Measurement of the wave height producted by the forced heaving of the cylinders

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T 'I



With Compliments

Measurement of the Wave Height Produced by

the Forced Heaving of the Cylinders

By FukuzÖ TAsAI


Reprinted from Reports of Research Institute for Applied Mechanics, Kyushu University


Reports of Research Institute for Applied Mechanics Vol. VIII, No. 29, 1960



By Fukuzô TASAT

ABSTRACT. We measured the two dimensional progressive wave height produced by forced heaving of the miscelanous cylinders, and then compared the amplitude ratio .i with the F. Ursell's results [1] and auth-or's theoretical calculations [2]. In general, the measured A was in good coincidence with that of the theoretical results. Then the author studied the Wedge effect, the effect of the variation of the section contour from the

Lewis form and the effect of heaving amplitude. These are shown in many figures in this paper.

1. Preface

It is essential to know the sectional damping force for the calculation of the damping force of heaving and pitching of a ship by the Strip Method. This

sectional damping force may be easily given if the progressive wave height pro-duced by the heaving of the unlimited cylinder is estimated, provided that neglect-ing the frictional force the dampneglect-ing force is proportionate to the heavneglect-ing velocity.

Namely, taking the amplitude of the progressive wave height by heaving amp'

litude by S, the ratio between them is indicated by A =- /S and using the ratio

the sectional damping force is calculated by the equation,

N=-. A2

where p = fluid density,

g acceleration of gravity,

= wave circular frequency, N = damping force/heaving velocity.

Theoretically accurate calculation of A for the circular cylinder was given by F. UrseIl [1] and for the Lewis form by the author {2J respectively. In order to calculate they linearised the free surface condition by neglecting the viscosity and surface tension and by assuming the forced heaving amplitude being small as always done at the wave theory. Therefore the difference between the theoretical value and actual A having various finite heaving amplitude is determined by the

experiment. Lewis Form calculated by the author in [2] was wall-sided on the

water line. However there are many cases of the non-wall sided in ship-shaped


Photo J. General view of the water tank.

Photo 2. Forced heaving apparatus.

280 F. TASAI

section. Generally it is very difficult to treat them theoretically. Damping force N of a certain section could be calculated approximately from [2], supposing that

the value of N is equal to that of Lewis Form whose area, draught and the width of water line are respectively equal to the value of that section. But in this case the influence of variation from Lewis form over the value of A is not yet eliminated.

To detect these influence, two-dimensional progressive wave height which

was produced by forced heaving of the cylinders, was measured. H. Holstein [3J

is the only person that have done those measurements before. He, using the wave profile board scored with 5 mm section, measured the two-dimensional progressive wave height which was generated by the forced heaving of the rectangular cylin-der with rounded corner changing the draught variously.

In this experiment he used a warter tank the length of which was very

short (L = 3.00 'n, B = 0.7 m, D 0.5 rn), so that the reflective and the standing waves would have exerted great influence over the wave profile. But it is not clear how he had treated them. So his measurements are supposed not to be ade-quate to compare with the theoretical one. To compare it with the theoretical cal-culation accurately the author had done the experiment changing the amplitude

of the plunger variously, first, on the circular cylinder. And to know the

influ-ence above-mentioned the test was exerted on the trianglar cylinder, rectangular

cylinder with rounded corner and that having ship-shaped section. Now I should like to present the results of these experiments.

2. Experimental Apparatus and Measuring Method

The test was held in the water tank being L = 60 m, B 1.5 m, D = 1.5ni







Gr de r



Fig. 2. Forced heaving apparatus.


--- Motor and heaving Cylinder , L.L2 Wove absorber - - -Plate tor meas4jring wave profil

- - - float for mesuriri9 wove heihf i---- Flop type wove nicker

Fig I. Water tank and general arrangement of experimental apparatus

2375 -E 60m -WL 2 2, Cylinder

Fig. 3. Heaving cylinder (duralumin).

019m W.L


Tc ri k wall Lo 5m Gea r Box Floor o 370 -r--- 370


'Weld k


-e--24 in 475m-- - 1825m T 3.0.


282 F. TASAI

The explanation of the experimental apparatus and measuring method is briefly written in those that follow.

2. 1 Forced Heaving Apparatus

Fig. 1 shows the general arrangement of all the apparatus. Fig. 2 and Photo.

2 illustrate the forced heaving apparatus. Plunger P was operated by the

elect-ric motor Al, eccentelect-ric apparatus E and connecting rod R. The amplitude could be verified by 70 mm. The wave making cylinder shown in Fig. 3 was made of duralumin of 150 mm in semi-diameter and 1 mm in thickness. The dimension of the other cylinders is elucidated in the figures of the experimental results. The

clearance of 2 mm - 3 mm was found between the cylinder and the water tank

wall. These formed a plunger type wave making machine. 2. 2 Wave Damper

Being elucidated in Fig. I and photo. 3a. 3b, there are wave-dampers at

Photo. 3 (a) Wave damper. (L1)

both ends of the water tank. L is ready made one operated fairly well because

of the whole surface of which being covered with hemp-palm brushes. The

appa-ratus L1 is the latticed one which is newly settled for this exreriment. The ap-paratus is in the following: The front wall of 11 is made of the hemp-palm

bru-shes packed in the frame of 60 cm in height. A board is put on the wall with the inclination of 12°. And thert square timbers of 30 mm is arranged like the lat-ticed frame, having the clearance of 10 mm between the board above-mentioned.

As the reflexion experiment, produced waves were recorded by operating

the plunger a few times.

After a group of the waves passed, water surface became still. Soon after

reflective waves with long period appeared gradually lessening its period. Though

wave height of the reflective wave was generally very small, wave height having

the period near the plunger period was higher than the other part. The time req-uired for the appearance of the main reflective waves was almost estimated from the travelling distance of the waves and the group velocity of the wave with plunger period. Travelling distance of the wave from plunger to measuring float

via the apparatus L1 was about 42 m 50 m.



Table I indicates the time required for the appearance of the main reflec-tive waves on the three plunger periods.

s p1 Pen oscI Io. wooden girder Tonk W.L <wall F

Tw= Heaving period of the plunger

t1, t2 = The time required to produce the main reflective waves after operating the plunger

t1 is the measured value, 12 the calculated value

Main reflect ive wave height generated by the operation of L1 was under

5% of the theoretical regular one for that period. For the L2, the travelling dis-tance of waves being about 53 m--60 m, the reflective waves at T = 1.0 and 0.636 by L1 and L2 was supposed to be recorded separately, but we could merely obtain

the record by L1. Reflection by L2 is supposed to be very small.

2. 3 Wave-Height Measuring Apparatus and Measuring Method.

Two methods were used for recording the wave height. The first one was

float-type wave height recorder which was small one (15cm X 2 cmX2 cm) of

plastics. Fig. 4 and Photo. 5a shows its appearance. As illustrated in Fig. 4,

Fig. 4. Apparatus for measuring wave height.

frames D1 and D2 were respectively revolved round the hinges i1 and j. Float

F with vertical axis R1 nearly took heaving motion. Then a fine tungsten wire

( -- mm) and a pulley was used to combine the rod R1 and the pen P1. The

wire was, moreover, pulled by the spring S1 with statical tension of 2gr. The

measured average free heaving period of the float was 0.255 sec. The float having

t1 (sec) 40-53 58--52 95 100

t (sec) 44-51 54-64 87-100

Table 1.


2 7.85.. T. 0531e

23: 9.5 - -' t: 2.120

¡ 0.9263

¡:0.9101 (corrected)

I Sec

Circular Cylloderl NO. 85

Then turn to the second method.

Illustrated in Photo. 7, wave profile scale board (200 cmx 30 cm) was hunged in

284 F. TASAI

such short period was designed taking into consideration the freiuency of the

forced having in our test, namely, wave frequency. Heaving displacement of F

was recorded by P1 through the tungsten wire and the pulley (See Photo. 5b).

Pen P1 was desinged not to move circularly but linearly. In the range of the

Photo. 5 (a). Float for measuring wave height. Photo. 5 (b). Pen Recorder.

heaving displacement of F in this test, the dynamical change of tension was so negligible that pen P1 worked very good in recording the displacement of the float

F. The recorded wave height was divided by mangification factor Y and

reduc-ed to the correctreduc-ed value. Some of the measured ones are shown in Photo. 6

and the i (corrected) is the one divided by Y.


I 8.0... T... rO.7946

2523.5- J,o.5o6

2 0.7CGO

¡ 0.7584 (correo ted)

lCircubar Cyhoder INC. 90


"-t.55.2 25 4.2 A 0.8018 A 0.7939(C0#RECT0l 7, :


ICirc.ilor Cylinder j NO. 10)1




t' V V

T.0.533l 2Sr 21.5 ¡ .209 ¡. I.IeS (CQjcfQd) , 1.9Ç'I sec1 fCW1 No.321


r- Period H Pflotogroph


r Is e c





Wove height obtained by Hoot (corrected)

lo 20 30t..

Fig. 5.

3. Summary of Experiment

In the calculation [2], A = /S is indicated as the function of =

The value B denote the breadth of the section on the water line, the maximum

of which was 30 cm in our experiment. The experiment was limited within 4< :12, taking consideration of the depth of the water-tank (1.3 m), natural heav-ing period of the float F and <2.5 which is sufficient for the calculation of the

damping force of a ship. The heaving amplitude was varied widely from 5 mm

to 35 miii. At the first place motor was operated with constant rotation before

the clutch of the plunger was put on. Then the plunger was worked with

con-stant period and amplitude. While time and period of the plunger began to be

286 F. TASAÌ

the water tank apart from the tank wall by 10 cm. And photographs were taken. Then projecting the nega film on the screen we could obtain the wave height by using the vertical scale, though accurate data could not be given owing to the enlargement of photo. But the difference between the value and the measured one

by the float was merely 2 - 3 (See Fig. 5). The data furnished by the float

was used for the measurement of the wave height, white the photographs were employed for the sake of checking the experiments and observing the waveprofile. The measuring point of the wave height being 15 m apart from the plunger, the

influence of the standing wave was so small as to be negligible in practice.



marked on the recording paper. Long waves were begun to be recorded

gradual-ly and the waves with the pluoger period appeared soon after. And finally pho-tographs were taken after the wave height record had been obtained with

regula-rity. These measurement should be done before the arrival of the main reflective wave for each . lO minutes was taken for the interval of each test in order to calm down the tank water. In this case the reproducibility of the experiment was very good.

4. Results of the Experiment

The length of the generated progressive waves were within 3.5 m>À>O.4 m and as shown in photo. 6, 7, they generally approximated to the sine wave.

Photo. 7. Photograph of wave profile.

4. 1 Circular Cylinder

Fig. 6 (a) shows the reault of the experiment on the semi-circular one. When the amplitude was small the value A by the experiment showed fair coin-cidence with the Ursells one. But with the increase of the amplitude the value

A got a slight less value on the average at large

Now let us turn to investigate on the cylinder with segmental section, the


l0 08


4 2 Io 08 06 04 02

Theoretical Results by Ursell

Experi nients

-0 -


Fig. 6a. Cylinder with segmental Section

H.=15, I30.722

Theoretical value obtained fron,

the reference (2)



H 1.0

Fig. 6b.

s = Heaving cmplitude of the cylin der T o= i50)

n 2S=ll.7,..,,

O---- 2S=20,


S= Heaving amplitude ot the cylinder

5 0,,

1.3 .2



dimension of which being T = 9.3 cm,-Ç = 13.85 cm, that is B/2= H0 = 1.5, 0=

0.722. The main results of the experiment could be summarized in Fig. 6 (b),

288 F. TASAI

Serrii- Circular Cylinder

04 02 25= 9.S,er X 2S=l0.5... .. 2S=25 o 2S=4l.2,,. - 0,9 08 0.7 o 05 1.0 IS 2.0 05 IO I5 2.0


1.4 .2 Io 08 06 04 0.2

Cylinder with sernentaI sect ice


H.2O, 0O697

A C e 1.0

-Vclue ci.itoined frorn(2J (See appendix et this paperi erim ests 0 Fig. 6c. Photo. 8 (b) Cylinders. .3 1.2

which was much similar to the curve of A obtained from Fig. 7 of the author

[2] by means of interpolation. The accurate value calculated by Grim [4] (Bild

[7]) was too large. Approximate value was also too large at < 1.0 but too

small at 1.0. And it has been observed for this cylinder that the value A

clearly got hump and hollow with the increase of . , which correspond to the

hump or hollow varied with the amplitude of the plunger.

Next 1 should like to mention on Fig. 6 (c), which is the result of the

ex-periment of the segmental section whose dimension is T = 6.0 cm, - = 120 cm,

H0 = 2.0, e= 0.697. Exact value of Grim [4] correspond closely with the

expri-mental value, however the tendency of the curve varies with the increase of


290 F. TASAI

We have known from the experiment of the circular cylinder that the experimen-tal value shows good coincidence with the Ursell's theoretical one. But a slight difference is found between the experimental value and the theoretical one, name-1y experimental value A has always undergone wavy variation with the increase of È and that the variation of the plunger-amplitude brings forth diversity of for hump and hollow. lt was also elucidated that the calculated value [2] was

sufficient one for the segmental section.

4. 2 Triangular Clylinder

Experiment was done on the three cylinders with the apex angles of which were a = 60, 90, 120, namely H0 = 0.5774, 1.0 and 1.732 respectively. Whereas

o was 0.5 for each cylinder. The values of Ho and a has been always constant in spite of the variation of the plunger-amplitude during the experiment. Fig. 7 (a), 7 (b) and 7 (c) indicate these results. Now let us consider of the cylinders






Triar utar Cylinder

o60, H. 0.58 oO.5

Theoretical value obtained from the reference (2 (by the interpolation)

Ex peri me nts

Fig. 7(a).


a = 60 and a

90. Great hump and hollow was found at large e0, though the

variation of the curve A by the amplitude of the plunger was small. These

pris-matic cylinders were non-wall-sided on the water line, but each of them had the

angle of inclination,


respectively. In this case A might generally

dif-fers from that of Lewis form of wall-sided. This is called Wedge Effect in this

report. On the value À of certain section there is generally found a difference

between the experimental measurement and theoretical calculation, which is as



1.6 .4 1.2 Io A 0.8 0.6 0.4 .6 .4 .2 .0 08 0.6 04 0.2 Lewis form TrianQular Cylinder d-90. H.I.0, O'=0.5 Trionijlar Cylinder rj=I20 Hl.732.G05 b


iu 8

--. r

Theoretical value obtained tram the reference (2) by the interpolation)

Expri ni ents

Fig. 7(b).

O 2SI29,..


Theoretical value obtained tram the refererce (2)

by the ext rpolation)

Ex perime ns flg. 7(c). 5 2.0 1.6 1.4 1.3 - 1.2 1.7 1.6 1.5 .4 1.3 .2


0.5 .0 1.5 '2$ I O 2S=20.6, .0 0.5


292 F. TASA!

Inadequacy of the linear theory when the steepness of the generated waves are large.

Wedge effect

Influence of the large amplitude of the plunger

Variation of the section contour from the Lewis form in spite of the same

II and


71 S

Steepness of the generated wave was il = =Ac°0 B Then í9<0.075

was in case 'o=9OE. When a=60 and a=90', the value A obtained from [2J by the interporation of H0 and a correspond closely with the experimental value.

Then the item (1), (3), (4) and Wedge effect was assumed to be small. Next

examine the case a= 12OE. Known from 7 (C), the theoretical curve of [2] was

coincide with experimental one till And then experimental one decreased

about 10% to 15% with the increase of e0. This was resulted from Wedge effect

that had a little influence at small but the effect increased gradually with the

increase of It is also important that the Wedge effect gives less value than

the theoretical one obtained from [2] for the same H0 and a

4. 3 Rectangular cylinder with rounded corner

Experiment was held on the four sections of H0= 1.0, 1.25, 1.5 and 2.0,

changing the draught of the cylinder with rounded corner which radius of curva-ture r=46 rum.

The value a was 0.98, 0.975, 0.97 and 0.96 respectively. For the small we used

Rectariqulor Cylinder 1.0 0.8 0,6 0.4 0.2 (H.= 1,25) (H. .50) (H,= 2.0) ., w8 B H, .0 cT-0.98

theoretical valuo obtained tram the reference l2



by the extrapolation0.5 - 0,4 DI 0.3 T 0.2 0.5 1.0 i.5 Fig. 8 (a). Experiments X'''' 2sI0.S..,. O 2sll.0,.,. 2s =20.5e.' 2s33.3,.


0.8 0.6 0.4 0.2 A t-J 0.5 A Rectanular Cylinder B H, .25 6--09750

Theoretical value obtained from the relererice Z)

by the extrapolation 0.9420


w' B


-2 Experiments O 2S9.8-. X 2SlO5." 2Sl9.8'-

25=33.0--x .. . .by the small model

1.0 1.5

Fig. 8(b).

Pectanqulor Cylinder

B3--- H,= 5, '=09700

Theoretical value obtained from the reterence (2

2.0 0.5 .0 y the extrapolation Ex per im e nt o ...2S= 9.4.,,,,, s ---2 S=19.8,,,,,, Fig. 8(c).

a similar model of ½ scale, to take off the influence the shallow water effect.

These cylinders are of course wall-sided. Fig. 8 (a), 8 (b), 8 (e) and 8 (d) show

the results of the test.

On the cylinders B1 and B0, the theoretical one obtained from {2} by

ex-trapolation corresponds closely to the experimental one generally. But as the case

of the circular cylinder, the experimental value slightly decreases with the increase

by the small model


.5 20



0.6 0.5 0.4 0.3 D A AO 6- 0.9468 o o 1.0 0.8 0.6 0.4 0.2 -0.7 -0.6 0.5


12 lo 0.6 0.6 0.4 0.2


Z ß

Thooretçal valuo obtained from (2) (see appendix of this paper)

E xperirn e nfS

(small model) t lore model)

S--- 2S=9.8, s--- 2S2I.0,,,,,


0--- 2Sl0.Q,

of e0. In regard to the cylinder B3 the experimental value decreased abut

5-10% at C0>'0.5. And hump and hollow were observed clearly in case of B2 and

B3. H. Holstein [3] also pointed out the phenomena that the minimum and

ma-ximum were observed in the experiment for the rectangular cylinder. In the

ex-periment of B4 very large hump was found at C 1.85. And yet I should like

append the value A and K4 for H0=2.0 of Lewis form in the appendix [II]. 4. 4 Cylinder A

This cylinder has the section which is similar to that found at the rear of the ship, which is given in Fig. 9 (a). As the figure indicates, cylinder A1 was

designed to be used also as Cylinder A2 by adding the projecting part under the

water surface. Then experiment was done by varying the draught in three ways:

A1W5, AW2 for the cyinder A1, and A2W1, A2W2 for the cylinder A3


Both A5, A2 have the same H01.0 on the water line W1 and the same II= 1.25 on W2. The condition of the test being non-wall sided either, A1W1 and A2W1 had the inclination of 60 which is the same condition with the

pris-matic cylinder of =6. So that Wedge Effect was assumed to besmall. Figure

10 (a), 10 (b) show the result.

Either of them had great hump and hollow 4 at >-1.0. And also it was

manifested that the value of which correspond to the hump and hollow was varied with the amplitude of plunger. On the section A1W1 the theoretical value agreed quite well with the average value of the experimental one, while on the

section A3W1 the theoretical one was less than the experimental one by 19-20%.

294 F. TASAI Rectangular Cylinder B4---H.2.o. G 09600 0.5 Io Fig. 8(d). 1.5 2.0


12 IO 08 06 0.4 02


A2 - Cylinder 05 A1 - Cylinder Lw Experiments l'o Fig. IO(a). 2_ 2Si26,.,,., 2Sl9Sm,.' Li -12 1.1 Io 0,9

Theoretical value obtained fron, the reference (21

(by the interpolation)

The value o of A2 was larger than that of A1, therfore A obtained from the

cal-culation [2] was smaller than that of A1. The section A2-W1, however, took

greater variation of section contour from Lewis form rather than A1-W1 and pa-ticurally nearby the water line it varied finely and at the lower part fully.

For this reason when the value becomes large, that is, as the progressive

wave length decreases, the upper part of the plunger near the water line exerted

greater influence on the value A than the lower part. This is why the experimen-295 C1,C2 Cylindr 0.6724 812 T lSO.,,.,.,I150,.,.' 1.5 ao


.2 1.0 08 0.6 0.4 wI 0.72 94 B/2I50 T cçJ _____-- wt B


o 0.5 A-09

Theoretical valuo obtained from the reference f2)

by the interpolation) 05 lO Fig. 10(h). 4- Cylinder Wa Experiments 3 .2 IO

Theoretical value Obtained from the reference (2)

(by the interpalation)


E.,-2 IO Fig. 10(c). .5 2,0 15 20 13 1.2 l_l I-0 A o 296 F. TASAT

tal value is larger than the theoretical one. It is most probable that the section

A2-W1 gets large A not for the Wedge effect but for the variation of the section

coutour from the Lewis form.

A2-Cylinder L 25 0.71201 II5._,I 92,,, H, o-1 B12 T O 2S=I2.5, A 2S=22.5, D ---Experi monts O---2S'12.3,,,,, A 2S'22.6,,, o ---25'39.l


Now compare the cases of A1 - W2 and A2 - W2 which is shown in Fig. 10 (e), 10(d). Generally both experimental and theoretical ones correspond closely

0.8 0,6 04 0.2 4 .2 08 06 0.4 02 A2 - Cylinder C - Cylinder

Theoretical value obtained from the reterence 12)

Fig. 10 (d). G-0.3933 8/2 i o o,,, T 50,,,, Experiments O 2S=12.3,,,, - - - - 2S'235,,,. 05 Io

Theoreticol value obtained from the reference 12) (by the extrapolation)

Fig. 11 (a). O---2Sb08,. A - - - 2 S'2l.5.5 O--- 2S'620,.. y ao .0 09


H, 8/2 w2 07950 120,,. 96,,,, .25 D 2S=404,,, 20 .5 WI 0.6667


.2 IO OB 06 04 0.2 04 0.2 0.5 6 03988 2 O 50 B/2 70,,,, o 25'-12.0,,., A 2S'2O.O,,m o 2S58.2,,.,, Io Fig. 11 (b). C Cylinder T 20,,,,,1 Fig. 11(c). 0---2S'-112,,,,, A--- 2S'-3I6,,, O---2 S'-660,,,,,

"Theoritical value obtained from Experiments the reference (2)

IO 09 08 0.7 r s 1, 14 2 IO A C2 Cylinder 12 Il 10 09 WI G'- 8/2 1/2 06667 0.4893 i00, ISO,,,, o 0 Io 298 F. TASAI

in each cases. On the section A1 - W2 the experimental one was a little large

at > 1.2 under the influence of the variation of the section contour from Lewi

form. While the section A2 -- W2, being closely resembles to the ellipes Ile = 1.2 both theoretical and experimental ones agreed quite well.


Theoreticol value obtained tram tIre reference (2J (by the interpolation

2.0 5

O. I 0.2 03 04 05 0.6 0.7 08 0.9 I-0



4. 5 Cylinder C

The section of this cylinder is similar to that of a ships fore and after part. (See Fig. 9 (b)). Section C2 was the one which was swelled under part of

C1. Draught was varied by two times, W1 and W2. The section C1 - W1 and

C2 - W1 has same H0 = and o = 0.3933 and 0.4893 respectively. Also ri = 0.3988

and 0.5690 on each section C - W1 and C2 - W2 (in this case H0 = 0.583). Fig. Il(a), (b), (c), (d) show these results. Ori the water line W1 and W2 the sec-tions C1 and C had an inclination of 45V, the Wedge effect was, however, sup-posed to be small judging from the experimental results of trianglar cylinder of a = 9. These experimental value showed hump and hollow, while theoretical

value agreed quite well with the average one of the experiment. The experiment

was done at the large amplitude 2Ss70mm, in due consideration of the pitching

of the ship. But no visible variation was found on the average value of A.

C2- Cyhnder Io 08 06 04 02 4. 6 Cylinder D

For the eight ship models K. Kroukovsky [5] had made calculation of coupled oscillation of heaving and pitching by Strip Method. And had compared

it with the experimental one. Then he said that the theoretical calculation had

failed for the Yacht-model 1699 B, 1699 D and concluded that Strip Method was

inadequate for these type. The fact is that the amplitude of heaving and

pitch-ing obtained from theoretical calculation is too small. These yacht-shaped model

has fine section and sloping sides at the water line. K. Kroukovsky [5]

calculat-ed the damping force from Grim [4]. However Grim [4] had large error for


w2 H 0.5830 8/2 T 70_, G-0.5690 .1 Expenments Fig. 11(d).

Theoritcal value obtonea frani tOe reference (2] o 2SIl.I,, .i 2S208,,, 12 ) 2S723,,, 09 08 07 - 0.6 08 01 02 03 04 05 06


o o


the fine sections, which was alluded to by the auther in [2]. It is supposed that the failure of the calculation for the yacht-model was mainly due to the damping

force obtained from Grim [4] being too large. Then the author took the

measure-ment of A on the cylinder D whose section was similar to the midship section of

the yach-model. Fig. 9 (C) indicate the section of the cylinder. D2 was the

one which was swelled the under part of D1. D1 - W1, D2 - W1 was Ho = 1.0 and


WI -0 04773 8/2 T I50._. l5O, '5 o -Cylinder Ci 'H =125, O=06 /

TteoreficoI olJe obtained from

Oie reference 121 (by the interpolator) Experiments O 25=102 .\ 25=26.0,,. Fig 9(d) H.' LO, ci'-03543 -16 (5 14 13 12 300 F. TASAI D D2 Cylinder 05 IO 5 20 Fig. 12(a). IS 4 LO 08 06 04 02


14 D2 -Cylinder

io .2

2 B 2

-.,-- volue obtoined from Grim (4)(Bild IO)

Theoreticol volue obfained from the reference (21

(by Hie infer polotion) Experimenis 2 S i 2 0,,,, - 2S2I3m,. Hi25.G-=07 io i


-iM I3 .2 C.- B/2 T 05618 ISO,»,., i50,,,,,


i) 05 iO (5 20

Fig. 12 (b).

o = 0.4773 and 0.5618 respectively. Fig. 12 (a), (b) indicate the results of the


In the range of

the value of A obtained by the experiments cor-responds closely with the theoretical one obtained from [2] but hump and hollow

was generally observed at o>0.8. On the D2 Wi, especially, the large hump

was found at == 1.8.

In fig. 12 (b) we inserted for reference the value obtained from Abb. 10 of Grim

[4] by interporation of is. The values are very large, especially in the case

D1 - W1 they are too large to be drawn in the figure. The average value of

the experimental one is smaller than the theoretical one at o>0.8. Principal

cause for this is supposed as follows: The angle of inclination being about 80

Wedge effect would be supposed to be negligible. The section contur of D1 -W1

and D2 - W1 vary from the Lewis form, especially fully near about the water line

and finely about the lower part. These were just against the case of A2 - W1.

That is, the larger become the value o, the more great influence exert the

upper part on the value .4. Therfore the experimental curve is lower than the

theoretical one. When is large, namely when the produced wave-length is small,

the lower part of the cylinder do not exert any influence on the wave-making. Then let us discuss on the section without the small swelling part of the

lower part. If the section is showed by the dotted line in Fig. 9 (e), the section

D1- W1 takes JI=l.25 and o#0.6, D2- W1 takes H0=l.25 is0.7. The value A calculated from [2] for that H0 and o resulted in approximate quantity to the experimental value as was illustrated by the chain line in Fig. 12 (a), (b).

On the section D1 - W2 the experimental value is genrally lower than the IO





1.0 o Wz D - Cylinder 0.5 DZ Cylinder


-I0 Fig. 12(c. W2 H0 .25 04510 B/2 25,,,,, T 00,,,, o 2sI0.9,,,, 2$ '21 I Experiments Q... 2S=I0.8_.. --- 2S=19.6.,. Experiments (.0

Theoretical value obtained from the reference (2]


Theoretical value obtained from the reference (23

(by the ext rapoiction)

4 .3 .2 1.0 302 F. TASAL 05 Io .5 2.0 Fig. 12 (d). T H0 B/2 88.. (.0 0.3543 Ba... 1.6 .4 .2 I.0 08 06 0.4 0.2 1.4 .2 .5 20 0.8 0.6 0.4 0.2



theoretical one as was shown in Fig. 12 (c). The decreasing percentage from the

theoretical value of A is remarkable with the increase of , namely, 13 % at

=0.4, 16% at =0.7 and 20% at 5 = 1.0. This section contour varies finely at the upper part and fully at lower part from Lewis form (See Fig. 9 (d)).

we judge from this view point experimental quantity is assumed to be slightly larger than the theoretical one. However, actually reverse is the case. This is owing to the Wedge effect. The inclination on the water line was about 260 which corresponds to the prismatic cylinder of a = 128V. Then it is suppos-ed that the experimental value became lower owing to the Wsuppos-edge effect which

is larger than the case a = 12O. The next one, D2 - W2, the experimental value of A was lower than the theoretical one from the same cause as D2 - W1. We denote the angle of inclination of the section contour on the water line by O.

When O is small, the Wedge Effect correction are obtained approximately from the experiments of this paper. lt is shown

in Fig. 13. On the other hand, the influence of the variation from the Lewis form such as A1-W1, D1-W1, D2-W1 could not be formalized. For a certain cylineder,

how-ever, concidence between the experimental 2

A and theoretical one could be abtained, if we calculated it by taking many terms of C in the equation (4) of [2] and using the

niathemetical representation more approxi-mate to the section of the cylinder. But real-ly for the section which has great displace-ment from Lewis from, we could correct the

value obtained from the equation [2], taking into consideration the experimental result in this paper.


We could gain the following conclusion within the limit < 2.0 from the results above-mentioned.

An sufficient average quantity was given by the theoretical calculation of F. Ursell [1] and the author [2] for the progressive wave height generated by the

forced heaving of the cylinders whose section was semicircular and resemble to

that of Lewis form.

The A of the measurements give rise to hump and hollow, especially with

the increase of they becomes greater.

In this experiment the influence of the amplitude of plunger over the value

X was generally small. But great influence was exerted over the 0 at which

hump and hollow was forced to be generated in the case of high heaving


Judging from the test of triangular cylinder and so on, Wedge Effect was

200 300 Fig. 13.


304 F. TASAI

small till the inclination of the section contour on the water line getting 45. So

the calculation of [2] was good for the cy1inder above-mentioned. Wedge

Ef-fect however came to be large in the case a = 12ff and V-type section opened widely as the section D1 - W2.

Owing to Wedge Effect the experimental value was lower than the theoretical one [21 for the same q and H. PiiculaIiy the tendency was remarkable with the increase of o.

The different A from the calculated value of [2] was found at large when

the variation from Lewis form was conspicuous, even if H0 and are the same

ones respectively. The fact that A was higher or not than the theoretical one

corresponded to whether the upper part of the section under the water line chang-ed finely or fully from the Lewis from.

Wedge Effect and the influence of displacement from the Lewis form was

small at the points where was small.

We can compute from [2] the wave producing paticularity of plunger type

wave making machine.

My grateful thanks are due to Dr. Y. Watanabe and to Professor M. Kuri-hara for the helpful opinions on my research. I am especially indebted to Mr. Awaya for considerable assistance in the preparation for recording apparatus

and to Mr. Ikeda and Mr. lnoue for the helful cooperation in the preparation

of the driving apparatus.

[i] Paticurality of the Measuring Float

1. Heaving Equation of the float and its solution

Heaving motion of the float is supposed two-dimensional because of the

float being slender. Fig. 14 shows the section of the float, in which x is taken

along the horizontal axis, y along the vertical axis and z-axis is taken

right-angl-ed with the paper. The symbol W W denotes the two-dimensional wave towards

x-direction. Half width of the float, in the figure, is denoted by a, draught by T

My debt to Mr. Arakawa for his unfailing effort and assistance at all sta-ges of this research cannot be adequately expressed.

List of Reference

ji] F. Ursell Quart. J. Mech. Appl. Math. (1949, p. 18)

[2] F. Tasai J. of the Society of Naval Architects. Japan

[31 H. Holstein W. R. H. Dec. (1936); s. 375-389

[4] 0. Grim Jahrb. Schiff. Tech. Ges. (1953)

[5] K. Kroukovsky S. N. A. M. E (1957)

[6] F. M. Lewis S. N. A. M. E (1929)

[71 C. Taniguchi, M. Shibata Seibu-Zoesenkai-Kaiho no. 9 (1954)





Fig. 14.

and length towards z-direction by I. Then put ,- the amplitude of the surface

wave, the isobars are given by


,= rWe cos(k0xwt) (1)

where w wave circular frequency.

In practically a=lcm and wave length used for the experiment is À>4Ocm so that resulted into a/2 <0.025. Therefore external force by the wave is obtain-ed accurately by putting x = O into the equation (1).

The followings are the symbols used in this paper.

y heaving displacement, V0 = displacement volume of the float

A = water plane area = 2a1

a sectional area coefficient = H0 a

2aT T

S,, immersed sectional area of the float

M0 = heaving mass of the float

Equivalent mass of the pen P1, M= M0 + m' 4M= added mass of the float, ír = added mass coefficient

= heaving parameter of the float = a

N = damping coefficient, Fy = wave force = spring constant of S1 (Fig. 4)

4E = dynamical change of the wire tension

Now the equation of heaving motion of the float is expressed as follows:


+ 4M--(y--0)+N

-- (yw0) + pgAy=F0-AE



denote the value at x =0)

If movement of the pen were taken into consideration

4E= ni' ± k,y (3)


4M p Vo


and put

then it reduces to

b Tm fo/fo,2 + fA2




Therefore magnification factor of the heaving amplitude Y = -.-- is given as

fol-lows: - k0T Fy =


cos cút (5) (6) 1 2 2 I -


A21 (7) Jo Put 2h=


4M=pV0a0, (8) 4M e)2 cØ

M+ 4M .

2 - H0

ca0 . Therefore the equation (5) reduces to

j+2hj±vo2y=fi coswtJ2sinwt



/ fi = o rin e Ho ( I - . c a0 110




denote the forced heaving displacement of the float, we obtain



± 4h2w2 (I)

= --,

2h vo fA JI


I fiji LT 0 ca0 (10) (12) 306 F. TASAI

M0 ± ni'=M the following equation would be given.







in this equation y indicates displacement from the equilibrium condition

where the wire is suffered from the statical tension E0==2gr.

The whole tension of the wire E is written as





Fy and N are given as follows:

we have





2hoi Noi

fA- KA u02 = fgA+k,

As the k, being small, regarding as k, = O, the equation

Noi . A2

pgAH- k3


is obtained.

Phase lag is given by the equation




-i --


¡cA JA



The wire tension E should always be positive in order to operate the wave measur-ing apparatus on the above mentioned condition, namely

E0 ± -i- k8.y>0

therefore next equation is given

2. Free heaving Period of the Float

It is observed from the free heaving experiment that the first one cycle shows complicated motion but since second one it damps with constant period. These constant period is denoted by T,. Natural frequency of the heaving V) is

deduced from 2 pgA + k M-f-p Va0 a0 is computed from [2] p ira2 2 C0-ira


- 4T

.K4 --(16)

For the section of this float, C0 '4= 1.4 is obtained from F. M. Lewis [6].

Then putting VoS 25.4cm3, T=0.86cm, =0.98 H0= 1.16 and a=lcm into the

equation (16) the following one

(l'o = 1.305 K4 (17)

is given. Frame D1 and D2 exert the rotation, so that the equivalent mass have

to be considered for the heaving of the float. Also taking the m' of the pen into

consideration, (M0+m')g=M.g'4=27gr is given. Being k=l.2gr/cni and 1=15cm,

(13) where



308 F. TASAI

V a is calculated by the next equation:

pg2a2l ± k.a 30.2

- Mg±pgV4ao - 27 ± 25.4 x 1.305 K4

The equation denotes the ralation between o and K4 which was obtained

from the definition of the natural heaving frequency.

Since the second cycle, on the other hand, K4 is given approximately as the function of by using the calculated value of [2] for the forced heaving, assuming that it is nearly close to the regular heaving condition. Fig. IS shows

(18) 08 07 06 05 04 rom K4 (18 05 06 07 08 Fig. 15.

the equation (18) and the value of K4-0 obtained from [2]. From the

inter-secting point of these curves it is obtained 0.633 and K4 = 0.625, and then

Vn = 24.9, T, = 0.252 sec could be obtained. Generally is approximately given

by the above-mentioned method using the calculated value of [2] for the

com-putation of the two-dimensional cylinders. T is resulted 0.25-0.26 sec in the

free heaving experiment of the wave measuring float.

3. Calculation of Y

Magnfication factor Y is computable from the equation (lO) (12) and (13), if the paticulality of the float (H4, o, i, T, V0), and circular frequency of the forced

heaving are given.

Table. 2 illustrate Y and Z= 1/Y for the 2.0, 1.5, 1.0 and 0.5 which are

heaving parameter -= . of the wave-making circular cylinder. This is also

given in Fig. 16. u in the table 2 denotes the heaving magnification factor used

in the List of Reference [7].

That is, y0 = = const. is put without computing it for each . And the


cal-I0 099 098 09 7 096 --z 0







culated from 7 =


r ,,('\2Ì2 -- , assuming that added mass and

damp-V i - yj j


ing force have constant value without concern to the frequency of the external

force. Using = 24.9, A r=0.5 we obtain J = 0.197 for this float. )it is the magnification factor which is derived from by putting e°°T = I and by neglecting the relative motion of the float and the wave in the equation (2).

Now if X1m denotes the relative displacement between the wave and the point x = O of the float, relative variation ratio X is computed from the equation

X=,/Y+l-2Ycos o


We obtain XTm = 1 mm when Tm= 20 mm.

In this experiment as mda ± ky was much smaller than Eor=2gr, the

float operated sufficiently well.

309 Table 2. w 11.43 9.9 8.1 5.7 0.1333 0.1 0.0667 0.0338 1.3803 1.3030 1.2250 1.1280 1.2280 1.1640 1.1070 1.0520 fo 0.8900 0.9170 0.9430 0.9720 0.8330 0.8520 0.8740 0.9130 t/f2+f42 1.025 1.018 1.010 1.0000 Y= Iz.fol./f°+fA2 Z-: 0.9756 0.9823 0.9900 1.0000


6 Ho=2.O O43 14 2 0.6 oQ3333, a=O. ((1=07854) a103667, aI.((1o6I0O) a=O3,O-0I (0=o9596) 4 B >0 .9 2 05 0 5 20 25 Fig. 17.

H 20


310 F. TASA!

[11] Additional calculation of A and K4 for the Lewis form section

We computed A and K4 at H0 = 2.0 from [2J as the data for calculating the damping force and added mass at the light loading condition of the ship.

Then it is published here additionaly (See Fig. 17, 18).

05 b-0 5 20 25 Fig. 18. (Received March 26, 1960) 8 16 4-b2 I-0W Os-os 04 02





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