# The theory of ship motion in two-dimensional irregular waves with particular reference to ship speed

## Pełen tekst

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B.S.R.A. Trnslrrtion No.

TI THEORY OF SHIP MOTION IN TWODIMENSIONAL IRREGULAR WAVES !ITH PARTICULAR REFERENCE TO SHIP SPEED

A A. Kurdpmov

Trans. Leningrad Shipbuild, Inst., No.33 (1962), p. 97

The linear theory of ship motion in twodimensional irregular waves has beer. developed, considering this as a steady process, in sufficient

detail; certain clarification is, however, necessary as regards allowing for ship speed.

Since reference [2, by Denis and Pierson, was published determination o f the spectral density of the exit process has been considered most

important; when a ship is in motion, this density is determined as a function of the apparent wave frequency. The correlation function of

the exit process is determined by the usual equation, integrated by the apparent frequencies*. This method of calculation is quite cumbersome, since the relationship of the true to the apparent frequencies is not simple. ?7hen the dispersion of the exit process was determined in. reference [1], the equation was integrated for the true frequencies, but there are no good grounds for this and objections are sometimes raised.

It is proved, in this article, that to determine the correlation function for the exit process with a ship in motion it is dufficjent to know the spectral density of the waves as a function of wave frequency or wave shape frequency (the wave number), and a conversion function,

expressed in terms of the same variables, and to integrate either by

(2)

2

wave frequency or by wave shape frequency. The spectral density of te exit process may be required when the Oppo8ite problem, that of ìetera.ining the spectral density of waves from the correlation function for the exit process, is being solved.

1

Let us regard twodimensional irregular waves as the result of superimposing a series of sinusoidal waves with random amplitudes; the following equation fcr the wave surface oan be written, on the fixed system of coordinates and ç ¡

where the frequencies

### w>

and are related by the following

equation, one of the accepted wave theory eqtions:

4)

(2) g

g is gravity acceleration, represents the random amplitudes, and the wave outline moves in the direction of the axis,

Equations (i) and (2), used for depicting wave systems, correspond to the assumption that

M O,

### (3)

and the random amplitudes of the waves at different

frequenoies are not

i.ntercorrelated, i,e at j j h M rh - (4) while D

### (5)

To make further transformations

easier, complex wave novement is

* The prosont situation regarding the theory of ship motion in

irregular aves was considered in detail at a seminar, held

in

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brou.ht into consiLeration:

,

so that

- Re

### (7)

and is a material random quantity.

The random function (6) depends on the two variables and t, and two correlation functions, for time and for the coordinate, can be

calculated for it.

The correlation function for a complex wave system by time is determined by the equatiun

3

t +

### f.e

e e j h

If we alter the order of summing and determine the mathematical expectation, which is quito justified since both operations are linear, and put the product of the coordinate functions after the mathematical

expectation sign, using equations (4) and

we get:

k01

### Doe

(8)

i'e. the correlation function by time does not depend on the coordinate.

The correlation function by time for an aotual wave system is

deterr.ined using the equation

- D p.cos .

¡3 3

(4)

### 4

and the correlation function by the coordinate is determined using the e qu io n

003

equation

### (io)

is obtained in the sarte way as equation (9), and the correlation function by the coordinate does not depend on time,

and

### (io)

are even functions of the variables t and respectively, and for the actual stationary function this must be so, Representing the correlation function in the form of equations

and

### (io)

is called tcanonical transformationfl*.

Since the frequencies and may in general vary between O

and co , if we substitute

### Dp

S1R))d,) (ii)

in equation (9), and

D

### - S2()dc

(12)

in equation (10), we get the following expressions for the correlation

functions:** co

(13) o k2( )

### Sd

(14) o

1e will cdl the function S1 ()) the "spectral density of a wave system by time", and the function S2 () the "spectral density of a

* The condition of evenness would not be true if we took the

random ir.itial phases of tLe elementary waves into accounts

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wave system by the coordinate".

Equations (11) and (12) for dispersion must be the same for a

particular wave; using equation (2), we can getz'

.s2 (-e--). (i

### 5)

The correlation function (13) can be determined by means of the erodic theorem; recordings made with a wavemeasuring float should be

used for this purpose; the function (14) can be obtained, on the baai8 of the same theorem, by analysing stereoscopic film. The speotral densities are obtained by means of Fourier's cosine transformation.

Let us consider the motion of a ship under way in twodimensional

irregular waves. To simplify the calculations wò.*ill assume that the

foreandaft plane of the ship is normal to the wave orests.

The complex ecuation (6) for a wave surface is written, on a moving system of coordinates, in the following form:

A i(

### .%-Ii

3

where the wave frequency cA and the apparent frequency are related as follows:

### -5

here ii is the speed of the ship, and is positive when ship and wave

cirection are the same.

coiplex exit process can be found by integrating the equations for motion in complex regular waves and summing the solutions

(iv)

* A.I.Vozneenskii found that equation 15) existed1

(6)

6

obtained for all the frequencies. In this manner we can obtain

. )¼

## (

U)(1, X )e

3

where (A)', k),

### X )

is the absolutely definite funotion of these paraieters determined by integrating the ship motion equations.

TJsin equation (4) and assuming to be material, the nelation function for the exit process is determined by the equation:

C

Q¼(( )(k)

C J j 3 C

### i

and the correlation function for the actual exit prooess is determined using the equation

D

(k), )

where

) øi.

(c (k) ). (20) o

j '

### i

, 0 3' 3

This equation is called the "conversion function", and is obviously a

real function.

o determine the conversion function it is suffioient to put the results of calculating ship motion in regular waves into the form

'k

= (e.,

### ,x) cos (3 Çt + E),

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3 J J

nd the conversion function can then be determined without diffioulty. Using equations (ii) and (12), the correlation function (19) can be

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A

(22)

A 2

008

### k).d

(23) (w o

The inference is that in equation (22) integration must be carried out

for the variable ) , ide. that / and must be expressed in terme

of 3, and that in equation (23) integration should be by the variable

(k) A

i'e. tnat must be expressed in terms of

(&.), the spectral density by time, is used in practical calculations, and the dispersion of the exit process can thus be determined using the

equation:

### a3) Z) s1(d)

, dc». (24)

If, for instance, the need arises for determining the spectral

density S1 (cc) of a wave system exit process from the exit prqoesa

correlation function kb () for a ship in motion, the integral in (22) should be transformed to the integration variable and the Fourier cosine transformation can then be used. We will not dwell in detail on

this matter, since it does not interest

### us

Equation (24) corresponds to the calculation system propo8ed by V.V.

Ekirnov, since it was clearly stated in reference [1] that if

X and y are fixed it is necessary to make calculations for several wave length values in order to determine the convers.on function.

### If, according to

ecluation (17), we substitute

### in

equation (24), this latter equation becomes

2

### (k))(k)xj s1[)J1

(25)

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-8-and tnis corresponds to the method worked out in reference [2]. Since

(k).

the relationsnip of (4)to Q is not simple, we have not given the integration limits.

Obviously calculations made by equations (24) and (25) must provide identical results, but calculating with equation (24) is incomparably

easier

REFERENCES

1 V.V.EKIMOV. Application of probability theory methods to general

ship strength problems. Trudy NTO sudostroitel'noi omyshlennosti, vol. VII, No II,

### 1957,

Sudprorngiz

2. S.N.DENIS and W.I.PIER3ON. On the motion of ships in oonfuaed seas, 1953, TSN.IE, vol.

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