Japan: Shapes on and echnical e) and M. Dpeller, itects of and Y. Highly Marine No. 1, e Effect Surface Society .panese) Highly Marine No. 9, Propel-av and iety of )75. (in
* Naticual Defense Academy
*4 Tsu Laboratory, Nippon Kokan K. K.
NISCHE UNIVERSITEIT (aboratorIum voor $cheep3hydromechanja Archlef Mekeiweg 2, D Deift 11L
On the Complete Reflection of Water Waves
Masatoshi BEssno*, Zl4cmbcr, Yusaku KYOZUKA*, il[c;nbcr, Osamu \TA\IAaIOTO** Ii1ember
(From J.S.N.A. Japan, Vol. 151, July 1982)
Summary
This is a theoretical and experimental study of the water wave reflector which is oating and moored.
Solving motions of moored floating body among waves by the two-dimensional wave theory, we may predict the amplitude of reflected and transmitted vavc, so that we may easily find the wave-length at which the incident \rave is reflected or transmitted completely. This wave-length depends mainly on the spring cowtantof its mooring system. Fiente, adjusting this restoring force, we may compose a complete reflector or transmitter for an incident wave of arbitrary wave-length.
Then ve describe conditions of complete reflection anti transmission by making use of phases of wave exciting force and mechanical impedance of mooring systeimi in the complex plane. Thms description enables us to estimate easily the necessary spring constant at an ar-bitrary wave-length.
Experiments on a semi-submerged vertical plate and a circular cylinder show good agree-meats with theoretical prediction.
According to this theory, it becomes possible to design a reflector of long wave ever ten timne5 of the breadth of floating body.
1. Intr3duction
Various kind., of wave breaker hìave been developed and have received practical applica-tion in recent years. Principles of wave
break-ing applied to these wave breakers can he
roughly classified as follows, reflection of an incident wave, conversion of wave energy into eddy energy in tluid, and combination of these principles. \Vave breaking by reflec-tion of an incident wave can be easily predicted by using a potential theory, and SO the
poten-tial theory lias usually beco applied for com-parison with model experiments and so on. But, in the case of conventional wave breakers, ideas of form or configuration of a wave break-er usually comes first, and a theoretical
investi-81
gation on wave breaking may not be carried out sufficiently. This fact leads to an inferior performance of wave breaking for waves with long wave lengths, and therefore results in the scaling up of a wave breaker for improve-ment of the performance.
On the other hand, many studies on wave power absorption have been carred ont
vigorously due to the recent
fossil energycrisis. Recent studies show that the maximum efficiency of wave power absorption attained by using a single mode oscillation of a two-dimensional floating body is only 50% hut it becomes 100% if a complete wave reflector exists at the 1cc side of time body. Therefore,
it
is obvious that a reflector with superior
performance can be effectively utilized for the PUrPOSe of the wave power absorption by a floating body.
relee-Fig. i Coordinate system
By using these expressions, equations of the motions in three degrees of freedom can be expressed as foflows:
Wk(--OG).
= F13 +F13 + + F131 + Fiw(3)
f Vkt F22 +F2R +F2M + F2 ( 4 )WkÖ1-Ik3
= F33 + F31 + FIR + F3 + F31. + F3w(5)
Where 0G denotes the distance of time center
tion of an incident wave arc derived from
a theoretical investigation based on the two-dimensional wave theory, and an attempt is made to realize complete reflection at arbi-trary wave periods by tuning the mechanical impedance of mooring systems. Consequently,
it is
found that complete reflection of an
incident wave can be attained by using a
dc ice with a simple mechanical configuration and that it can reflect not only short waves but also long waves in high performance.
2. Theory
2.1 Problc;n formnulatwn (or Equation of i;io-tío n)
The coordinate system is shown in Fig. i
We denote an incident wave
swayingmotion j and heaving motion ?Jz of point O and rolling motion 1J3 around the point O as follows:
j=l,2, 3
of gravity of the floating body from coordinate origin, W and I arc tile weight of the floating body and moment of inertia around point O respectively. is the hydrodynainic force in the i-tim direction caused by j-tu mode motion. By using an added mass coefficient fije and Kochin function Hj+, it can be
expres-sed in time form
(6) F2R and F3, denote hydroclynamic restor-ing forces for heavrestor-ing and rollrestor-ing motions, and they can be expressed as:
F2= -jigLB
(7)
F3a= pUVGM
(8)
where L, B and V are length, breadth and
displacement volume of the floating body respectively.
F1 and F1f( are restoring forces in i-th
merle motion of a mooring system caused by j-th mode motion. If the number of the
moor-ing lines is n and the moormoor-ing lines of sprmoor-ing constant Rn are stretched symmetrically about the center plane, these become:
F131 = - cos = - nkn sin 0
F331= i1kJf(x312 sin 0+ yn2 cos F13 =ii Rn Cos OY;r3
Elm = nkir cos 0y;r
ITere O shows the angle freni the horizontal plane to the mooring line, and Xjt and ya are the coordinates of the points where the moor-ing lines are connected to the floatmoor-ing body.
By the use of the Kocliin function, a wave exciting force in the i-tu direction can be ex-pressed as following:
F=-1xjHjL
(14)2.2 linj5cdanccn
With an assumption that the amplitude of an incident wave is unity, and by substitu-tion of Eqs. (6) (l4) into Eqs. (3), (4), (5), the equations of motions are replaced by the following: Zi11+ZL23= j:
Z2:2= -pgIi/i
Z31-i+Z33= i
1-lere Z0, what is cí ance, is expressed as fZ11=pH1ÏU+i
uRa ces O wL fll?.4 5ri O wL Z33 = f3Û)HlH3+ L imk11(x312 sin Z13=Z31=pwlH1 .Ç / TVO L Û) I qLFor t1e convenience rewrite the impedan phase as follows: Z11 p(aHiII1
Z =pwIIÏf
Z33 = pwH3T1 Z13 = pw1Hihii 2.3 Condition of coi Complex amplituc a transmission wal/Hz
H + ¿k2R I /II2 H AT= + j/?2II2\Ve will also rewrit 82 Masatosixi Basano, Vusaku KvoZuzA, Osanin YAMAMOTO
4 ly from coordinate ght of the floating a around point 0 Tdrodynarnic force
d by j-th
mode mass coefficient it can be expres-rodynamic restor-i rolling motions, s:gth, breadth and
he floating body
ing forces in i-tb
system caused by rnber of the moor-¡ng lines of spring rnmetrically about ne: ijr2cosO)Ç3 (11) m the horizontal md X3r and are s where the moor-the floating body. i function, a wave rection can be
ex-the amplitude of and by substitu-Eqs. (3), (4), (5), -e replaced by the
ZiipuHiJIj+sec/Jiei
On the Complete Reflection of Water Waves
pci.)II3H3' sec fia
Z11 pcolHi sec fi 3V
2.3 Conditio,'i of complete reflection31
Complex amplitudes of a reflection wave and a transmission wave are expressed as follows:
I H. 1J
AR=(+
+ ik(1+l3)H1
+ikII1
(26)(27) We will also rewrite the Kochin function by
amplitude and phase as follows:
(281 By substitution of solutions of Eqs. (1 i (17) into Eqs. (26), (27) and by the use of }', (52)(25), (28), Eqs. (26), (27) are rewrjtt1. as
AR= __C2'22) +---e'''>
(29,A ' = - -h-e a2(-,î) __Ci21_1') (30; where fi' is
fii'=Arg{sec/]ie1 secfiaci'3scclfiIlen1} (31) The first term on the right hand of Eqs: (29), (30) shows a symmetrical wave and the second term an asymmetrical wave respectively.
Since complete reflection means
ARI 1 (32)
a condition of complete reflection is derived from Eq. (29) and is expressed as follows:
121'fiz±(2fl+l)-_
(33)On the contrary, in the case of Ar = 0,
which reans complete transmission, a condi-tion of complete transmission is derived from Eq. (30) and expressed as in the following.
12fil'fi2±fl7T
(34) In the same way, a condition which make-s D transmission coefficient smaller than 0.5 isobtained as follows:
(ci2)-(35)
2.4 Exam lc of a umerical calculation 2.4.1 \Tertical flat-plate
We will consider a vertical flat plate oscil-lating only in swaying mode-. Complex response amplitude of swaying motion ran be expressed by using the expression of impedance shown by Eq. (h 8) as follows: pgLH1 20.1/11 (36)
Zii+Zua= pçiHi4]ico
(15)Zazz pgIJ/iw
(16)Z3ia+Zt= pgH3Jiw
(17)Here what is called Mechanical imped-ance, is expressed as follows:
¿11=poH1+IL++{oi(L +1)111e)
--
IV fl/ar COS O (18) oL Zz2=pwHH3++i{ / JV+pf2zc) nR.;z sin O pf7B) (19)-
wLZ3=poH3Ï[+i fw(i-+pfac)
sin 0+ Th12 cosO) pqvGM (i)L Z3 =Z31 =PWIWIL+.L+ WOG COS O qLFoi- the convenience of later sections, we will rewrite the impedance Zi by amplitude and phase as follows:
81
Numerical values of i arid /J ± (2n + 1)17/2
calculated by changing the spring constant of the mooring system are shown in Fig. 2. As shown by Eq. (39), the cross point of curve and /Ji±(2n-+- 1)17/2 curve is the point where complete reflection is realized. kx' in Fig. 2 means a nomdimensional value defined by the following expression
Masato1ii I3assno, Yuaku I'vozUIA, Osaniu YAMAMOTO
(38)
(39)
(40)
i
Fig. 3 Phases of Koch for a swaying a
Fig. 4 Mooring
condition of compiE plicable to this cas' (33) already. Resu
)
and {/J1fl2±(
Fig. 5.
In thc case of
/(fli/Jz) curve cros point, while, in ca these curves occur fore, complete reti the frequencies of
case of RM'=3.9, ti t) curve and
(flu-IT/fl in ka ranging fi
derived from Eq. coefficient takes a
Moreover, in the Ci ence between
(t
curve bccomeS ab ramige, it follows fiFig. 2 Phases of IÇocliin f iction luid impedance for a swaying vertical cylinder
The Koch iii function of a vertical flat plate is, as shown by Kotik°,
H±--
Il(/?d)+Ll(k4 (3)k K1(ka) + uni (ka) /
where 11(k5) and
Ki(k)
are moditied Bessel functions of order 1, and L1(ka) is a Struvefunction of order 1.
Complex amplitudes of reflected waves and transmitted waves can he rewritten, by sub-stituting Eq. (36) into Eqs. (26), (27) and by using the expression of Eqs. (22), (28), as in the following:
1IR
AT
=ij_ei2(il)
22
From time Eq. (38), the condition of complete wave reflection becomes
(41)
It is clear froni Fig. 2 that there is not any
frequency where complete reflection can be realized in the frequency range if the spring constant of the mooring system is zero, and that there is only one frequency where com-plete reflection can be realized if the spring constant is a finite value.
2.4.2 Half immersed circular cylinder
We assume that a half-immersed circular cylinder oscillates without interaction be-tween swaying and rolling motions. Then complex amplitudes of swaying and rolling motions can be expressed in ternis of the impe-dances of Eqs. (18) and (19) as follows:
-
(42)i(0Z22 (43)
Substituting Eqs. (42) and (43) into Eqs. (26) and (27), and using the expressions shown by Eqs. (22) and (23), Ecjs. (26) and (27) can he replaced by the folliwing expressions.
From Eq. (44), a condition of complete rellcc-tion is derived as in the following:
iaiflifiz± JilT
(46) Fig. 3 shows the result of calculation on (and (/11 -ß ± n17) in ti le case of a half-immersed circular cylinder moored by hori-zontal mooring lines. The definition of kir' shown in Fig. 3 is the same as Eq. (41). Fig.
3 shows that
the intersectionof (an)
curve and (/]t,82±017) curve occurs at two point in the frequency range shown in Fig. 3 if the spring constant of the mooring lines is tuned at time appropriate values, and this fact means that complete reflection can he realized at two frequencies of these intersections.
2.4.3 Floating body of Lewis form section As an example of a general floating body, we consider the case as shown in Fig. 4. TIre AR= _C122_i lei2(l_,,I) (44)
i there is not any
reflection cali be ange if the spring rstem is zero, and uency where coto-[ized if the spring sr cylinder
immersed circular t interaction be-motions. Then aying and rolling terms of tue
impe-as follows: (43) into Eqs. (26) tressions shown by 3) and (27) can be pressions. (4.4) of complete reflec-wing: Liculation on (i
-e cas-e of a
half-moored by bon-definition of Ä' as Eq. (41). Fig. tionof (az)
ve occurs at two shown in Fig. 3 mooring lines is ues, and this fact )n can be realized :tersections. s form section al floating body, n in Fig. 4. The '(/2 XFig. 3 Phases of Kochin functions and impedances for a swaying and heaving circular cyltnder
Fig. 4 i\louring of a Lcis form cvlitidcr
condition of complete reflection which is ap-plicable to this case has been shown by Eq. (33) already.
Results of calculation on
(-a) and {/h /I ± (2;i.+ l)n]2} arc shown in
Fig. 5.
In the case of ktz'=O, (i--)
curve and(fit
ß2) curve cross each other at only
onepoint, while, in case of /nr'3.9, crossing of these curvos occurs at three points. There-fore, complete reflection
can be realized at
tite frequencies of these three points, in tite case of k1'=3.9, the difference between (rei-ot) curve and (/Jt/32) curve is smaller than
m116 in I/a ranging from 0.2 to 1 .2, so that it is
derived from Eq. (35) that tue transmission coefficient takes a smaller values than 0.5. Moreover, in tite case of kM' = 13, as the
differ-ence between (1-2) clurve
and (/3i7)
curve becomes about x/2 in a wide frequency range, it follows frani Eq. (34) that complete
On the Complete Reflection of Water Waves
o i.o T O.75 Ho 0.65 G- O.857 55 = 0.34 a =Q3 B X,, 0.4 0.55 o 't o '(/2 -R:
Fig. 5 Phases of Kochin functions and impedances for a Lewis form cylinder oscillating in three dcgree of freedom
transmission is almost realized in the frequency range.
3. Model experiments 3.1 Verticaiflat plate
A vertical
flat-plate used in
the model experiment is sltoovn in Fig. 6. This plate is attached under a slide guide which is installed on a narrow channel of 4 ni length and 0.61 breadth, temporarily constructed in a wave basin of L ><13 xH=9 xl .2 xi m. In the model experiment, heaving and rolling motions are restrained by tite slide guide and only swaying motion is allowed. Waves arc measured by three wave-height-meters, and one of themis installed at
lee-side and the others
at wheather side. These wave signals are ana-lyzed by a Fourier series expansion and ampli-tudes of an incident wave, reflected wave and transmitted wave are obtained from tite0.01
Fig. 6 Dimension and figure of tite vertical plate for model test
p, 6.5 t 1421 OCn= .0 l.o <,j 2.0 O c'1.o
S6
Fig. 7 Swaying motion of the vertical plate
Fourier series.
The amplitude response function of swaying motion is shown in Fig. 7 and the reflection coefficient in Fig. 8. In these figures, small rowid marks show results of the model experi-ment and solid and brokcn lines show results of theoretical calculations. The broken lines are calculated including the effect of mechani-cal friction, in which case impedance is expres-sed as follows:
ZU j%JiIiI+/Lf
j (TV
'\ 11kM)mt
t --rpfitc1)--- oiL (4/)
where p is the frictional resistance coefficient. lt is clear from comparing tise results of the model experiment with the theoretical calcula-tions that the reults of the model experiment almost coincide with the theoretical calcula-tion which includes the effect of mechanical friction. As shown in Fig. 8, the reflection coefficient calculated without mechanical
fric-tion becomes almost 1.0 at K4=0.5, while
both the results of the model experiment and the theorettcal calculation with
mcchanical-friction show Cr=0.7 at K4=0.5, so that it
is concluded that complete reflection can iot
be attained in our model experiment on a
swaying vertical flat plate. Generally,
if a
frictional force exists, a dynamical system be-comes dispersive and energy conservationMasatoshi Bassuo, Yusaku KYOZUKA, Osamu YAMAMOTO
Fig. 8 Reflectioo coefficient of the vertical plate
law's do not hold well. As this fact inevitably leads to CR< 1.0, it is proper that complete reflection can not be realized in our model
experimen t.
3.2 ¡JuIf immersed circular cylinder
The fIgure and dimensions of a circular
cylinder arc shown in Fig. 9. Model experi-ments are carried out by using the cylinder which i 2.97 ni in length and 0.6 ni in breadth and is half-immersed in a wave flume of L x B xH=60 x3 x 1.5 ni. \Vave measurement and analysis are carried outdn the same way
as that used in the case of the vertical flat
plate. Motions of the cylinder are measured by a motion measuring instrument equipped with
B =0.6 T =0.3 HO .0 O. =0.7854 55 0.075 ç =02 X= 0.272 b..- 0.127 e o
Fig. 9 Dimension and figure of the circular cvlin-der for model test
4 f. 3 2 o o o
Fig. 10 Swayingmotion
poten Lio-nieters. Moorin wires and coil Springs, and with the cylinder at a posi the water surface and stre Amplitude response fun and heaving motions are and
il,
and transrnjssio reflection coefficients are and 13. Smallround niark these figures show resultsperiments and the
theorerespectively in the case of l triangular marks and brokc of K71'=2.8. These figure results of the model ex coincide with the results o
0c
.5 '.0
64 20
1. <4 20
f the vertical plate
his fact inevitably cr that complete ;cd in our model
)'iin der
Os of a cirrular
9. Model experi-sing the- cylinder 1 0.6 m in breadth ave flume of L > ve measurement
in the same way the vertical flat are measured by nt equipped with B =0.6 T =0.3 Ho .0 0 =0.7854 65 = 0.075 a =0.2 X 0.272 M 0.127 e - o
the circular
cylin-.4 a 3 2 o C 'b o 'B
Fig. 10 Swaying motion of the circular cylinder
1.5
¿fl\
N
KO 2.0
Fig. li Heaving motion of the circular cylinder
potentio-inetcrs .Moor ing lines consist of wires and coil springs, aoci these are connected with the cylinder at a position 0.127 m under the water surface and stretched horizontally. Amplitude response functions of swaying and heaving motions se shown in Figs. 10 aoci 11 and transmission coefficients and
reflection coefficients are shown in Figs. 1 2
and 13. Small round marks and solid lines in these figures show results of the model ex-periments and the theoretical calculations respectively in the case of k.V=9.l, and small triangular marks and broken lines in the case of K.u'=2.8. These figures show that tue results of the model experiments almost
coincide with the results of the theoretical
Fig. 12 Transmission coefficient of the circular cylinder
Fig. 13 Reflection coefficient of the circular cylinder
calculations. A slight discrepancy is observed in the reflection coefficient in Fig. 13 in the case of K.Tr'=9.l, and the experimental values are smaller than the theoretical values.
Al-though complete reflection is
not attained
perfectly, it can be recognized that a result close to complete reflection is attained in the model experiments. Fig. 12 shows that the transmission roefficient becomes zero at three frequencies in the case of K,1' 9.1. This fact is different from the result shown in Fig. 3,
but it
is proper because both swaying and rolling motions interact witheach other in
tins case, in the case of Kr'=9. 1, the trans-mission coefficient becomes about 0.5 or less in the frccuency range of K40.3 and shows good perfurnmance of wave reflection.
As clarified by these resuhis of the model experinients and the theoretical calculations, the tuning of mechanical impedance by chang-ing the sprchang-ing constant of the moorchang-ing lines leads to change of the natural frequency of
20
On the Complete Reflection of \Vater Waves 87
.5 1.0 1.5 64 2.0
1.0
88
swaying motion, and results in higher
per-formance of wave reflection in tite liigi t
frequ-ency range above the natural frequfrequ-ency as
shown in Figs.
12 and 13.
Consequently, cxcelient performance, where tite reflection coefficient is larger than 0.9 and tite transmis-sion coefficient smaller than 0.2, is attainedeveir in the case where the wave length is
longer than ten times the breadth of the
floating body. 4. Conclusion
The performance of wave reflection of a
floating body equipped with a mechanical
mooring system was investigated by using the two-dimensional wave theory. The concept of impedance was app lied to the iiytlrodynaniic property of a floating body, then rctiectcd and
transmitted waves were described by the
phases of the impedances and tite Kochin
functions. Moreover cbnditions which should be satisfied b:T the pitases of tite impedances aoci the lochin functions for complete reflec-tion were clarified. Numerical calculation on these pitases were carried out with respect to a flat plate, a half-immersed circular cylinder and a Lewis form cylinder. Results of the calculation showed that complete reflection can be realized at mere than two frequencies if tite impedances arc tuned properly by
ad-justing the spring constant of tite mooring
system.
Furthermore, model experiments were cat:-ned out concerning tite flat-plate and tite
circu-Masatoshi BEssno, Ynsaku KY0ZuKA, Osarnu YAMAMOTO
lar cylinder, and it vas confirmed that results of tite model experiments almost coincide with the results of the numerical calculations. Moreover it was verified
that a long wave,
more than ten times the breadth of tite float-ing body, cari he effectively reflected in the
case of the circular cylinder by tuning tite
impedances.
For the practical application of the results of titis paper a long length floating body causes
problems such as structual and sea traffic.
But the same property could he expected by
a linear array arrangement of man.y short
boches in the antenna theory. References
M. Bassno: On tite Theory of Rolling Motion of Ships among Waves, Scientific and Engineer-ing Reports of tite Defense Academy. Voi. 3, No. 1, May (1965), (in Japanese)
M. BEssHo: On tite Theory of Rolh.tg Motion of Ships among Waves, Scientific and Engineer-ing Reports of the Defense Academy, Vol. 3,
No. 3, Jan. (1966), (in Japanese)
M. BESSHO: Introduction to Water Wave Engineering, Bui. Soc. Naval Arch. Japan, No. 634, pp. 9-20, April (1982), (in Japanese) J. RoTta: Damping and Inertia Coefficients for a Roiltng or Swaying Vertic,ti Strips, J.S.R.,
\i 7, No. 2, I'I 19-23, Oct. (1963)
T. KtMUR.\, A. H0NMA: Study on Wave Power Absorption (on a device of complete reflection or transmisaion), graduation thesis of the Defense Academy, March (1981), (in Jpanesc)
i
The after vater plane is upper vater pl near the water section of ship impact pressur on the stern fis The f tamed from th of tite coefficiet 'fhe i to 11 Beaufort is smaller than sure is large. tion induced b S.
Oni
J' 1. -Introduction Wheti ships are they sometimes exp ing green sea impcand the how fiare
pressures cause not
structure but
alsovibration. Because by whipping vibra water bending stre atress, it is necessar these impact pressul ing the longitudinal fore, many studies
this problem untiE Car ferries, pure ships with stern rae aft bodies in order td * Hiroshima Univc Kawasaki Heavy
