Systems Analysis of Tides

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Report No. 13-83

Systems analysis of tides

J.P.Th. Kalkwijk

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Laboratory of Fluid Mechanics Department of Civil Engineering

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Delft University of Technology

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1 -I

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SYSTEMS ANALYSIS OF TIDES

J.P.Th. Kalkwijk

Laboratory of Fluid Mechanics Department of Givil Engineering Delft University of Technology

Lecture notes UNDP-course at the

Central Water and Power Research StatioI November 1983

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1

Contents I. Introduction

2. Some remarks about linear systems

2.1. Systems with one input and one output signal

2.2. Systems with one input and one output signal, with noise superimposed

2.2.1. Direct determination of the.input response function 2.2.2. Direct determination of the frequency response function 2.3. System with two input signals and one output signal with noise

superimposed

3. The system according to M. and C. 4. The input signals

4.1~ The tidal potential 4.2. Solar radiation

5. Requirements as to the response functions 5.1. Maximum sample interval- of the irp's 5.2. Duration of the irp's

6. The determination of the response functions 7. Some remarks about results obtained for Honolulu 8. Prediction

9. Extension of the linear theory to weakly non-linear systems

ID.

Example of a weakly non-linear system 11. Procedure of M. and C. for Newlyn

2 3 4 5 6 7 8 10 II 14 17 17 19 19 20

2

9

29

34 37 References Notation 41 42

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.

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~

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-]-Systems analysis of tides 1 • Introduction

Quite a long time the analysis of tidal observations has been based on the

knowledge of the frequencies of the tidal components. These frequencies

follow from an analysis of the tidal potentialor the tide generating force. This analysis also yields amplitudes and phases, but they are not equal to those of the real tidal components, called tqe tidal constants. The

latter, namely, have to be derived from observations and the method of the

least squares is a suitable tooI for that purpose. Another method is the Fourier. analysis (transformation), which yields information about which components are important, and application of the method of least squares afterwards,but this method is used less frequently. One of the·reasons

is, that gaps in a registration cannot be handled by Fourier analysis, but the least squares method can overcome this difficulty. However, knowledge

about astronomical frequencies remains required.

In a way these methods for tidal analysis are rather conservative compared

to the.method of time series analysis, wh i.chis now widely used in many

branches of engineering, physical sciences and economics. One important

aspect of times series analysis is spectral analysis, such as has been treated in chapter 4 considering deterministic signals ~n particular. Munk and

Cartwright were the first to apply this theory ~n behalf of tidal analysis. In this connection it means that the output of a system, consisting of a water level registration, for instance, is correlated to a series of inputs,

sueh as the tide generating force, solar radiation and possibly meteorological influences. Schematically this system can be represented by a·block diagram

in which (x](t), x2(t)...• represent the respective ~nput signals and xn(t) the

output signal, consisting of the summation of output signals from various sub-systems. The output signal may contain noise, which has nothing to do with input or output signaIs.

x] (t) 'Ik](t)~ x2(t) ~ x3(t)

>[k/

t

)

t-.

-, x (t) / _n xL1(t)

)

I

k4(t)}

(5)

--I

-2-I

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Time series analysis aims at the determination of the properties of the (sub-)'systems, either in the time domain or in the frequency domain. An extensive set of mathematical tools has been developed in this respect in

particular for so-called linear systems.

M.and C. pay much attention to the case of a linear system and give an extension for weakly non-linear systems. The latter is certainly necessary as the tides at the continental shelf contain non-linear influences. At the same time the non-linear behaviour reduces the attractiveness of the method, since the method loos es its theoretical elegancy taking,account of such effects.

The article of M. and C. is not easy to read, as a thorough knowledge of time series analysis is required. Moreover, several, sometimes unnecessary,

artifices are applied which hardly contribute to the insight'. As, furthermore,

the calculational procedures are rather time-consuming, one cannot say that this method is daily practice. For physical interpretation, however, it might be quite useful.

In the following some basic theory will be treated before the actual method 1.8 discussed ,

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'2. Some remarks about linear systems We sha1l consider the next cases

a) systems with one input- and one output signal

b) systems according to a), but now the output signal also contains noise; on1y the superposition of output signal and noise is known

c) systems with two input signals and one output signal plus noise: this can be easi1y extended to system with more input signals.

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In all cases attention will be paid to the way how the system properties

can be determined from input- and output signals.

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_2. 1 •

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.

1 I

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-

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-3-Systerns with one input and one output signal

Schernatically such a system can be represented as below.

If ket) is the response to an impulse (delta-function) as input signal at the time t=O, then the response of a linear causal system to an arbitrary.input signal J.S: .

"

xI (t --T)k(T)dT

(I

)

According to the causality 'principle k(t)=O for t < 0 and therefore the lower integration boundary can be replaced by

-

00 :

-T )k(1')dl' (2)

The equivalent in the frequency domain is:

(3)

where the capitals represent Fourier transforms of the corresponding time functions:

x

(

O

)

_

00

J+oo .x()t e

-2n

i

o

t

d

t

Furthermore, in the following ket) and his transform will be cal led impuls response function (irp) and frequency response function (frp), respecti:vely.

A simple method to determine the response functions would he to carry out the calculations in the frequency domain and to use (3). The frp, however, can also he derived via the power spectrum and the cross spectrum. These ones can he calculated hy rneans of the covariance functions.

*

This expression can also he written as:

x2(t) = _ooJt x1(T)k(t - T)dT

(7)

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_--

.

_.

1

1 I

I

1 I

I

1 I

I

,

1 --4

-So, the autocovariance function of a function x1(t) !s: autocovarianc e c Cr)

=

E[X Ct) x Ct 'IT)l 1) )' 1

J

I " I rm 21' T-)-OO (5

where E

=

mathemati.ca1.expectation 2T time interval This yie1ds for the power spectrum: ~ll (9")=1im ~T X~,~ (o)X) (a) T-t= 1 ,i.m21T

1 ()1

X] a2 .=. rea1.

*

T-)O) (6

The cross covarianee funetion between input- and output signa1 1S by definition:

(i

so that the cross spectrum is:

(t

From (6) and

(8),

the power and cross spectrum, the frp of the system

ean be determined:

(~

2.2,Systems with one input and one output signa1, with n01se superimposed

Schematica11y this system ean be represented as in the seetion before, but now:

(

.'~heequiva~ent of

X2(a) (complex)

j+oox(t)x(t - rjdt 18 D1:the frequeney domai.n equa1 to -00

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-5

-zet) represeilt a disturbance of random nature, which behaves as white

noise. This implies, that the noise will have a band structure in the frequency of interest.

Further, it is assumed that the nOlse lS not correlated to input or output signal.

Again, the problem lS how to determine the system proper ties out of input and output + nOlse. The direct determination of.the f.r.p. according

to (3), i.e. the direct transformation of xl(t) and the measured x2(t)does

not yield a correct result since x

2

(a) now also contains noise. Therefore a procedure must be followed which allows the elimination of the noi se, In p·rincipletwo methods can be applied, viz.:

1 determination of the irp in the time domain by means

of the method of least squares

ii determination of the frp in the frequency domain by means of XI (a) and X2(a) •

In 'the next sections this will be treated.

2.2.J.Direct determination of the impuls response function

The method implies, that the integral (10) is discretized according to:

I

n

=

I

k x

C

t -

nillT)lll m=O m z (11 )

The weights k are parameters, that have to be determined using a least m

squares procedure. The way of computation is as usual.

Jenkins and Watts (pp. 422) make the following critical remarks ln this connection:

1 The number of points of the lrp, ket) to be taken into account

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1 I

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l

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6

-ii The estimates for points of ket) lying closely to each other are strongly correlated. This implies instabie behavi9ur (ill-conditioned

matrices) and large variances. Therefore of ten the method to determine the frp, as described in the next paragraph, has to be preferred.

2.2.2.Direct determination of the frequency response function

This method is based upon the fact, that noise and input are not correlated, 50 that the covariance between these two signals is equal to zero

cov [x(t+T), zet)]

o

voor alle T (12)

This property can be used to det.errni.ne the frp of,the system, by maki.ng such calculations, that (he noise is reduced by averaging by using

systemati.calLy cova'ri.ance.functions.·' So:

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Using the followinz relationship:

XI (o)K(o) + Z(o) (14)

this changes into:

~' :;; GIl (o)K(o) +

U

rn

X~~ (o)Z(o)

T-+oo

(15)

..

The equivalent of 'the second term at the

RH

S

in the time dornain lS:

lim ~T

_ct/tm

xI (t)z(t +,)dt

=

0 T-+oo

(10)

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1

1 I

'

I

1 I

I

1 I

I

I __

I

_

~

_

-

7

-lim ~T

x~~(

a

)z(

a

)

=

0 T-+«>

Changing (15) into the known expression:

Cl I

(a)K(

a

)

(9)

that also holded for the noisefree sign~l: So, use of the crosscovarianee function CI2(T) implied, that the noise vanished.

,

2 .

3 "

System with two _input signals and one output signal with noise superimposed

Schematically this system can be represented as below.

The expression for x3(t) is:

+ zet) (16)

The equivalent in the freque ncy domain is:

( 17)

To determinè the frp's from the input and output sismals again the

covarianc~ functions of the various signals must be used:

The cross spectra are then given by:

lim Xcc

(

a

)x

3

(a

)

T-+«> IT lim X2Icc

(

a

)x

3

(

a

)

T-+«> (I9) ( 18)

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--,.,....-·

1 .

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-8-I

beforeSubstitutingagain cross spectrathe result forofX3the(ó) noise and the input signals arLse.according to (17) into the expressions These cross spectra can be set equal to zero, so:

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. (20)

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(21)

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A special property is that C 12(0)

=

C~~(0) • The two equations yield K1(0) and K2(0). By Fourier transformation K1(t) and k2(t) can be

determined again. Of course these irp's can also be determined directly in the time domain by means' of the method of the least squares.

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It is clear, that the system as considered in this section can be simply extended for more input signaIs. The number of equations with a structure like (20) and (21) remains equal to the number of frp's to be determined.

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In principlethe set-up by Mthe. theory treated briefly beforeand C. It will appear that someis sufficientsimplificationsto understandare

possible, but this is not essential.

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3. The system according to M. and C.

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The tide on earth is generated by the tide generating force, as caused

by the gravitational interaction of earth, moon and sun.Lnt.he astronomical analysi -+

of ten use is made of the tidal potential V. The tide generating force F

is then defined by -+

F = grad V (22)

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This implies, that the surface of equal potential at the surface of the earth yields the equilibrium tide. In principle the tidal potential depends on time and on the location on earth (latitude 8., longitude ~ :

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V = V(8,À,t)

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One may put forward, that t~e real tide on a certain location is influenced not only by the potential on that locatiorr, but also by the potentials at other surrounding locations. In other words the potentials at all locations influence the tide at one location. The potentialof each location contributes in some way to the real tide, but we do not know how. If the system would be linear this can be represented schematically as below,

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VI

=

V(8),Àl,t)~ I--~---' )h (t)

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V(83,À3, t)--7 1--_---'

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Or, using the convolution form (noise neglected):

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h(t)

/OOV (t-1)kl(1)d1

-ro I

(23)

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The numbersignais, can be very high,of sub-systems in thisresultingscheminto ae, eachsysternwith is unrnanagablewith their own input because of the high number of irp's to be deterrnined.

It appears, however, that the potential can be written as a sum of terms, where each term consists of a product of a time dependent part and alocation dependent part:

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V(O,À,t)

2

n=O

(24)

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The functions a(t) and bet) in this expression do not dep end on the location and U and V do not depend on time.

Substituting the expression for V according to (24) into (23) yields:

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· I

+

OO

P

.I

oo m In

+

b:<t-t)V:<B1 'ÀI~ kj (rjd-r + h (t) a ₍_t-t₎_U_{n (}₆_{1 '}_À_{j )} -00 n n=O m=O

~/oo

r

[

00

L

_am (t-t)Un(m 62,À2) + bn(m t~t)Vnm(62,À2) ] _k2₍_t_)d_t + n-=Om=O n +00[

J

k3(t)dt +

-

J

. -00 ••• etc_-_{_}. ₍₂₅₎

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Collecting terms with the same a and b one obtáins:

I

-I

+ J 00

Ï Ï

b m(t.-rr )

[

tf

ffin(e 1-,À 1 )kI (T) + ••••••••••••••••••• ] dr - 00 n=O n=O n (26)

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The expression above aga1n contains a number of convolutions. The parts between big brackets in (26) are functions of time and can therefore

be considered as irp's of systems with aCt) and bet) as the input signais:

I

h(t)

00 00

_ ooJ+oo '\/.., /..,'\ a (t-t)u (t)dtm m

n=O m=O n n (X) - 0:> _~J+oo '\ '\ m TIl + ~ /.., /.., b (t-t)v (t)dt n=O m=O n n (27)

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At first sight this result looks complicated enough, but nonetheless

there is significant _gain as only a few coefficients a and b are important.

.01 d 3 . - bOl 2 d 3 d

For 1nstance a2, a2 an a2 are very 1mportant, ut a3, a3, a3 an a3 an terms with higher subscripts are almost negligible. Moreover aCt) and bet) appear to be dependent which also simplifies the problems.

4.The input signals

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j

l

-i

l

~

I

_, -;

I

'

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-11-One hand tidal potentials as determined by the influence if moon and

sun and on the other hand solar radiation. In particular the introduction of the latter aspect was rather new, but it 1S of limited significance.

From the formal ipoi.nt of view also meteorological effects can be taken into

account. This has been done by Cartwright (1968).'

4.1. The tidal potential

The gravitational potentialof a mass M (moan or sun) 1n an arbitrary

point on earth P(a,8,À) is:

v

CM

_p (28)

,where

C

=

gravitational constant M

=

mass~celestial body

p distance between the point Pand the celestial body

M

ve rna].'êquinox

r .

Byimaans of the .parallax

S

::

R

(r

₌

OP, R

₌

distance of center of

gravity

of

earth to celestial body) the potential can be written as

V =-

CM

V

1 II cosa 'R 2 1-2E~\1-~ co

s~

CM

L

(ll) =- n R n=O (29) where Pn (11)<represent Legendre polynomials, e.g,.:

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-12-I

Po = P = ₁₁ I 3 2 I P2 =

"2

)1 2 5 3 3 P3 2 Il 2 Il

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In (29) the first term n=O) has no meaning. The second term (n=l) represents a constant force ~n the direction OM, which is equal to the

force needed for the motion of the celestial body in its orbit. The

third term (n=2) bec0111esof real interest. Because of the relationship that exists between the universal gravitational constant and the

acceleration due to gravity g:

r

l

I

-I

G gr2 H a

I

11 -: ma ss earth a r = radius earth

!f -;" acceleration due to gravity

I

(29:)changes into:

I

00 V 11

1

n+l P (Il)

- =

r M ~ g _._n=2 n a .oi?

[

ru

n+1 M -n+l =

!

K P (p) with K =:' l' H ~ n n n n=2_, a

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with R = OH = distance between centers of gravity of the respective celestial bodies

I

-~, R = mean

E:,

R.

This is the usual formulation (see Schureman, Doodson) •

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'

i

l

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-13--Values for K are g~ven ~n the table below.

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Moon Sun :

H

/

H

0,01229 332700 a ~ 0,01659 4,2635 x 10-5 K2 0,35735 m 0,16427m K3 0,00593 m 0.000007 m

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This table shows, that terms with subscript 3 are very small w.r.t. terms with subscript 2.

Equation (29) yields the equilibrium tide and so it is a function of a. For a fixed location on earth (coordinates 8,À) this angle ean be expressed into the variable coordinates of W, the projection of the celestial body on the celestial sphere.. By means of so-called associated Legendre polynomials (29) changes into (z

=

co-latitude, L

=

length w.r.t. vernal equinox):

I

,

I

L U~(8,À) - ~ P~(Z)sin L V~(8,À)

I

\({i

2 2

VJ;i

2 . V;(6,À)] + 15 P2(Z)cOS 2L U2(8,À) ₊

15

P2(Z)s~n 2L + (30)

Cî[Ii-

R 41f 0 0 + K3

R

7

P3 (Z) ';-1

.

U3(8,À) ~I 1

\f!fi

.

1 ( . 1

- ₂₁- p (Z)cos₃ L U3(8,À)

_-

_ZlP3

_Z)S1n L Y3(8,À)

ffir2 2L 2

'!fi:

2 . 2

+

,

2:10P3(Z)cos U3(O,À) +

2ïO

P3(Z)s:J.n2L V3(8,À)

\fJ-s3 3 ~ 3( ) . _3L

V

;

₍₆_'À~

,.;. P3(Z)cos 3L U3(8,À) _- _31S_P3 _{Z si,n}

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-1

4-I

in which the coefficients P, which only depend on Z, are:

I

pO

3 22

I

pO

5 3 3

Z

cos

z

_I

ces "

-

_"2

cos Z

-

_"2

cos

I

2 3

p

I

p

I

p

I

3 _Z₍_S 2 _J) S1.nZ 3 S1.11Z cos Z ==

_"2

81.11 cos

Z-I

2 3 p2 _r: ₃ _S1.. ₁₁2_Z p2 ₁₅

.

2.

_Z == S1.n Zeos 2 J p3

_-

_I

_S

_S_J. 3_._U _Z 3

I

Considering the structure of (30) eaeh term contains a Pcos nL (n=I,2, ..•). Here cos

nL

represents the rapidly varying part, deterrniningthe character of the tide, i.e. diurnal, semi-diurnal, etc••.• ; the coefficient p

depends on Z, taking account of slower variations. Furthermore there is a slow variation in R, that also should he taken into account. By g1.v1.ngZ, Land R as functions of time yields a form corresponding

with eq. (24). The expressions for Z, Land R can he found in the original artiele of M. and C. Fig. la gives an example of the time functions.

Decomposition of eq. (30) is not necessary: for this method it is sufficient to know the input signaIs, not the typical frequencies playing a part.

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4.2.Solar radiation

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M. and C. also propose an input function that takes account of the

radiation hy sun. They propose a.function

r

as follows:

I

_r

== S~_p cosa

*

0 <

lal

<.-'IT d~y .:

~ 1T

lal

< night <, ==

°

"2

< 'IT where S

=

constant (31)

I

1

*It seems more logical to take the radiation proportional to l/p2

instead of proportional to lip, since it can he assumed that the energy flux produced hy the sun is transported through a sphere with area 4'ITp2

I

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,A,

A

D

I

A

A A

/

\

A

fv '<1\,0,/\

1\ /

\

.

~~

,

D

I

tI.

A A

t

,

A

(\

f

\

.

~

A

A (\

A

1\A

/

:

,

/'0

n

.)

i\Î\

!

\

1lJl.A

il11J\M/\

AA

l~flJl.f1MLflM'u\ flllllAllfI..fJJJ.il11..fl.n_f

[L'lflflilJtfl111lillJJ~~AA

nMfl_filL~JL~Mllll

flUfl1l. AfItJ..AflAflJJ.fJLflll!W1l.H+H--__

~J JV

VU

VV

\JIJ

V I

lVVVVTJlJ

VIJVVVV'r

V

VV

VVV

V

'VV

VVVVVVVVVVVVvrrJVVVVVlJIJVUIJ

IJ

VVVvVVVVVVV

V

VVIJ\JVVVvm

g~

hN\

f

\NiA

V

~

~VtN\AJlMfM

WM~

Jv

4;

l

v

T/

W

tr

W

V

v

V

WWr/tJ

\frMMANWvANWVVtM

_~AfWfJ~tWMVWWvWNVWV*\WtM

NVMAAJv

"--

(19)

I

-16-I

rt

I

r

. 'day 1 --",.CI:

~

----

night

---~>

~

I

This function of the radiation lip ~s again expandéd us~ng polynomials of Legendr~, yielding the result

I

r

K P (p) n n

-RIl

=

S R (~+

I

p + ••• ) (32)

I

::-In (32) K represents a constant: n KO =

_!_+ .'!_

₄ ₆

S

Kl =

_"2

+1.

₃

s

5 1 K2 ;::

-

+

-s

16 3 r (t;;;:: -) R

I

The first term (n=O) in (32) represents the mean radiation. This one does not contribute to the generation of the tide. The next term (n=l) does, which is contrary to the tidal potential which begins to contribute for n=2.

As in the case of the tidal potential the polynomials ~n (32) can be written using associated Legendre polynomials.

The final result is:

I

+ second harmonie

I

Primarily this concelOns a diurnal tide as determined by s~n Land cos L. Furthermore, long-periodic influences are present.

A point of concern will be how the various frp's of the (sub-)systems with the tidal potential and the solar radiation as input functions have to be determined, as ~n both input signals the same frequencies occur (e.g. S2).

I

(20)

I

11

I

-17-5. Require~ents as to the response functions

Since the input signals do not contain all frequencies, it is not possible to calculate an irp or frp applicable to the entire frequency range. The functions can only be determined for those frequencies occurring in the input signal. In view of efficiency of the calculations it is recommandable to carry out a minimum number of computational elaborations. In this respect two points are important, V1Z.

1 the max1mum admissible sample interval of the irp's

11 the minimal time interval over which the irp's have to be determined.

In the next paragraphs more attention to these aspects will be devoted.

5.1. Maximum sample interval of the irp's

The execution of convolutions ~n the time domain in behalf of tidal prediction must be based on the use of discrete points of the irp's.

The lower the number of points of the irp's the more efficient the computation, but a large sample interval may lead to shifted spectra of the irp, that influence the original one. So, the choice must be such that the operation in the frequency domain is also correct. A first guess may be the same as for the observed signal, the output.

In view of the required length of the irp's this results into a large

number of points, close to each other. Moreover this choice

(e.g. ~t

=

3 hours) proves to work bad, as the matrices to be inverted show instable behaviour (ill-conditioned).

A larger sample interval proves to work better" but it must be chosen such that the shifted spectra do not influence the original one.

Selection the proper sample interval is based on the small band width

of the input signals. So the diurnal components contribute in the range

0,8 < 0 < 1,1 (c/day) and the semi-diurnal components in the range

1,75 < 0 < 2,05 (c/day).

If the sample interval of the irp's is ~t, then after Fourier transformation of the irp's additional spectra in the frequency domain arise shifted

(21)

I

1

-'

18-I

.

-a

2

-a

+20

-a

+ 20 1 2 - f 1 f

1 I

The figure above illustrates this f?r an arbitrary spectrum with theoretically

only values at the interval ~ (01,02)' To fulfil..the requirement, that the original spectrum is not disturbed by the shifted spectra it can be shown that of must fulfil· the condition:

This yields for the diurnal tides (0,8 < a < 1,1 c/day) a largest ~t of 1,25 < ~t < 1,35 days. M. and C. have selected ~t= 2 days.

This implies, that the theoretical spectrum should lie in the range

0,75 < a < 1,0 c/day (n=3) or 1,0 < a < 1,25 c/day. Consequently the

ehoiee of ~t

=

2 days seems not so good for diurnal tides. Nonetheless the final results are quite reasonable. The explanation is, that at the

range 1,0 < a < 1,1 e/day the input is quite small.

The semi-diurnal tides 1,75 <

a

< 2,05 c/day are adequately eovered by ~t = 2 days. Theoretieally a range of 1,75 < a < 2,0 e/day is possible and this agrees well wi th the real range.

1

a2 <

af

n+1

1

1 I

I

1 I

n 1, 22, 3, (33)

I

Sumrnarizing it ean be put forward that a large sample interval for the

irp's ean be used; anyhow it ean be much larger than could be expected

(22)

I

-19-I

_{from the} _{normal condition that} _the _foldin_g _{frequency must be larger}

than the highest frequency occurring in the spectrum. Here this is

possible "because of the small band widths .of the diurnal and semi-diurnal input signals.

I

5.2. Duration of the irp's

I

The duration of the irp's follows from the criterion of Rayleigh about the resolution in the frequency domain. If it is assumed, that the frp's

behave smoothly (creqo of smoothness) the duration of the irp's can be limited. M. and C. put forward that oscillations (wiggles) in the frp's

with a IIperiodll smaller than I::.cr

=

t

c/day are unlikely. According to the

criterion of Rayleigh then the minimal length of the irp's is 1/1::.0

=

6 days. In view of the lag I::.T of 2 days this yields only 4 weights per input in function. peculiarly enough M. and C. present results with a maximum of

14 weights (7 complex weights). Moreover their results clearly show

'oscillations' in the frequency domain with a period of about 0,2 c/day, which is rather close to their assumption.

Calculations at the Delft Univ. of Technology indicate that an optimum result is obtained for a certain number of weights of the irp's. This

number corresponds with the number of tidal constants which can.be derived with the harmonic method. For instance,.a month analysis yields about 8 tidal components (= 16 tidal courtents) for the semi-diurnal species. The method with the weights gave also a number of 16 for the semi-diurnal

input (P~). In particular, more weights caused ill-conditioned matrices.

I

6 • The determination of the response functions

I

Actually there are two methods, by M. and C. denoted with lumped and

sequential analysis respectively.

I

lwnp

e

d a

na

Zy

s

is

This method is carried out ~n the time domain and a~ms at the determination of the weights of the irp's ~n one computational procedure, using the

method of least squares.

I

se

q

uential anaZys

i

s

As indicated by the name this ~s a step by step analysis, which is carried

out in the frequency dornain. The sequential method is necessary as there

are inputs with components with the same frequency. The output does not

indicate directly which parts originate frornthe respective inputs.

Therefore first an analysis has to be made with e.g. P~, the most important

input. The obtained frp's are smoothed in a way (elimination of peaks) and

I

(23)

I

.7.

I

-20-subsequently the residu~l of the output is determined by subtracting the

m

results for P2' Next the analysis is continued with another input, e.g. m

P3'

Of course, both methads are closely related. If the irp's have been determined their Fourier transforms must agree with the calculated frp's.

The way M. and C. apply the methad is rather (unnecessary) complicated.

SA they combine the dependent inputs a(t) and bet) to farm a complex input

c(t)

=

a(t) + ~ bet), which means for a specific input for example

P~ cos 2L + i P~ sin 2L. Theoretically a(t) only would suffice. This g~ves

r~se to a number of complications, which all result from this approach

(complex input yields realoutput). 50 the weights are also complex. One complex weight has a real and imaginary part. In essence, however,

their methad is as described befare.

Same remar.ks about results obtained for Honolulu

M. and C. apply the method for two stations, namely Honolulu and Newlyn. Honolulu is attractive as there the non-linear influences are almast

absent. The following figures are adopted from their artiele.

fig. spectra and admittancies (frp's for input signals

0 1 2 _r_es_olut_ion _cycl_e_/_mon_th

P2' P2' P2' fig. 2 idem for P~, 11 cycle/year fig. 3 11 11 1 11 11 11 P2, fig. .4 11 11 P~, 11 11 11 fig. 5 11 11

p~,P§,P~,P~,resolution cycle/month fig. .6 11 11 _sol_a_{r radi}_a_tion _input.

table Honolulu prediètion variances table 2 Honolulu weights.

M. and C. abandon the principle of causality by admitting irp's for negative time, i.e. ket)

f

0 for t < O. The weights are symmetrically

(24)

I

-21-1

I

il

I

_!

.

11

1

2

~--

P~

p~

pi

I-< '"

~ S

0 v (.)

I

.

1

I

III

a

~ s

I1

... bi) .-2 :> 0

.!

ro-I-<"__'" bi) -4 1 o 0 0

i

m~m

m

~~~~

~

ilili

~~~

~~

r

~m~

ï

W

~~~

m~~~[~

il

~~~~

~

---"d ---

-'"

-2 _~

E

'" o

E

0

bO

-4 (.) 0 0

bD

-....__. 0 ~ 2

-

....__. I-< _>.. <I) _.::: ~

~~

~~mm~

~~~

~

m~

!~

~m~~~~

~~~~

~~

~~~

~

~~~

~~m

r

~

Cl) <I) 0 _~ ro <I) V' "d Cl) _>.. -2 bi) I-< <I) c:: -4 <I)

I

1 I

I

1

<I)

-

:

~

]

x (,)

~ <I) c::

M

-0. ...ro

E

.

:::

o

E

(,)"'0 ro <I) <I) 2

W

1

"'0 Cl) ~ ro ... ..c:: ;..::::0. 0 ~ 0."'0 ~

E ~

ro ro _-₂ 0 0·5 1.5 tyclesJday

I

1 I

FIGURE 1. Honolulu tide spectra at 1cim resolution for spherical harmonies of degree 2. P~, P~, P;, refer to tidal species 0, 1, 2 respectively. The upper panel shows the energy of the gravitational equilibrium tide at Honolulu relative to 10-4 cm"; some 'Doodson group numbers~:è:~;itte'iî

below the columns. In the ncxt two panels, the observed sea spectrum is designated by the total height of the columns. In the upper of the two, the height of the Jilled portion designates

the energy coherent with the equilibrium tide; in the lower, the height of the unfilled porti on designates the noncoherent energy. The left scalc refers to energy per column, the right scalc to energy per cid. In panel 4, thc filled circles rcfer to the real part, the open circles to the

imaginary part of the admittance. In panel 5, they refer to [admittance] and phase lead in radians, respectively. The vcrtical lines show the 95

%

confidence limits of the circles. Thc curves represent the admittance functions derived from thc convolution process.

(25)

-~ o c::: t""'CZl~ ("D 0 I:'l

~ a

t-:I i:l ("D • 0.. ~ .... tJ

::r:

Cl> s;» 0 o ':! i:l & ~.5> -("D ;:l C ::l ~ 8" <; Cl> tn· '< ,..,. Cl ::! ... 0-1 0.. ~ 0- ("D o Cl>

5'

1;;"] ::n s;» o (Iq ;:l ::j C 0.. ~ ..., ~ Cl tJ ~ _0 o -0..('")_{tI)_} 0-< ;:l ""I ~ ("D ('") ti) o

2-i:l C en

.... ....

....

.

....·0

5'

i:l ("D Ö'

g.

""I i:l .... C

:::r-a

("D 0-0' ~ ~

'"

~ s;» ""I ~ ""I ("D p.. ("D"ê o ...Cl "j _C-;:l_('")i:l co C -< ':l. 0.. ti) ("D'"O ;-11 0.. :::r-~ ... ("D o i:l ::1. ::J ...._:::r-s;»o :'1 ("D -(1), c :::r-:-11 '"0 s;» '"0 ""I ("D

a

""I 0 '"0

8.

s;» ('") ~ ~ :-- ~o

-.Q

('")

-("D en

_

0.. ~

-amplitude and phase

~ 0 ~ - -1 o o o '" ~ \=' ;;;

-.:.

complex admittance o r--- -1 Til

-;. o

sea energy (log10cm")

~-r-,....--r -r----'--, ~-~ C:===:====:J

=

= =

=

'"

o ~

energy density (loglOcm2 (CJd)-I)

;. _.:. .---, ~~ ~~ ___, = = =

=

₌

= =

=

= .:. o

- -

-gravity energy (loglOcmê) o

..

, o ~ ~---f- ---1---,---. lI) 00' __ t:) •• 2 ~ Q OI ..2~

-.

,

0

'"'!" ~ ~ ~ 02..2_ 1 N N 1 0' e- _ ~ 03·2 -

.

030 _ 0',,2 _ 0" 0_ '"

(26)

- - - -

-I-:j

....

Q c:: ~ t"l ~ ~ 0 ::l 0

-e

-e ...

...

0.-0 ti> "'0 0 n ...

..,

p) p) ...

...

n n

-'<

_..,

_'< e, 0 (1) en _en

s..

-

0.-e p) ... '<

....

.

0 ::l Ö'

..,

... ::r' 0

0.-....

.

c

_..,

-~ ::l p)

-0 en "0 "'0 rt ::r' (1) 0 0.-::l. o Hl p) ti

-0 ::r' S p)

..,

ti

3

(1) 0 Hl ::l C;" ~ o ~ o 8 o i:: s ~

- -

_

.

-amplitude and phase

,

., 0 ., r---r----,r---~----_r---, s

-complexadrnittance .!. ...",...

j

-seaenergy (IOgIO cm")

.

~ ., ,---,~~.--T- r I 1

=

= =

=

=-= =

=

= = = = = = ~ =

==-=

=

= =

=

=₌

=

c= =-= = = =

=

= = =

=

.

= = ~

=

= =

=

===-= =:-= =

=

= ==-~ =

=

= = = = = =-

=-=

=

= = =

=

==-=

=

= =

=

= = = = =-= = =

.

~

.

~ :., = =

=

-=>

=

= =

=

= =

=

= _,

=

-=

=

= =

=

= _,

-=

=

= -=

==

-=> = = = =

=

= = = -=> -= __,

=

= = =

-=

=

= =

=

--=>

=

= = =

=

-=

=

= =

=

= = =

-=

=₌ .:. '0

-

gr

-

avityener

- -

gy

-(IOgIO cm")

.

~ ... 0 ~~r-I--I---' 1""0• I·'" _ t..) 1·'0_00

-1-)2_~ t-a e ---1-22 5:> ~

,

.

;;;;;;.-

---I N W I

-1-11_ _S> 10·1

,

..

-102 _ _j:: 11

'

-10-)

111=\

!:f> ::.0 ~ 12-2_ 12. _

s-1'-'

".

--

<=> _<=> 140_

(27)

-~ o, o "0 rr (t) c, Hl M o S M (t) Hl

-

_j

~

-) '.~

~

1

'1 -( J > '}. :..<

:

_--.;

~

:,1 " "

-I<:j .... Q C ~ M !"'"

::r:

0 ::I

a

:::

-s:: ~ 5: ~ Cl> "0 ~ Cl ~ "1 f>l f>l ~

-Cl

-'< "1 Cl ~ ~ Cl>

a

(;"' s:: Cl>

-~

0.-....

.

0 Pl ::I _'< Ö' "1 ,.... ~ o Cl> ~ S 5: i::' "1 ::I

e,

Cl> "0 ~ ~ ::1. Cl Pl

-

~ f>l "1 S 0

2.

Cl '"t1 ..,..,

-•

•

'" -e -c '"

-;,

amplitude and phase

o ~ b

...

i;: o

- - -

complex admittanee

..,

...., o -e-o

- - - -

;. o sea energy (logl0cmê) o :. .----r--T-l -1-.1 1

=

= =

=

= =

=

====-= =

=

= = =

=

=-==;=> =

=

= = =

=

= = ====-

₌

= = =

=

== c -=, C:=:::_::J =

=

₌ -=

=

₌_~_~

=

= = = ~ c...--:----, c:_~

=

=--==::1

=

=-:c;..~__ =

=

==-= = = = = = = = =

=

₌

=

= ==> :. ~ ;. _:. c=:=:3 I I~

=

= = = = -==

=

-=

=

= = = = = = =

=

= = = =

=

₌

_~ = = = =

=

= ~=====::I -= ~-='

=--=

= =

=

=---= =

=

-= -= = -=

=

= =

=

c::=--= C==':"'--==::.J --= =---= = = = = = = =

~=

= =

-= = =

=

= ~-, =

=

₌ c:..:-=::::l =

=

₌

_~

energy density (logl0 cm2(cfd)-I)

.:,

-

""

(lOgl0

e

cm

.

-)

-;. .;. o r---'---'I--'--,- T--1 2-'0_ 2-'2_ ~ 2'1 0 ;:;;;.. ...~--- -2·1' .. _

s

, 00

=

==

:..

_.;..---I N ~ I

-'0: _

J:.

, 10 - >'

-

...

-.

212_

s-.;-;

,

.

)

=

=;.;.

-

-:-

.'

"

o

_-

"-

""I

~--

~

"22 •. ~

,

,-,

-

_, 230_ :

..

.

,

.

2'0_

(28)

-I-:'j amplitude complex gravity energy

(5 and phase admittance sea energy (logIOcm") (logIOcm-)

c:: , _, _" " '. en 0.. ~ '" 0 \ft 0 CA 0 (Jol 0 ... ...., 0 ...., .... tV 0 ...., .. N 0 ~ lij' t::I "0 0 ~ ~\J1 "_

~~.

~

.

~ ~ ~ • wo ~ lij'~ 1 '<::»0 == ::s::s =

=

PJ ~ ~ c::::=:=::::J - C r::::=:::=::=:J c:===:::::l o - c:=:=::::=::=:J c::::=::::::::= (JqC_o

=

0 c:=:=:=:=l ~ c,_ i.,.. ~ ~ c::::::=J ~ 0'· = =

=

~ ti) c::==::=:::=::J c:::::::=::=J ::r'''O

=

_

-,

::»n

=

e

::s::»

.

a

A

7<

=

=,

~

-

1 =, - "'0 =:n ~ c::::::=:=:=J --===::J 1 0 _ w _ aqt""'t' - _ s:: c::=::=::=:J c:==:::=:=:::J , .,..., c:=::=:::=:=:=: ~ n

=

I-"_Q_ t::=::::=:l ~ I v

8 ~

~

N n n = = V1

1=iri~

~

I

~[~

~ ~ ,_~~. ~ ~ ,...c. 0.. "'" ~ c:::=::=:::::J ::r'o::» =

=

;:;'::s '< ~ ~ p,.ö> ~ ~

03:

(Ï) ~

~

t=

.

V

'

=

--=m~

-

_ o ~_~("O r=====:==-=" 7 0 :---_{_} ~ ("Ij\,) t""'t::::! • ...., _ ... n t:::==:::J c::==::=::::J o..~

=

("b ~ c::===::'J ~ en

=

~::r'::r'

_~~

=

₌

=

₌

o <,""1

=

"'d

na

~

~(D erO

=

= ::s

=

0... ~ - • c::::=::=J c::::==:=l ::sn 0:"

=

~ en ... c::=:::::::=::1 ~ r-j rio ~ ~ o

a

"'"

=

9 op,. ~ ~ ~ ~r~

Jl~

r

~

cl,

c,v~ __ ~ ~ ~n

=-

_,

,

.

"

""1 ~ c==:::=J • y u o w ~ .. c=::==:=I ~ _

a

,_]

w • ., LI11111 ,...~ N 0 fit.) N 0

(29)

20 G.> >< U G.> s:: ·0 Jegr~e ... CIS o...~ 'eo I -.30 ·.97 E ... imoginory -;51 '.49 o E.-20 • u"O CIS -40 0 0.01 cyclcsJday

FIGURE 6.Honollllu radiational tidc spectra at cJy resolution for spherical harmonies of degrees 1

(left portion of split columns) and degrcc 2 (right portions). Thc upper panel shows thc encrgy of the radiational potcntial at Honolulu rclativc to lO-7 unit. For species 0 and 1 numerical values of the admittanee (assumcd constant) are entercd. Circles in parentheses for species 2 have confidcncc limits too wide to be reliably computed.

I

-~

I

-~",C'"f,'-,

"

~"

'-',,

"

--

~...

o -2 -4

-

26

-":' ö 0 (0)

+

(0) •.34 •.98 ~ • -.U '3.54

;

0·99 1.00 1·01 1.99 2·00 2-01 adopted from ref, 1.

(30)

I

species 171 method 0 response

I

response

I

harmonie response 2

I

harmonie response 3

I

total[ response response harmonie

-2

7

-TABLE 1. HONOLULü PREDICTION VARIANCES FOR 1938-1957

cHO,±1, ±2, ±3),cHO±I),Xl.xà,~·c~ As above, without xl,

x

~

,

~.c~ c~ (0, ± 1,±2) 2Q,Q,p,0,M,P,S, K,], 00 c~(O,±1, ±2, ±3),cHO, ±1),X~,~·c~ As above, without ~.c~ 'HO, ±1, ±2, ± 3), cHO, ± 1) As above, without d(± 1) c~(0,±I,±2) 2N, P"N, v,M, L,S, K

c3

(0, ±1)

ei

all variables listcd above

minimum number of variables listed above

minimum number of variables

no. of station observedj constants <~2)m 7 23-16 variances (cm2) predicted residual ratio u~ q~Q. u'!Jn.l<0m 10·20 12·!){1 0'5GO 10·02 13·14 0·567 9·64 13·52 0·584 8·59 14·57 0·629 8·72 14·44 0·624 154·70 0·08 0·0005 154·()3 0·15 0·0010 154·61 0·17 0·0011 154·38 0·40 0·0026 157-50 0·13 0·0008 157·40 0·23 0·0015 157'37 0·26 0·0016 157-35 0·28 0·0018 157·30 0·33 0·0021 157-11 0·52 0·0033 0·008 0·053 0·87 0·008 0·053 0·87 322·41 32·78 0·0923 320'51 34·68 0·0976 320·21 34·98 0·0985

I

t

For example, cl(0,±1) designates the sixvariables.e] (t-Ár), aà (t), aà (t+ LiT) b~(t- Ár), b~ (r), bà(t+ Ár). In all casesÁT= 2 d.Variables

without parentheses wc:e uscd with unlaggc~ tonly..X.Verers ta the radiational potentials a::'+ip,::.

!The 'observed variances ' are the energres of Wi Inthe frequency ranges (0to 4 cIm) (1 c/ld ±4 cIm), (2c/ld±4 cIm), (3 c/ld± 2cIm),

respcctivcly. The 'total' observed varianee isthe overall varianee of the lowpassed series ~(t), inc1uding inter-tidal noise,

variablest

ag(0, ±1),a?, ay',ag, ag'

Asabove, without a~(±1) a.~,a~',ag,a~'

et

Y,

a

y

'

harmonie Sa, Ssa

I

variabie

a

g

I

xl

X~ ·1

re

'

!-' ~ 2

I

4 2 4 26 20 10 20 24 22 20 16 10 16 6 2 63 24 40 154·78 157-63 0·061 355·19 TABLE 2. HONOLULU WEIGHTS s U, v, variabie s U, v, 1 -0·163524 ,2₂ ₃ -0·032468 -0,050293 0 0·078005 2 -0·238275 0·292670 -1 -0,089109 1 0·728548 -0·014861 0 -4·94637 -8'87672t ;" 0 -0·505637 -0·853476 -1 -0·004338 0·850862 0 -11·99387 -13·50501 t _-₂ ₀_·₃₂₆₅₈₂ _-0,38₁₁₀₈ 3 0·015010 0·035523 -3 -0·140442 0·033328 2 -0·035527 -0·072475 ,2 ₁ -0,533411 -0,640720 1 0·229742 0·101201 3 0 0·141800 0·729601 0 0·067604 -0·499019 _-1 ₀_·264934 _-_0·₀₉₂₁₄₉ -1 -0·037543 0·218337 -2 0·056136 -0·094427

x

~

0 -1·98122 -1·36855 -3 -0,018781 0·017824 ~c~ 0 -0,00023 -0,00102 1 -0·402419 0·202837

,

3

₃ 1 0·007453 -0·240591 0 0·394974 -0·254192 0 -0·101228 0·127434 -1 -0·150418 0·107273 -1 0·053084 0·027965' 0 0·15069 -0·09649 0 -0·69386 1·58026 0 -0·00314 -0·00005

-adopted from ref. 1.

(31)

I

-28

-I

I

.

I

Several remarks with respect to the figures can be made.

I

a) There 1S a plateau of n01se 1U the power spectra of the recorded water level over the entire range of frequencies considered, strength

2

about 0.01 m day.

b) The coherent euergy pattern shows much agreement with that of the

input signaIs; the noise increases somewhat (tidal cusp) of places where strong components such as M2 occur.

I

d)

The relatively large partion of non-coherent energy Ln the solar

groups (11) and (22) has to be attributed to solar radiation. All frp's show a srnooth behaviour.

m

The energy of P3 1S about 4 order of magnitude smaller than that of

P~;

the frp's of p] are quite different from those of

PZ'

Solar input at low frequencies (Sa) and the diurnal tide SI 1S strong

compared to that of the gravitational input.

g) S2 is as far as generated by solar radiation has an amplitude of 1,8 cm

c)

I

e)

I

f)

11

at Honolulu.

I

(32)

I

, 9 •

-29

'

-8

.

Prediction

M. and C.'nave compared their results with .thoseof the usual harmonic analysis for the "linear" station Honolulu. The data for both methods are derived from the same series of observations. The predictions which

ean be done using the harmonic and the response method respectively

do not differ very much. The response method requires slightly ,fewer

constants, but the differenees are hardly significant. For instance, the

harmonie method using

4

0

station constants

(

20

tidal components) gl.ves a

variance of the resid

.

ual of

34,98

cm

2

, whereas the response method with

2

4

constants gives

3

4 ,

68

cm2 (total observed varl.ance355,19 cm ).2 The

m

contribution of the input P3 proves to be very small. In the harmonic method it is even not taken into account.

Extension of the linear th~ory to weakly non-linear systems

In particular on the coastal shelf the tides are influenced by non-linear

effects as friction and convective acceleration. If these effect do not dominate it may be attempted to apply a systems analysis which can take account of the non-linear processes. Basically the problem is this. The

input signals do not contain components with frequencies higher,than about 2 c/day (P~ neglected). In shallow wate~ higher frequencies are generated by the non-linear effects. This process cannot be described by a linear system,. since the basic idea is that a sinusoidal input will generate a sinusoidaloutput (with other amplitude and phase). For example in shallow water we find components such as M4, MS4 and components with higher

(33)

I

-30-I

I

_{In the purely} _{linear case this is} _{not possible,} _as _immediately _follows

from eq. (3) of this chapter, holding for the frequency domain. In order to extend the theory we will go back to (2):

I

+00 x

2

(t) =_00/ xj(t - T)k(T)dT (2)

The integral ~t the RHS cau be approximated by:

+00

L

i

=

-

00 I

+00

L

x(~~(t)k(i)

i

=

-

oo

+ ~•••

-I

I

Lnvt.his case x2 as__a 1i:n"ear. combination of many functiöns of time. More general

~t

can be put forward that x2 is an arbitrary function of

(1). (2) (3) .'. . . x ,x ~.x , ••. (subscript ••~ omifted):

I

( ) ( (0) () (L) ( ) ) x2 t = ~J X t, x t, .••••••

I

An approximation for x2 can be obtained by expanding tbe RHS US1.ng a mor-e-d imeasional Taylor series in whieh x(0), x (j) , •••• are considered

the independent variables:

I

= 1/1(0, 0, -0, ••• ) (0)' - (I) + x Ct) 1jJ0(0, 0, .•. ) + x Ct) IjJl(O,

p, .. )

+ ••• I

f,[

(0)

J

2 (0) (I)

+

21U

X

(t) IjJOO(O,

0, •• )

+ x (t)x (t) IjJOj(O,

° ..)

+ + x(O) (t)x(2) 1jJ02(0, 0, •• ) + x(l)(t)x(O)(t) IjJjO(O,0, •• ) •.

J

I

+ 3 rd order terms

I

where ~J• 1. _dXei) dX(j)

(34)

I

1 I

-I

I

1

1 I

I

1 I

I

-31-Af ter summati~n this eau be written as: --

.

- Ij;(0, 0, 0, •. ) +

+

f

X(i) (t) ~).(0, 0, 0, ..) r,

~

=

-

OJ

+2T

+00

I

xCi) (t)x(j)Ct) lji.. (0,0, 0, •• ) ~,]

i=

-

oo

j

=

-

oo

+

3!

+ ..•.•••

lji(O,o.,o, ••• ) c<l:nbe consi.de.red as the noi se and if -iti-s set_equal t.o.izero , then the_~_ontinuous-equivalent-of the aboveexpression change s into:

(35)

+ ••••

expression for a linear system; cal led the bi-linear contribution, the tri-liriear one, etc.

Accordingly k(Ll,L2) is called the bi-linear irp and k(Ll,L2,L3) the tri-linear irp, etc.

Eq. (35) shows, that now also higher frequencies ar~se ~n the output signal. Consider for instanee, the bi-linear part contain'ing the product Xl(t-'q). X2(t-L2)' If xl(t) is a harmonie with frequency a, then the

output will contain a component with frequency 20. If there are two components in xl(t) with frequencies al and 02 respectively,

then tha"output has components with frequencies (01+02) and (01-02)' This implies

_,

regarding tides, that the bi.-linear part will have a H4

(H2 x M2) or a MS4 (M2 x S2) etc, but not

a

M6 (M2 x M2 x M2) or a 2MS2

(M4/S2). Such non-linear contributions have to be generated by the tri

-linear part.

The first integral ~s identical to the The part with the double integral ~s the part with the three-fold integral

(35)

-32-I

I

The equivaleuts iu the frequeney domain eau_also be determiued. If the

bi-response is deuoted with 'x;Ct) then:

I

(36) Sinee

I

( )

i'":..

(

)

Zniot do '. xI (

=

-00 xl 0 e

I

(36) eau also be writteu as

I

·

1 I

2 . x2 (t)

I

,. . -2niCo' T + 0 T ) J+OO J+OO, , ). -'. 1 J' . 2- 2 • -00 -00 k('[J 'T i.: e .

I

The last iutegral presents the two-dimensioual Fourier transform of '

k(Tl, T2) : ( ) /" Ji<x>k( ) .21Ti(oJT'l'·+°2T2)" dT KOl' O2

=

-<>0 -00 TI' .T2 e Q.l1. 2 (37) ,

I

Heuee_:

I

(38)

-Fourier trausformatiou of this expressiou yields:

2 C

)

J+oo J+oo ~

C

O

X2 0 = u -00 -00 1 (39)

I

.

(36)

I

-33-I

I

cr_I +

°

₂ = u

I

2

TIlis expreSSLon indicates, that the calculation of one value of X2(0) the

integral must be calculated along the line °1+°2=0. If a limited number

of components is'involved,as in the case of tides, than those components gLve a éontribution'of which the sum or difference of the frequencies is equal to o. In particular the last expressions makes clear that the interaction between the (tide) components with certain frequencies generate other components with other frequencies. The magnitude of their mutual interaction is determined by the bi-admittance K(01,02)'

In general the interaction of components .ina non-linear system also yields

components with frequencies already .present. in the linear part of the spectrum. Therefore the operations in the frequency domain loose their straightforwardness, since now the output signal consists of several parts:

I

In these expression

X~(

o

)

can n~t be separated from

X;(

o

)

on objective grounds. Furthermore, the bi-spectrum

X~(

o

)

is the result 'of an integration procedure

in which there are contributions of interactions of different components. This means, that even in case

X~(

o

)

would be known, the bi-admittance

K(01,02) cannot be calculated. Consequently the operations must take place in the time domain.

I

(37)

I

-34-I

I

10. Example of a weakly non-linear system

I

The theory presented in section 5.9 is rather complicated. -Therefore it

will be illustrated with a simple example, though the results are complicated

enough. We will consider a mass-spring system which is subcritically damped.

The spring constant C is weakly non+Li.near.according to:

I

_C

₌

_c₍_I _- _EX₎_: €. « I

I

"The mass-spring system 1S excited by an arbitrary forc.e f(t)";with f(t)=O for

t < O. -The equatio~~escribing the motion of the mass is:

I

a?

+

bx

+ c(x - Ex2) = f(t) (40)

I

For negative t the system is

at

rest, x

=

i

= i=

0 for t <.O. Solutions will be tried of the form (asymptotic expansion):

2

x

=

Xo + EXI + E x2 + •••

I

The zero-order system becomes :

I

f (t)

I

The solution for

X

o

(4])

I

~here .k(T) represents the i~p of fhe zero-order system. Note that the upper boundary can be replaced by t since f(t)=O for t < O.

The first-order system is:

I

yielding the:solution:

I

x

(t)

=

I _-took_.₀ (T')CX~(t - T')dT' (42)

I

(38)

I

i

-I

I

'

I

_

I

__~_.-

~

...~

-35-In this special case the irp's of the zer o- and first-order systems are the same.

,By means of (42) ~q. (39) can'be written as:

:t<X> J k(T')clT'

o

+co +00 J k(T)f(t-T' - T)dT J k(T")f(t-T' -T")dT"

o

0 Introduce T' + L :::-T1 so that:

'In this expression the integration with respect to T' can be

executed (formally):

+co

c

r

k(T')k(T' - T)k(T' - T2)dT'

o

(43)

so that (see also (36))

xl (t) :: C , (44)

In thi~ case the irp ket) of the zero-order system for

a2 ::4ac - b2 > 0 (damped oseillation) is: bt

-

_

ket) 2 e sin at

_Za

t > 0 ket)

=

0 t <

0

so that

B

e

ëY?

t - È_(3T'

-f

e 2a

o

. cr' . a(l'-Tl) si.n (tel

'

-ez

.1 dr SUl - s~n .L • 2a 2a 2a -(45)

This integration-can be carried out analytically, but the final result ~s not very illustrative. For the purpose of interpretation the response to a unit pulse at t ::.0 will be cànsidered. Using eq. (44)