,(n) contributions of the plane waves for wavenumbers

B ---

-

-

-

-

--

b A ------

r c

CL --

x ~

Figure 2. Effect of a plane wave on the vehicle tracks. the vehicle tracks.

Full lines represent

wave crests; the broken

lines are

ISOTROPIC

ROAD

ROUGHNESS

135

for the range InI 5 no. Hence the cross-spectrum must be independent of the auto-spectrum at lower magnitude wavenumbers. Equation (14) provides some qualitative insight into the relationship between the spectra. It expresses the cross-spectrum as a weighted integral of Sb(n,) from n, = InI to 00 and shows that S,(n) is most strongly influenced by regions of S&n,) with large derivatives: i.e. those areas where the auto-spectrum is changing rapidly. The weighting factor Jo(27rcF n, - n ), plotted in Figure 3 for a particular n value, is a weakly damped oscillation of alternating sign and approximate period l/c. A one-way change in SD(n,) has a large effect when it is concentrated in an interval small compared to l/c; its inlIuence is then greatest at n values for which it aligns with the weighting function stationary points. A change in S,(n,) over a wider interval has a smaller effect: positive and negative contributions in the integral (14) are generated which tend to cancel one another. The most rapid change possible in S,(n,) is a discontinuity, which can be represented by a delta function in Sl,(n,) at the jump point.

Figure 3. Weighting factor J,,(~vc.+&~)

us. n,. n =0.5/c.

The results of section 3.1 have two important implications for the calculation of cross-spectra from an experimental S,(n). Firstly, S,(n) must be measured at wavenumbers higher than those of interest in the computed S,(n). Often it is reasonable to assume that S,(n) is monotonically decreasing at wavenumbers higher than no, the upper limit of measurement. For this case a useful error bound follows simply from equation (14): the contribution of the unmeasured part of S,(n) to the cross-spectrum at a particular wavenumber cannot have magnitude greater than S,(n,). Secondly, it is clear from equation (14) that the resolution of the spectral estimate (i.e., the minimum wavenumber interval over which changes in the spectrum can be detected) must be considerably finer than l/c, the period of oscillation for the weighting factor. Finer resolution may of course be required to distinguish features of the auto-spectrum itself. 4. CLOSED-FORM

SOLUTIONS

FOR

CROSS-SPECTRA

Table 1 lists functions commonly used to describe single-track spectra and some references where they have been discussed or applied. Solutions for the corresponding cross-spectra are obtained in this section; note that functions 1 and 2 include factors of

A. N. HEATH

136

TABLE

Functions

1

used to represent single-track

roughness

spectra References

S,(n) 1.

Clnl-‘”

Power law

O
Cln/nJ’“‘l

2. Split power law

0

ClTI/?IJ2”” /c

3. Rational function

n, slnl
n2+a2 C

4. Rational function

(n2+a2)2

Note: C, w, w,, w2, n,, a and p are real,

[I, 191

W, 211

0 slnl
[221

Ocln(
[211

P2)2+4n2cu2

(n2-a’-

[If31

0 s/fll
C(n2+a2+p2)

5.Rational function


positive constants

2 in the exponents to simplify the writing of results. The formulae here may also be used to generate cross-spectra for single-track spectra which can be expressed as the ratio of two polynomials, the numerator having lower order than the denominator. These rational functions arise in the modelling of road roughness as the output of a linear filter excited by white noise [23,24]. Functions l-4 are monotonically decreasing with In( and therefore satisfy condition (7). It is shown in Appendix 2 that function 5 is admissible provided (Y2 /?. Note that the processes represented by functions 1 and 2 are non-physical mathematical abstractions: like white noise [ 121 the integral (5b) does not converge for y = 0, implying infinite variance R,(O). Nevertheless, the processes yield physically reasonable results when applied to systems with suitable frequency response characteristics (see for example reference [5]). As well, cross-spectra defined by equation (12) exist and have the expected properties. 4.1.

CROSS-SPECTRUM

Substituting

FOR

function

THE

SIMPLE

1 of Table

POWER

1 in equation

LAW

(13) and integrating

by parts yields

Sx(n)=2Cw Upon using the integral

given in reference

[ 171, page 434, this gives, for IArg (n)l < 7r/2,

SX(n)={2C(7rc/n)“/I’(w)}K,(2~cn),

(15)

where K,( . ) is the modified Bessel function of order w, and IY * ) is the gamma function. As before the result for unrestricted n is obtained by substituting InI for n, so that after normalization g(n) = {21~c~I”l~(w)}K,(I2~cnl).

(16)

It follows from the properties of K,( *) [25] that g(n) is a positive function with g(n) + 1 as nc + 0, and g(n) + 0 as nc+ 0;). Some solutions for various values of w are shown in Figure 4, plotted against the non-dimensional wavenumber nc. (References [ 171 and [25-271 are useful for the evaluation of Bessel functions.) They appear similar to the solutions of reference [9], calculated by a numerical technique. Some features of the solutions can be predicted directly from equation (14). In the range of the integral the spectrum changes most near n, = InJ, where the derivative S’,( n,)

137

ISOTROPIC ROAD ROUGHNESS

flC

Figure 4. Normalized cross-spectrum to bottom, w = 1.5, 1.25, 1.0, 0.75.

us. non-dimensional

wavenumber for function 1 of Table 1. From top

is negative and the weighting function is large and positive. This indicates that the cross-spectrum will be positive. For smaller c the region of greatest change becomes more concentrated with respect to the oscillations in the weighting function, so that g(n) increases with decreasing c. 4.2. CROSS-SPECTRUM FOR THE SPLIT POWER LAW The single-track spectrum may be written as the sum of three terms: S,(n)=C~n/n,~-2W~+CH(n,-~n~)~n/no~-2”~-CH(no-~n~)~n/no~-2”~,

(17)

where H( 0) is the unit step function. The cross-spectrum may be found by applying equation (12) for each term and summing results. The solution for the first term follows directly from the previous section. The second and third terms have the same form; series solutions may be generated by substituting the power series for the Bessel function [ 171 in equation (12), and integrating each term separately. This yields, after normalization for (n] < no, g(n)=l+~$~2~w1-w2~{~K,(121~)-1} +E I n0 I

*w, m c (- P2C2n~)m+‘I~m~~nlno~*, wJ-E,[(n/n,)*, m=O m!(m+l)!

w*]},

(18a)

and for InI > no,

where l--x En(x, w) =

I0

(U+X)-WUm d”*

(19)

Note that equation (18b) is the power law solution (16) with index w2, and that g(n) + 1 as nc+O. A recursive scheme for evaluating E,,,(x, w) is described in Appendix 3. Some solutions calculated by a truncation of the infinite sum are shown non-dimensionally in Figure 5, with the parameter grouping nc, n,c, w, and w2. For n,c < 0.5 (the usual

A.

138

N. HEATH

Figure 5. Normalized cross-spectrum us. non-dimensional wavenumber for function ~,~=0.15,~,=1.5,w,=0.75;---,n,c=0~2,w,=1~0,w~=1~25;-----,n,c=0~25,w,=0~75,w~=1~5;~~~~~~. n,c=0.8,

w, =O.l,

2 of Table

1. -,

w2= 1.25.

range of interest in vehicle problems) partial sums converge quickly: six terms are generally sufficient to achieve an accuracy of 0.001% . The cross-spectrum is positive for the usual parameter range, but large not and w2 with small w, may lead to negative values as shown by one of the curves. The negative results may be explained by using equation (14). For w? < w, the greatest change in S,( n,) over the range of the integral occurs near n, = Inl, so that as for function 1 S,(n) is positive. When w2 is sufficiently larger than w, there is however a second region of substantial change at n, = no, which leads to negative S,(n) values in the range InI < no when n,c is large enough to allow the region to align with a negative loop of the weighting function. . 4.3.

CROSS-SPECTRA

FOR

RATIONAL

FUNCTIONS

In this section cross-spectra are considered for SD(n)‘s expressed as the ratio of two polynomials, the numerator having lower order than the denominator so that S,(n) + 0 as 1nl+ 00. It is assumed that the single-track spectrum is admissible; this must be checked in each case. Since S,(n) is even and real, only even powers of n appear and complex roots of the denominator polynomial occur in conjugate pairs. By using partial fractions and grouping appropriately, S,(n) can therefore be expressed as a sum of terms with the form Re{F/(n’+GY},

(20)

where p is a positive integer and F, G are constants (possibly complex). The cross-spectrum for a single-track spectrum given by expression (20) is first considered. By summing solutions of this form the cross-spectrum for a general rational function can be found. The function within the curly brackets of expression (20) is the same as function 1 in Table 1 when ( nz + G)“’ , p and F are replaced by n, w and C respectively. Upon choosing the principal branch of the two-valued function 2”’ (i.e., IArg (z”“)I < 7~/2) a derivation similar to section 4.1 applies, so that S,(n) Applying

equation

= Re {[2F/r(p)](7rc/(n2+

G)“2)pK,[2rrc(n2+

G)“‘]}.

(21) to function

1 and normalizing

one has

3 in Table

g(n) = 27rcJn2+

a2K,(277cJn2+

cw*).

(21)

(22)

ISOTROPIC

ROAD

139

ROUGHNESS

From previous discussion g(n) is positive and tends to zero for large nc. Some solutions are plotted non-dimensionally in Figure 6 for parameter groups nc and UC. For function 4 in the table the result is g(n)=27r2c2(n2+cx2)K2(2&n2+a2).

(23)

Again g(n) is positive and tends to zero for large nc. Solutions are shown non-dimensionally in Figure 7 with the same parameter grouping. The general features of functions 3 and 4 and their cross-spectra are similar. Both auto-spectra are monotonically decreasing and change most near wavenumber (Y. For wavenumbers much greater than (Ythe auto-spectra decline as negative powers, so that the S,(n) solutions resemble those of section 4.1. For wavenumbers smaller than (Y,the major contribution to the integral (14) comes from the region near n, = cq where Sb( n,) is large and negative. Provided (YCCl/c (i.e., (YCCC 1) this region aligns with the first positive loop of the weighting factor JO(27rcJn: - n’) and a large positive g(n) results. When (YC

nc Figure 6. Normalized cross-spectrum VS.non-dimensional to bottom, CYC = 0.01, 0.05, 0.1, 0.2, 0.5.

wavenumber

for function

3 of Table 1. From top

wavenumber

for function

4 of Table 1. From top

nc Figure 7. Normalized cross-spectrum to bottom, CYC = 0.01, 0.1, 0.2, 0.4, 0.8.

OS. non-dimensional

140

A. N. HEATH

is larger the region of large derivative spreads out to more loops of the weighting function; its effect is diminished by cancellation so that g(n) is smaller, though still positive. Finally consider function 5. Using partial fractions, one has S,(~)=[CM/~(Y(~‘+M~)]+[CM*/~~(~‘+M*’)]=R~(CM/~(~~+M~)}, where M =a+@ normalizing gives

and * denotes

the complex

conjugate.

g(n)=Re{[2mM(n2+M2)“2(n2+M*2)/a(n~+

(24)

Applying

equation

(21) and

M’)“‘]}.

a’+P’)]K,[(2~c(n’+

(25)

Some solutions are shown non-dimensionally in Figure 8 for the parameter grouping and /3/a. The function g(n) tends to zero for large nc and is positive for the usual range of interest, although large cyc and /3/a may lead to negative results for small nc as shown by one of the curves. nc, cxc

I.0

O-6

I.0

0.1

0.01

nc

Figure 8. Normalized cross-spectrum us. non-dimensional wavenumber for function 5 of Table 1. From top to bottom, crc=O.Ol, P/a =O.l; ac=O.OS, P/a =0.2; crc=O.l, /3/a =0.4; (~c=O.2, p/n = 0.6; ac = 0.3, /3/a = 0.8.

For P/(Y < l/v!? function 5 is like function 3: monotonically the change centred near wavenumber (Y. Positive cross-spectra expected. For larger /3/ (Ythe spectrum is no longer monotonically at wavenumbers below about (Y and falls thereafter. Negative smaller than (Y, and (YCis large enough so that in integral (14) spectrum tends to align with the first positive loop of JO(2rrc,/n: part tends to align with the second negative loop.

5. LIMIT

BEHAVIOUR

It is asserted in reference zero for large wavenumbers:

OF THE

[l] that g(n) i.e., lim g(n) =O?, “-a

NORMALIZED

approaches hi

decreasing with most of values are therefore to be decreasing: it increases S,( n)‘s arise when InI is the increasing part of the - n2), and the decreasing

CROSS-SPECTRUM

unity

for small wavenumbers

g(n) = l?.

In this section it is shown that the limits are true for certain do not generally apply.

and

(2627) types of S,(n),

but that they

ISOTROPIC

ROAD

141

ROUGHNESS

Consider first S,( n)‘s which decline algebraically at large wavenumber: i.e., S,(n) + Cn-‘” as n + CD for some C, w > 0. All functions in Table 1 and all rational functions with denominator degree greater than numerator degree belong to this class. Then limit (26) applies, since g(n) approaches solution (16) for large wavenumber, which as previously discussed tends to zero as n + cc for non-zero c. Next consider S,( n)‘s which possess an algebraic singularity at n = 0, but are otherwise finite: i.e., S,(n) = C]n]-2w + t(n) where C, w > 0 and t(n) is a bounded function. The cross-spectrum is a sum of solution (16) and v(n), a function found by applying e(n) to equation (13). A simple argument shows that T(n) is bounded and it follows that limit (27) applies since ‘,z g(n) = lim

1

n-+0Clrtl-‘“+5(n)

2Cl7rc/nl” T(w)

K,(P~cnl)+

v(n)

(upon removing terms with finite limits in numerator and denominator), = 1, by the properties of K,( *) [25]. To show that neither limit applies generally consider a Gaussian single-track spectrum, S,(n) = exp (-nn*), which is admissible since it decreases monotonically with In]. The Gaussian function is its own Fourier transform [ll], so R,(y) = exp (-n-y*), and upon using equation (4), Rx(y) = exp (-~TC*) exp (--7~y*). Retransforming and normalizing one has g(n) = exp (-rc*), a constant independent of wavenumber which clearly satisfies neither limit (26) nor (27). Functions 3-5 of Table 1 also do not satisfy relation (27).

ACKNOWLEDGMENT

Financial support of this work by the Australian Road Research Board is gratefully acknowledged.

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142

A. N. HEATH 1983 Proceedings ofrhe 8th IAVSDSymposium on theDynamics of Vehicles on Roads and on Railway Tracks, Cambridge, Massachusetts. The artificial generation

10. D. CEBON and D. E. NEWLAND

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APPENDIX The inner

integral

1: THE

of equation

I = Re Changing

the variable I=Re

INNER

INTEGRAL

OF EQUATION

(9) may be written as ‘.Y exp [i2n( n,Jy’+ c’- ny)] dy --cc

(9)

.

of integration to u = sinh-’ (y/c), one has OCI exp [i2nc( n, cash u - n sinh u)]c cash 1.4du 11 -Ix-

(AlI

.

(‘42)

Two cases are considered. 1: n s n, 20. Putting 2~7cn = W cash X, n: and changing the variable of integration manipulation, Case

2ncJn2-

I=Re{(27rc2/W)jIi

exp (-i Wsinh

2ncn, = Wsinh X, so that W= to u = u -X yields, after some

.

v)( n cash tr + n, sinh II) dv I

ISOTROPIC

Now exp (-i Wsinh

ROAD

143

ROUGHNESS

u)n, sinh u has a real part odd in u, and so does not contribute.

Hence,

cc (2m2n/

I = Re

W)

exp (-i Wsinh

Changing

the variable

of integration

.

u) cash u du

I -cc

{

I

to u = sinh u one has 00

I=Re {

(2m2n/W)

= Re {(2m2n/ = (4&c2n/

I -m

W)27r 6( W)}

, I

(see references

[15,16]),

W) 6(W).

for W yields equation

Substitution

exp (-i Wu) du

(A3)

(lOa).

Case 2: n, > n 2 0. Rutting 2mn, = U cash V, 2mn = U sinh V, so that U = 2mJnfand changing the variable of integration to u = u - V, yields, after some manipulation, exp (iU cash u)(n, cash u+ n sinh u) du

I=Re{(2nc2/U)l_l

n2,

. I

Now exp (i U cash u)n sinh u has a real part odd in u, and so does not contribute.

Hence,

m I = Re {

(27rc2n,/ U)

= 2(2m2n,/

U)

I

exp (i U cash u) cash u du , I -a2 I

cc cos (U cash u) cash u du, 0 @c

=

(4m2n,/

sin (U cash u) du

U)(d/dU) {I

=

(4m2n,/

U)(d/dU){(7r/2)Jo(

for U yields

APPENDIX

U)}

(see reference

[17], page 180),

U)J,( U).

= (-2r2c2n,/ Substitution

, I

0

equation

(A4) (lob).

OF

2: ADMISSIBILITY

FUNCTIONS

5 IN TABLE

1

for which function 5 of Table In this appendix the parameter range is determined satisfies condition (7). Changing the variable of integration to u = 7 n, - n gives f(n)

= -(d/dn)

m S,(m)

du.

(A9

I0 Substituting

equation

(24) into this, integrating f(n)

and differentiating,

= Re {n~CM/2aN3},

1

one has (Ah)

where N = ( n2+ M2)“2 (principal branch). Multiplying top and bottom of the expression (7) is equivalent to s(n) 2 0 for n L 0, within the brackets by N*3 shows that condition where s(n) = Re {MN*3}. The lowest value occurs for n =0: s(O) = ((~~+/3*)((~~-/3~). It follows that function 5 is admissible for (Y2 B.

144

A.

APPENDIX

Simple

manipulations

N.

HEATH

3: THE CALCULATION

of the integral

OF E,(x,

(19) show that, for x > 0,

E,(x,w)=(l-x’_“)/(l-w),

w#l,

E”(X, l)=-Inx,

&(x, w) =

(A7, A81 w#m+l,

E,(x,m+l)= By using the relations

w)

E,(x,

m (1-x)’

c y-

,=I

-J

w) may be calculated

Inx

.

recursively.

(A9)

(AlO)