Journal of Sound and Vibration (1975) 42(3), 295304
TRANSIENT
SCATTERING
BY A CIRCULAR
CYLINDER
R. P. SHAWL Joint Tsunami Research Effort, NOAA, University of Hawaii, Honolulu, Hawaii 96822, U.S.A. (Received 14 June 1974, and in revised form 10 March 1975) The scattering of transient plane waves by a circular cylinder is studied by using the Kirchhoff timeretarded potential boundary integral equation method. Two distinct problems are solved: (i) surface velocity potentials (or pressures) arefoundfor rigidcylinders scattering ramp, rampstep and Gaussian incident potential (or pressure) waves and (ii) surface velocities are found for free boundary (pressure release) cylinders scattering ramp, rampstep and Gaussian incident velocity waves. The numerical schemes for both boundary conditions are very similar; since the same intluence coefficients are used they differ only by a sign in the final formulation. Numerical results are readily obtained for the first few transit times. This approach is complementary to the usual modal approach in that it is best suited to early time values where the modal solutions converge most slowly. 1. INTRODUCTION
The scattering of transient waves by circular cylinders is a classical problem in theoretical physics. The analytical solution of this problem, however, is far more difficult than the corresponding time harmonic case even with the simple geometry involved due to the additional independent variable. Friedlander [l] developed a Green’s function for a rigid boundary and a delta function pulse in terms of a Fourier cosine series in the angular coordinate, 0, and
a Laplace transform in time. The inversion and summations involved were possible only in the shadow region and numerical values were developed only for very early times (see, e.g., reference [2]). In several later papers similar methods were used on acoustic wave, elastic wave and acoustic waveelastic shell scattering problems (see, e.g., references [3][6]), the difficulty of solution in the illuminated region being resolved by various techniques such as direct numerical integration of the inversion integral [3], steepest descent approximation of the inversion integral [4], conversion to a convolution integral equation with numerical integration 151,etc. The bulk of these methods, however, require some form of modal solution to the problem after a Fourier or Laplace transform over time. Alternatives such as creeping wave theory [6] or geometrical diffraction theory [7] are also available, but these require a “tracing” procedure in which various distinct physical effects are followed and superposed. The requirement of a modal solution inevitably involves a summation over several modes to obtain reasonable accuracy (e.g., Geers [8] uses 8 to 10 modes, each of which requires a numerical or approximate integration of some sort) or some asymptotic solution such as that of reference [9]. It is shown here that an alternative method, already found useful in transient acoustic wave scattering [lo], can be used to develop numerical solutions valid for this simple geometry for the first several transit times. This method also has been used to examine wave front behavior in an elastic cylinder embedded in another elastic medium [l I]. This approach is based on the Kirchhoff timeretarded integral equation [12] which is equivalent to the wave equation. The form of the fundamental solution used requires that the problem be considered as threedimensional rather than two; this proves to be no real difficulty. t On leave from Faculty of Engineering and Applied Science, State University of New York at Buffalo, New York 14214, U.S.A. 295
296
R. P. SHAW
The original basis for this work arose from water wave theory and as a result only continuous incident wave fields were considered (the modifications for a discontinuous incident wave are discussed in reference [13] and in fact this serves as a check for later work [14], on inhomogeneous media). Some numerical solutions for the rigid boundary cylinder have been obtained by other methods (see, e.g., reference [15]), and can be used to check these results. Thus this work is offered as a convenient way in which numerical values for particular scattering problems can be obtained rather than a significant new contribution. However, it should be made clear that the present integral equation solution is best suited for moderately early time solutions, i.e., for the first few transit times, and this represents a region in which standard modal solutions are slow to converge. Thus this approach may be considered complementary to rather than competitive with the others. The governing Kirchhoff timeretarded integral equation [9], is written on 4 as either the velocity potential or the pressure; these satisfy the same boundary conditions and differential equation and are distinguished only by the initial condition (i.e., incident wave) :
with E = 0 if the field point is outside of the fluid (within the scattering body), E = 1 if the field point is inside the fluid and E = 3 if the field point is on the scattering surface in which case the integral is a principal value integral: i.e., the singularity point R = 0 is excluded. Here i; represents the location of the field point, T;,is the integration variable or source point and R = IP ?,,I. The time to is retarded from the actual time, t, by the amount necessary to travel Finally, S is the (threedimensional) the distance from the source point to the field point, R/c. cylinder surface with outward (from the wave medium) normal n and r$,,,is the (continuous) incident wave which contacts the cylinder at t = 0. 2. RIGID BOUNDARY SOLUTION AND RESULTS First consider a rigid circular cylindrical boundary with zero initial conditions ahead of the incident wave : a+lar=o on S (r = a), (2) t < 0.
4=Av,
(3)
The equation is nondimensionalized by using the radius of the scattering cylinder, a, as a length scale and the ratio of a to the phase speed, c, as a time scale. This essentially gives back equation (1) with c = 1 and S representing a cylinder of unit radius. Using the rigid boundary condition and allowing the field point to approach the boundary, one has Zn
m wo,
[email protected],
t)
=
dMt
t)
+
~
$ 0I/[ cc
to)
RZ
+
1
R
@(so,
to)
at,
aR an 3
dzod&. 0
(4)
On the cylinder surface, R = (20 + 4 sin2 [(0,  6?)/2]}““,
(5)
aR/an, = aR/aro = 2 sin’ [(do  0)/2]/{2; + 4 sin2 [(e,  0)/2]]l/2.
(6) Although &e,r) is independent of z directly, +(B,,t,) does depend on z. through the retarded time, to, within the integral and the problem is solved as a threedimensional one with $~(8,t) the same in all crosssections, and determined on the z = 0 crosssection. Note that the limits on z. are not actually infinite but depend on t in that points on the cylinder for which to = t  R < 0 cannot influence the field point. Thus the limits actually are determined by z. = *{t’  4 sin2 [(0,  f?)/2]}1’2.
(7)
TRANSIENT SCATTERING BY A CYLINDER
297
There are a number of methods available for the solution of equation (4) but the simplest is probably a direct numerical approximation, analogous to finite differences for differential equations, wherein the dependent variable is taken to be constant over specified increments in time and space. This leads to a set of successive linear algebraic equations. One identifies the constant value of 4 with its value at the center of an angular increment, At? = 2x/M, and at the end of a time increment, At. Take the index Zto identify the field point, 6’= (I  l)*AB (I = 1,. . . M), Jfor the integration variable, N the present time, t = N*At and N  K + 1 the retarded timesuch that spheres of radius R = K*At centered at the field point would encompass all points which could affect the field point by time K*At. Equation (4) then becomes t6(Z, N) = &(Z, N) + $
$ $ (WI,.& K)* &.Z, N  K + 1) + C(Z,J, K)* J=l
K==l
I= 1, . . . M
Kt I)&J,NK)]/At},
[&J,N
(J#Z
for
N= I),
(8)
with, by making use of the symmetry in z,,, 20”(K)
(J1/248)
2 sin’{(&  (I  1) A8)/2}
B(Z, J, K) = 2
j.
dOo. s
[zg + 4 sin’ ((0,  (I  1) A8)/2}]3’2dzo’
(9)
zoue1)
(J3/240)
e.g., (JlIZdO)
B(l,J,K)=
[
[t22(1
co~&)]~‘~t~dB~;
(JiDA,,
where the point R = 0 is to be excluded. z,,(K) is found from equation (7) with t = K*At. At must be chosen such that the spatial region neighbouring the region containing the field point does not influence the field point with its current values but only with retarded values. This uncouples the system of equations allowing a successive solution at each time step. The evaluation of these influence coefficients, B and C, is considerably simplified by the symmetry of the circular cylinder. They need be evaluated (numerically) only for the field point Z = 1; the remaining coefficients follow from B(Z,J,K)=B(l,]JZ)+l,K), C(Z,J,K)=C(l,IJI\
+ 1,K).
(11)
Then one has the successive set of algebraic equations (3  Wl,
1,1) + CO, 1,1)/dt)/W*
=b,(z,N)+$i)
$+(Z,J,K)*$(J,NKfi) J=l
+
&I, N)
K2
C(Z,J, K)* [&J, N  K + 1)  $(J, N  K)]/At} 
 C(l, 1, 1)* 4(Z, N  1)/4nAt.
(12)
298
R. P. SHAW
One may choose C#J~ to represent an incident velocity potential and therefore 4 to represent the total velocity potential, realizing that identical results apply if & is an incident pressure wave and C$is the total pressure field. Three types of incident wave field were chosen to illustrate this procedure. These are a ramp (I) 6, = 05(t  X)H(T  X) = 0.5 [t  (1  cosB)JH[t  (1  cosfq], a ramp step (II) Av = i
oqtX)H(ix),
tx<
1
0.5
tx>
1
f
and a Gaussian pulse, beginning at 3a (III) +,,, = exp [(t  x 
[email protected]/2&] H(t  x  3~). Figures 1 and 2 show the response of the rigid cylinder to the ramp incident wave at various boundary points for all time and all boundary points for various times, respectively. The steadystate solution parallel to the original incident wave is clearly developed within the first two transit times, i.e., by t = 4. The appropriate arrival times for boundary points in the shadow are found with some very small (< 0.01) precursors which arise from the numerical
Time,
I
Figure 1. Boundary velocity potential for rigid cylinder subjected to ramp incident wave for 0 = 0”, 90”, 180”; I = 0.0 (O.l), 10.0. Ramp: At = 0.1; A0 = 10”.
0
20
40
aI Boundary
80
[email protected]
wordinate,
120
140
160
180
8 bgreee)
Figure 2. Boundary velocity potential for rigid cylinder subjected to ramp incident wave for t = 1,2,4,8; 0=0(10”), 180”. Ramp: At=Ol; AB=lO”.
TRANSIENT SCATTERING BY A CYLINDER
299
cancellation of the incident wave by the integral up to the actual diffracted wave arrival time: e.g., t = 2.57 for 0 = 180”. Solutions were carried out with At = 0.1 and A8 = 10”for 200 time stepsonly the first five transit times are shown, however (to t = lo), since the steadily increasing solution is very well established by this time. ,
1
I
l.2+____ ___ l.Os” c 3 o+=
*
I
.A !
G
‘.,.
!
u 06G *
! 1
? :: 0.4
loo
Y. p .h. / +\..Y_.
_
/go0
//i&o’
s $
!
/
0.2 1 i
/if I:
:
0
_~~~~~~=~=_~~~~_=
/’
I
2
4
3
5
6
Tim,
7
8
9
10
t
Figure 3. Boundary velocity potential for rigid cylinder subjected to rampstep incident wave for B = O”, 90”, 180”; t=O.O(O.l), 10.0. Rampstep: dt=O*l; At?= lo”.
s” $I.
_ 0.8
..
2
\ . .
f
...A._,_. g
s 6 a a m
._
0.6;
P
i_
me

‘
0.4
A
1.
‘\\\
‘\ \ ‘,I=1 \
0.2
0

I 20
I 40
\
I 60
\
t=4
‘, ‘1
\ .\/=2 ‘C3
\
‘\
\ ‘\ ,‘\ 80
‘\.
‘\ ,
,
100
120
Boundary coordmate,
‘y 140
1 160
180
6 (degrees)
Figure4. Boundary velocity potential for rigid cylinder subjected to rampstep incident wave for t = 1,2,3,4; 0 = 0” (lo”), 180”. Rampstep: dt = 0.1;
[email protected]= lo”.
Figures 3 and 4 show results for the rampstep incident wave with At = 0.1 and Af3 = 10”. Although calculations were carried out for the first 10 transit times (t f 20), all values past t = 10 were within 2 % of the expected asymptotic value of 0.5 and therefore are not shown. (At t = 20, the maximum deviation from the asymptotic value was 0.1x.) The arrival times in the shadow region again are seen to correspond to actual diffraction arrival times (e.g., at 0 = 180” the arrival of the first diffraction occurs at 2*57), with very small precursors. In Figure 4, one can see clearly that the solution by the second transit time, t = 4, is already very close to the steady asymptotic value. Figures 5 and 6 show the results for the Gaussian input with Q = 0.521. Past t = 10 the calculated values were 2 “/”of the input peak, 1, and are not shown. At t = 20, the largest value is 0.5 y. of the peak. The curve for 8 = 0” in Figure 5 is in very good agreement with Figure 5 of reference [15]; the first maximum and minimum values are within 3 %. Figure 6 shows the
300
R. P. SHAW
Time,
t
Figure 5. Boundary velocity potential for rigid cylinder subjected to Gaussian incident wave for 0 = o”, 90”, 180’; t= 0.0 (O.i), 10.0; C= 0.521. Gaussian: At = 0.1;
[email protected]= lo”.
0
m
40
60 Boundary
80
loo
coordmote,
Ix)
140
160
I80
8 (degrees)
Figure 6. Boundary velocity potential for rigid cylinder subjected to Gaussian incident wave fort = 1,2,3,4; B=0”(10”),180”;cr=0~521.Gaussian=At=0~1;AB=10”.
development of the solution for all boundary points at several time values. Due to the time delay in the development of the peak input, the steadystate solution is established essentially after four transit times in this case. One may draw several conclusions from these results. The steady solution, asymptotic to some constant value or some constant slope, is well established within the first two to three transit times of the major portion of the disturbanceat least for continuous incident waves. In addition, asymptotic values are found directly from equation (4): e.g., for a constant asymptotic value of 4, the time derivative is zero and C#I is constant over all of the integration region of significance. Then ++(ASYM) = &(ASYM) + ; or 4(ASYM) = &(ASYM)
4(ASYM). (2n),
(13)
as expected.
3. SOLUTION AND RESULTS FOR A PRESSURE RELEASE SURFACE Next consider the complementary example of a free (pressure release) boundary with zero initial conditions ahead of the incident wave: $=O
on
S(r = a),
(14)
301
TRANSlENT SCATTERING BY A CYLINDER
b, = +,,
t < 0.
(15)
One again nondimensionalizes equation (1) as before. However, one now has a dilemma. Using the free boundary condition directly leads to an integral equation of the first kind in 2~= a&%: (16)
This equation can be solved numerically but careful attention must be paid to the position of the leading wave front as it envelops the cylinder (see, e.g., reference [IO]). This in turn distinguishes field points by their location relative to the incident wave front and destroys the symmetry basic to the previous development. While one could proceed in this manner (reference [IO] gives this procedure for the analogous case of a pressure release sphere) there is an alternative used originally in reference [16] which allows use of symmetry. Equation (1) may be applied to a field point just outside the cylinder and differentiated with respect to the normal to the cylinder to give an equation on V.One requires v to be zero at the leading wave front in order to avoid terms arising via Leibnitz’s rule by differentiating an integral with variable limits (see references [13] and [I 61). This then yields
aUpo, to) at, +Ti‘  ar fo=fRdS,. at, 1
(17)
These kernels are similar to those of the rigid case. If one now allows the field point to approach the cylinder surface, one has a contribution of +J from the integral at R = 0,leaving aR ar
dSo>
(18)
*0=,R
with R defined by equation (5) and aR
= ar
+2 sin” [(&  Q/2] [z; +
(19)
4 sin2 [(&  @/2]]“”
Then one can go through the same solution procedure as in the rigid boundary condition case :
+C(Z,J,K).[V(J,NKfl)v(~,l,_)]/dti,
Z=l,...M,
(20)
with B and C defined as in equations (9), (10) and (11). The successive set of linear algebraic equations on v is then equivalent to that on 4 given in equation (12) except for the sign of the summation term. The program for the rigid boundary case therefore can be rerun for the free boundary case with only a minor change. Figure 7 shows the normal surface velocity response of the pressure release cylinder to the ramp input in velocity ((I) with u, for &J for 8 = 0” and 0 = 180”for time from 0.0 to 10.0 in steps of 0.2 and d0 = 10”. An ever increasing velocity with a positive acceleration is found as would be expected from this ever increasing incident velocity wave. Beyond four transit times, t = 8.0, the velocity at all points is essentially uniform. The first diffraction effects from
302
R. P. SHAW
Time, t
Figure 7. Boundary velocity for soft cylinder subjected to ramp incident velocity wave for 19= 0, 180”; t = 0.0 (0.2), 10.0. Ramp: AT= 0.2; AC?= 10”.
the discontinuity in slope at t = 0 are felt at 6’= 0” at time t = 2 + rc, causing a slight “bump” in the curve for velocity, but generally the response follows a smooth curve. Figure 8 shows the response to the ramp step input (II), again for B = 0” and 180” over time from 0.0 to 10.0 in steps of 0*2 with A0 = 10”. Here the diffraction effects are seen far more clearly. For example, the response at B = 0” follows the initial ramp portion of the incident wave field up to t = 1 with a little increase caused by local diffraction, continues increasing up to t = 2 + n when the first diffraction of the leading slope discontinuity reaches back to 0 = 0” where it levels off until one unit of time later when the first diffraction of the second discontinuity reaches B = O”, whereupon it begins increasing again. Similar steps are seen in the response at 8 = 180” and may be interpreted in the same way. Finally, Figure 9 shows the response to a Gaussian input for 0 = 0”with e = 0.521. The same time step and angle increment as for Figures 8 and 9 are used here as well as a finer calculation with a time step of 0.05 up to t = 5.0. The finer grid gave the same values as the 0.2 time step and was used only as a check. Here one would expect a constant velocity field asymptotically; l&O
,’ /
1
I
16.0
14.0.
b
12.0.
i z _o %
Da
? 0 z a m
&O
GO
4.0
J
0
I2
3
4
5
6
7
8
9
IO
Time, I
Figure 8. Boundary velocity for soft cylinder subjected to rampstep 180”; t = 0.0 (0.21, 10O. Rampstep: AT= 0.2; A0 = 10”.
incident velocity wave for 0 = 0”,
303
TRANSIENT SCATTERING BY A CYLINDER
IO.0 
‘_
so
r” “0 :
6.0
,x B 2 m
4.0
I 2
I
0
I 3
I 4
I 6
I 5
I 7
I 9
I 8
IO
Time. I
Figure 9. Boundary
velocity for soft cylinder subjected to Gaussian incident velocity wave for 19= 0”; 0.521.
[email protected] =0.521): L~T=O~~(LU’=O~O~X);de= lo”.
t= 0.0(0.2), lO.Oand t = 0~0(0~05),5~O(x);o= I5
1
0
2
1
4
6
8
IO Time.
12
14
16
18
2(
i
I
Figure 10. Boundary velocity for soft cylinder subjected to Gaussian incident velocity wave for 0 = 0”; t = 0.0 (0.2), 20.0; a = 0521. Gaussian (0 = 0.521): AT= 0.2; A8 = lo”.
this does not appear to have developed within the first five transit times, i.e., by t = lO.Orather the field seems to be oscillatory. To pursue this further, a run was made to ten transit times, shown in Figure 10. Again an oscillating response is found, although these results should be viewed with some suspicion since the total time may be large enough to allow accumulation of errors in the numerical integration to affect the results. The oscillations do appear to have a period of approximately rt as might be expected for two waves, one in each direction, circling around the cylinder and the gross behavior may be expected to be correct but actual numerical values may be inaccurate at these later times. One may examine the same asymptotic formula for the free cylinder as was done for the rigid case. If such an asympote exists, it would satisfy +u(ASYM) = u,(ASYM)  ;
z$ASYM)* (2n),
which requires u,(ASYM) to be zero in order to have a finite although undetermined u(ASYM). Here the importance of the sign change in the two solutions is seen. 4. CONCLUSION An alternative solution form has been presented for wave scattering problems which takes a particularly simple form for this simple geometry. Although the numerical solution scheme
304
R. P. SHAW
used was straightforward, there are other and undoubtedly example a Galerkin approach to determine the coefficients of case, the approach is seen to be direct and useable, and has (see, e.g., references [17] and [18], commented on in reference scattering problems following essentially from reference [lo].
more efficient procedures, for “best fitting functions”. In any been applied to other acoustic [13]) and electromagnetic [19]
REFERENCES 1. F. G. FRIEDLANDER1954 Communications in Pure and Applied Mathematics 7,705732. Diffraction of pulses by a circular cylinder. 2. F. 0. FRIEDLANDER1958 Sound Pulses. Cambridge University Press. 3. E. Y. HARPER 1971 Journal of Applied Mechanics 38,190196. The scattering of shock waves by cylindrical cavities in liquids and solids. 4. F. GILBERT and L. KNOPOFF 1959 Journal of the Acoustical Society of America 3, 11691174. Scattering of an impulsive elastic wave by a rigid cylinder. 5. H. HUANG 1970 Journal of Applied Mechanics 37, 10911093. An exact analysis of the transient interaction of acoustic plane waves with a cylindrical elastic shell. 6. H. UBERALL,R. D. DOOLITTLE and J. V. MCNICHOLAS1966 Journal of the Acoustical Society of America 39, X4578. Use of sound pulses for a study of circumferential waves. 7. J. B. KELLER 1958 in Calculus of Variations and Its Applications, Vol. VIII, edited by L. M. Graves, Proceedings of Symposia on Applied Mathematics, McGrawHill, New York. A geometrical theory of diffraction. 8. T. L. GEERS 1972 Journal of the Acoustical Society of America 51, 16401651. Scattering of a transient acoustic wave by an elastic cylindrical shell. 9. Y. M. CHEN 1964 International Journal of Engineering Science 2,417429. The transient behavior of diffraction of pulses by a circular cylinder. 10. M. B. FRIEDMANand R. P. SHAW 1962 Journal of Applied Mechanics 29,
[email protected] Diffraction of a plane wave by an arbitrary rigid cylindrical obstacle. 11. W. L. Ko 1970 Journalof Applied Mechanics 37,345355. Scattering of stress waves by a circular elastic cylinder embedded in an elastic medium. 12. H. LAMB1945 Hydrodynamics. New York: Dover Publications. 13. R. P. SHAW 1968 Journal of the Acoustical Society of America 43, 638639. Comments on numerical solution for transient scattering from a hard surface of arbitrary shaperetarded potential technique. 14. R. P. SHAW 1975 Journal of Applied Mechanics Paper No. 75APMW40. An outer boundary integral equation applied to transient wave scattering in an inhomogeneous medium. 15. A. C. VASTANOand E. N. BERNARD1973 JournaiofPhysical Oceanography 3,40641&L Transient long wave response for a multiple island system. 16. R. P. SHAW and M. B. FRIEDMAN1962 Fourth U.S. National Congress of Applied Mechanics pp. 371379. Diffraction of a plane shock wave by a free cylindrical obstacle at a free surface. 17. R. P. SHAW and J. ENGLISH1972 Journalof Soundand Vibration 20,321331. Transient acoustic scattering by a free (pressure release) sphere. 18. K. M. MITZNER 1967 Journal of the Acoustical Society of America 42, 391397. Numerical solution for transient scattering from a hard surface of arbitrary shaperetarded potential approach. 19. C. L. BENNETTand W. L. WEEKST970 Institute of Electrical and Electronic Engineers Transactions AP18, 627633. Transient scattering from conducting cylinders.