ISSN 0704-3716

CANADIAN TRANSLATION OF FISHERIES AND AQUATIC SCIENCES

No. 4773

A study of the natural oscillations in the level of harbours on the Kuril-Kamchatka coast

by

R.A. Yaroshenya

Original Title: Issledovanie sobstvennykh kolebanii urovnya bukht kurilo-kamchatskogo poberezh'ya

From: Teor. Eksp. Issled. Po Probleme Tsunami: 153-164, 1977.

Translated by the Translation Bureau (GAD) Multilingual Services Division

Department of the Secretary of State of Canada

Department of Fisheries and Oceans Institute of Ocean Sciences

Sidney, B.C.

1981 •

21 pages typescript

1

DEPARTMENT OF THE SECRETARY OF STATE

TRANSLATION BUREAU

SECRÉTARIAT D'ÉTAT

BUREAU DES TRADUCTIONS

DIVISION DES SERVICES MULTILINGUAL SERVICES CANADA

] INYO - ER

ENGLISH

PUBLISHER — élJITEUR DATE OF PUBLICATION DATE DE PUBLICATION

Academy of Sciences of the USSR "Nauka" Publishers

PLACE OF PUBLICATION

LIEU DE PUBLICATION

Moscow, USSR

YEAR ANNE.E

l'177

VOLUME ISSUE NO.

NUMeR0

DFO TRANSLATION BUREAU NO. 554049 NOTRE DOSSIER NO

REQUESTING DEPARTMENT MINISTRE-CLIENT

DIVISION MULTILINGUES

e/%5 7 3 TfiANSCi.TED FROM TRADUCTION DE

RUS SIAN

AUTHOR — AUTEUR

R.A. Yaroshenya TITLE IN ENGLISH — TITRE ANGLAIS

A STUDY OF THE NATURAL OSCILLATIONS IN THE LEVEL OF HARBOURS ON THE KURIL-KAMCHATKA COAST

TITLE IN FOREIGN LANGUAGE (TRANSLITERATE FOREIGN CHARACTERS) TITRE EN LANGUE éTRANGERE (TRANSCRIRE EN CARACTÈRES ROMAINS)

ISSLEDOVANIE SOBSTVENNYKH KOLEBANII UROVNYA BUKHT KURILO-KAMCHATSKOGO POBEREZH'YA

RITERCNCE In FOREIGN LANGUAGE (NAME OF 13005 OR PUBLICATION) IN FULL. TRANSLITERATE FOREIGN CHARACTERS.

(117.FéRENCE EN LANGUE éTRANGe.RE (NOM DU LIVRE OU PUBLICATION). Au COMPLET, TRANSCRIRE EN CARACTÈRES ROMAINS.

Teoreticheskie i eksperimentarnve issledovaniya po probleme tsunami

REFERENCE IN ENGLISH — RéFéRENCE EN ANGLAIS

Theoretical and experimental research on the tsunami problem

PAGE NUMBEFiS IN ORIGINAL

NUMéROS DES PAGES DANS L'ORIGINAL

153-164 HUMBER OF TYPE() PAGES

NOMBRE DE PAGF.S

DACTYLOGRAPHI éES

21

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PERSON REQUESTING DEMANDÉ PAR

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May 8th, 1.981

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d 505-200.10.0 IRE V. 2/09)

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MULTILINGUAL SERVICES DIVISION - DIVISION DES SERVICES MULTILINGUES

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Translator (Initials) - Traducteur ( Initiales)

554049 Russian GAD i^^^ ^^ i,^ !•

Theoretical and experimental research on the tsunami problem, Academy of

Sciences of the USSR, Soviet Geophysical Committee, "Nauka" Publishers,

Moscow, USSR, 1977, pp 153 - 164.

A STUDY OF THE NATURAL OSCILLATIONS IN THE LEVEL

OF HARBOURS ON THE KURIL-KAMCHATKA COAST

by

R. A. Yaroshenya

r. C..

:J.D̂--•

By methods of spectral analysis of tide gage records, thenatural oscillations in level of harbours on the Kuril-Kamchatka coast

are studied. By means of a special procedure, the tidal component ofthe oscillations in level is eliminated from the records. In the

calculated spectra, a number of maxima, which can be associated with thenatural oscillations in level of the harbours, stand out clearly. For

tsunamis from remote sources, the spectra show a low-frequency componentwhich is absent in close tsunamis. For 39 harbours on the Kuril-Kamchatka

coast, an analytical calculation has been done of the periods of thenatural oscillations in level. The results of the calculations show

good agreement with the results of spectral analysis of tide gage record.

Long waves have the property of resonating with the natural /153*

periods of oscillations in level of harbours. The resonance phenomenon

is a selective process, that is, in the incident wave, consisting of

several simple oscillations, it is not the entire spectrum which resonates,

but only those oscillations, the periods of which are close to or multiples

multiples of the period of the natural oscillations of the basin. Due to

resonance, the heights of the tsunami waves vary. In order to determine

* The numbers in the right-hand margin indicate the corresponding pages

in the original. - Transl.

s^l? 944i' IIaMv. WIr!

2

the increase in the tsunami height as it traverses harbours, bays and

straits, it is necessary to determine the natural oscillations in the

level of the harbour.

In the'study of the propagation of long waves near shore, it is

very convenient to introduce the concept of "background noise". If we

exclude the energy of the tsunami waves, and other large disturbances,

then in the high frequency range, the energy level is determined by surf

pulses, and in the low frequency range by the energy of waves due to

meteorological factors.

Snodgrass, Munk and Miller [12] measured simultaneously the

"background noise" off the coast of California and at a point 100 km

from shore and calculated its energy spectrum. It turned out that

"background noise" varied at different points: the predominant noise

frequencies for one point differ from the frequencies at another point.

The energy level of the "background noise" is measured in 10 2 cm2 .

According to Munk [8], the most surprising property of long

waves is their absence, and only in the passage of large disturbances

(of the tsunami type) does the energy level increase markedly.

Moreover, the frequency range of "background noise" remains the same,

but the amplitude increases at those frequencies which are close to the

frequencies of the component oscillations of the tsunami.

Having studied the "background noise" of harbours, straits and

bays, and having determined the frequency range which is characteristic

of these basins, one can determine the frequencies at which the

components of the tsunami waves will resonate.

The periods of the natural oscillations in the level of harbours

were investigated by spectral analysis of available observations of the

3

change in level and by calculation using well-known analytical formulas.

1. Spectral analysis of observations by time series

Spectral analysis is widely used in the analysis of time series

[2, 4, 7-10, 12-14].

At present, there is a sufficient number of observations on

oscillations in the level of the water areas of the Kuril-Kamchatka

coast. Subjecting the available records to spectral analysis, we can

isolate the typical spectrum of the frequencies of oscillations in the

level of harbours, bays and straits.

In recent years, the Leningrad branch of the State Oceanographic

Institute of the Main Administration of the IIydrometeorological Service,

under the supervision of V.A. Rozhkov, has greatly developed the•methods

of the probabilistic analysis of wind waves. One of the necessary and

important stages of wave measurements is the calculation of the

statistical estimates of probability characteristics. A set of

programs, written in a code with a system of commands for M-220, BESM-3,

VESM-4 type computers has been published in a separate collection [2].

These p:-ograms for the calculation of the statistical estimates of the

correlational and spectral functions, as well as the programs to exclude

the tidal component, have been used to analyse the records of the

oscillations in level during the passage of tsunami waves.

Instrumental observations of oscillations in sea level are done

by means of self-recorders of sea level and are registered in the form

of curves on the tide gage tape. To calculate by computer the

statistical estimates of the probability characteristics of the

oscillations in level, the continuous record must be transformed into a

discrete sequence of values of the recorded process.

/154

4

A basic problem in the transformation of continuous information

into discrete information is the selection of the quantization interval

of the input data. The selection must take into account the structure

of the studied'process and the kind of functional transformation, on

which the algorithm of the calculation of the estimate of the

probability characteristic is based.

The main requirement for this selected interval is that the

estimate calculated for the discrete sequence of values must deviate

little from the same estimate, calculated for the continuous process.

On this basis, and in order to increase the series of

realizations, enlarged (2-3 times) photocopies of tide gage records were

prepared. The quantization interval was reduced from 10 to 2.5 and 1.7

minutes, and the series of realizations was increased 2-3 times.

The ordinates were read from the enlarged photocopies

automatically and semiautomatically with a device for the input of

graphical information into the computer. This device is designed for

the automatic and semiautomatic reading of the coordinates of the

points of any curve with subsequent transformation of the values of the

coordinates into a binary code on a standard punched card. The

coordinates of the read points were fed from the punched cards into the

computer, and the information in binary code was translated by program

into the code of the specific computer.

The long-period tidal component stands out clearly on the tide

gage record. It must be excluded from the records of the oscillations

in level. Methods of the sliding mean type are widely used in

mathematical statistics to exclude a tendency in dynamic series. The

random disturbances in the found tendency can be eliminated by the

selection of an appropriate smoothing operation.

An effective example of the calculation of a tendency is the

polynomial smoothing of the results of the sliding mean of observed

values of a studied process. In this case,

012

(t).=

is taken as the approximate value of the tendency. This is refined by

means of an operation of the kind

n (i)=-*Pbo(i) ± k (s) bk (I s) as, ..0

where a (s), g (s) are weight functions, g o = g (0), bk (t) is the

coefficient of a polynomial of degree n, which approximates e(t) in the

interval [t, t + p]. The expressions contain the values 6, p, n as

parameters. The parameter 6, connected with the period of the

fluctuation process, is best taken as equal to three-to-five mean wave

periods. The par.ameter n which depends on the supposed change in the

tendency on the analyzed section of the record, is taken as equal to

unity. The parameter p is specified so that in the interval (t, t + p), /155

the mean line varies linearly; in particular, for tide gage records with

slow change in the mean line, p = (5-7) 6, while for wave records

with variable mean line p = (3-5) S.

Fig. 1 presents the real tidal component and the component

excluded by the method of the sliding mean.

The selective properties of a filter, as is known, are evaluated

on the basis of the amplitude-frequency characteristic R(w), which

characterizes the ratio of the amplitudes of processes which differ in

frequency of harmonics.

o

6

At present, the amplitude-frequency characteristic of the

studied process has not been established. Consequently, the calculation

of the statistical estimates of the spectral function Is done for

several combinations of controlling constants. The coincidence of peaks

on the curves indicates the reliability of our calcuLations. A

deviation was considered both as an error caused by a filter with a

combination of constants which do not exclude the tidal component.

For discussion, we present the spectra of the oscillations in

level of Kasatka Bay (Burevestnik settlement) during the passage of the

tsunami waves of 28.111-1.IV, 1964, 14-16.V, 1966, 16-18.V, 1968,

11-13.V111, 1969, 22-25.XI, 1969; the spectra of the oscillations in

level of Yuzhno-Kuril'sk Harbour during the passage of the tsunami waves

of 23-29.V, 1960, 13-16.X, 1963, 28.111-1. 1V, 1964, 14-16.V, 1966,

18-20.X, 1966, 15-18.V, 1968, 29-30.1, 1968, 11-13.V111, 1969; the

spectrum of the oscillations in level in Kuril'sk Bay (Kuril'sk) on

28-31.111, 1964. The spectrum is presented of the oscillations in level

of Yuzhno-Kuril'sk Bay for the period 12-14.11, 1961.

Fig. 2 shows the spectra for Burevestnik settlement, calculated

from 3024 points for the tsunami of 28.111-1.IV, 1964 and from 1074

points for the tsunami of 16-18.V, 1968. The period T in minutes is

plotted along the axis of ordinates. The spectra have been calculated

for seven combinations of controlling constants (see Table). The

coincidence of the peaks of spectra with different controlling constants

indicates the reliability of the periods found. The spectra of the

tsunami of 28.111-1.IV, 1964 clearly shows two peaks, corresponding to

periods of 99 and 43 minutes. On the spectra of the tsunami of 16-18.V,

1968, the main periods are 47.3 and 19.5 minutes.

7

Figure N p p e, ut nt s t à II,

.1

2,a: • / 30'24 10 20 71 300 300 250 1,00 0,1

2 3024 7 15 35 300 300 250 1,06 0,1

2 3024 10 20 as 300 ' ;100 250 1,60 0,1

2,6:

/ 8024 7 15 47 300 300 250 1,60 0,1

2 3024 5 10 47 300 300 250 1,66 0,1

2,a:

/ 1074 3 5 35 100 100 60 1,66 0,1.

:t 1074 a 10 35 100 100 60 1,66 0,1

Fig. 3 presents the spectra for the passage of a tsunami at

Burevestnik settlement. Periods of 44 and 46 minutes stand out clearly

on each spectrum. The period of 44 minutes is the largest for the

spectra of close tsunamis. On spectra of distant tsunamis, the periods

of 46 and 38 minutes are shifted to the right, while the greatest

periods are 107 and 84 minutes.

Fig. 4 shows the spectra for nine tsunamis and the spectrum for

the period 12-14.11, 1961 for Yuzhno-Kuril'sk. For close tsunamis, the

maximum energy occurs in the period of about 40 minutes (38, 44

minutes). In the passage of distant tsunamis, the spectra show a long-

period component - 62, 78, 104 and 115 minutes. In the absence of a

tsunami, the spectrum has a maximum of energy in the 41 minute period.

A long period component with T= 100 min appears in the spectra

for Kuril'sk (Fig. 5) during the passage of distant tsunamis.

For comparison, Fig. 6 gives the spectra of the same tsunami at

different points. The spectra differ sharply in shape. For the tsunami

of 28-31.111, 1964, at Kuril'sk, the main periods are 104, 41, and 26

minutes, at Yuzhno-Kuril'sk the periods are 104, 61, 49 and 40 minutes,

and at Burevestnik 107 and 46 minutes; the presence of a long period

/156

/157

/156

!ZO

V!/

17. !B Z1v J G' 9 t, vuc q,

I

Fig. 1. Comparison of the tidal component of the oscillations in level,excluded by the method of the sliding mean (1) with the observedcomponent (2).

a - hours.

a Gl

cov

I I^ ^^ I^r^ id 1 . ^;►I --- , . :='^3'9 IIJ .?2 2,9 /.Y

8

I^

I d C

/!// 4',) J.l" 79 /.Y T, ^niu

/dd /I I I ^ : -- -^

^ ^ III '

I^^^^.•^.. ^r--^^i^^,

--^--^---1---• --=-

!^! 3// /d T, ^^ti^!

Fig. 2. Spectra of tsunamis, registered at Burevestnik settlement, withdifferent values of the controlling constants;

a - b, 28.III-1.IV, 1964;c - 16-18.V, 1968.Values 1 - 3, see Table.

‘ot■ellb. Vi

ZS

a 7,5" 60

«41

72

d

j et 1 1

d

/5

2, 24

a

1.-QS• 1,->

7. mall

N.% 0

N'■

9

component - 104 minutes is characteristic for these spectra (a distant

tsunami was examined).

Fig. 3. Spectra of tsunamis, registered at Burevestnik settlement

a - 28.111-1.IV, 1964; b 16-18.V, 1966; c - 15-18.V, 1968 d - 11-13.V111, 1969; e - 22-25.XI, 1969. The period T in minutes is plotted along the axis of abscissae.

Fig. 4. Spectra of tsunamis registered at Yuzhno-Kuril'sk

a - 23.V, 1960; b - 13.11, 1961; c - 13.X, 1963; d - 28.111- 1.1V, 1964; e - 4.11, 1965; f - 15.V, 1966; g - 18.X, 1966; 29.1, 1968; i - 16.V, 1968, j - 12.V111, 1969.

h-

10

For the tsunami of 14-16.V, 1966, the main periods are 78 and 23

minutes in the spectrum at Yuzhno-Kuril'sk and 44, 18 and 12 minutes in

the spectrum at Burevestnik.

In the'spectrum at Yuzhno-Kuril'sk, the main periods are 71 and

25 minutes for the tsunami of 16-18.V, 1965. In this spectrum, the

energy occurring at different frequencies, is almost at the same level.

The main periods in the Burevestnik spectrum are 44 and 18 minutes.

In the spectra of Yuzhno-Kuril'sk and Burevestnik, for the

tsunami of 11-13.V, 1969, the main periods are very close to 44, 22 and

18 minutes.

Comparison of the spectra of oscillations in level of the bays

for close and distant tsunamis showed that a long-period component

appears in the spectrum for distant tsunamis.

The spectra are the same for one point for different tsunamis.

For different points, the spectra of the same tsunami are different.

Consequently, the nature of the record of oscillations is determined by

bottom topography.

2. Calculation of natural oscillations in level

There are many different ways of calculating the periods of

natural oscillations on the basis of analytical formulas. Of interest

is the method of the Japanese researchers Honda, Terada, Yoshida and

Isitani, which is one of the main methods used in calculation. The

other methods were auxiliary. However, the advantages and drawbacks of

the methods should be discussed briefly.

The formula of Merlan [1] is correct exclusively for a

particular case, but gives satisfactory results. Often the error in the

determination of the periods is rather substantial.

11

Fig. 5. Spectra of tsunamis registered

a - 13.X, 1963;b - 4.II, 1965;c - 28-31.III, 1964.

at Kuril'sk

Fig. 6. Comparison of spectra of tsunamis registered at differentpoints

a - 28.III-1.IV, 1964;b - 14-16.V, 1966;c - 15-18.V, 1968;d - 11-13.VIII, 1969; 1 - Kuril'sk;3 - Tiurevestnik.

E

a

u

o ^ ^ôâN

/O^.cHO-Nyf^nnar.,r E I

2 - Yuzhno-Kuril'sk;

I

^yPeasC,7,,,U,r - ,

12

The formula of Dubois [1] allows us to calculate the periods in

a rectangular basin with variable depth.

Kristol* [1] describes the natural oscillations by means of

differential equations, which are derived without allowance for friction

and the geostrophic acceleration. To solve the main equation it is

necessary to find the normal curve, which in nature has rather complex

outlines. For this reason, the periods of seiches are determined by

numerical integration. There are a number of methods of solving the

equations. The Kristol* method is very time-consuming.

Ertel's method [1] gives distorted results when the width of the

water area increases drastically.

Proudman's method [3] allows us to calculate the periods of

oscillations in irregular basins with variable depth. With drastic

change in the width of the bay, unsatisfactory results are obtained.

The method cannot be used for the determination of the period of

oscillations in straits which are open on both sides.

The main formulas of the method of Defant [5] are derived from

the equation of continuity and the equation of motion. The method gives

good results with any shape of the basin and even when the width of the

harbour increases greatly.

The method developed by Neumann [11] which he calls the method

of impedances, on the basis of the similarity of hydrodynamic

oscillations and oscillations occurring in electromagnetic circuits, is

the simplest of all the available methods of calculating the period of

natural oscillations in the level of harbours.

* Spelling approximate. - Transi.

13

Although the described methods were in fact developed for closed

basins, their application to open basins gives satisfactory results.

Honda, Terada, Yoshida and Isitani [6] studied natural

oscillations iz bays and harbours. They examined the following cases:

1) rectangular bay of constant depth; 2) bay of irregular shape; 3)

two basins, communicating through a narrow strait.

The calculation of the period of a free oscillation in a bay of

irregular shape amounts to the calculation of the period of a seiche in

a lake of irregular shape:

I 21 T (1 4- 111 -F- 112) '

V gh o

where

1 2rxn (x) cos ----I— dx,

11 2 7,77- As (x) cos t-12.1--9—xn dx, zi o 0

and n is the riumier of nodes of longitudinal and transverse seiches

respectively, b o and s o are determined from the conditions that

ilsdx 0,

if Ab..b(x)—bo, As--=s (x)—s, (1) . W is the width of a vertical

section, s(x)* is the area of a vertical section, h o = s0/b 0 is the depth

of the basin).

Considering that,1 is the length of the lake, then .1=2L where

L is the length of the bay. Then the formula for the calculation of the

period of natural oscillations of the bay will take the form

2 ex , 2 TGX A

7 r Ab cos ux S. As cos —, ux. vgh, 60 gh L so V gho o o

It follows from the expressions for II and 112 that the period

increases with reduction of width in the central part of the lake, and,

on the other hand, it decreases with increase in width. In connection

with bays, thege integrals are examined for a lake, the shape of which

is symmetrical with respect to the vertical plane. Let us determine the

sign of the correction

27cx 2nx Ab•Cos dX î As cos --e clx so

with widening of the lake in the middle.

We shall examine a bay, which constricts at the end. The depth

of the bay at the end becomes smaller, L is the length of the bay. We

construct the symmetrical reflection of this bay and we obtain a lake of /160

length 1 (Fig. 7, a). Let us examine each term separately:

i 27:a: 1 NCI 2.nx • cos -7-- dx = -b- .2j, Ab i cos --1-. Afc i 4—

, d=i

1 2.3.0 f 1 2r,x, x1 + — AL cos — 1-2 Ax ii .

2 1J

The second terni is always less than zero. Let us determine the sign of

each term in the total. In parts 1 and 3 of the lake, 1 b< 0 (bi<b0 ) and

2'xe<0 Ab>0, Consequently .-7 andcos— - >O. In part 2,e0s-7- 1

2r.x cos da; < O.

We determine the sign of the correction:

1 P—i

i 2r.x 7 2MX i A i

'--"" As cos •-r" aX =-.- •— •1 I As, nos ê0 so ,

0 J., 1 2zcx 2ra

-I-- As.u cos —9- ba + As„ cos ----It A;, , I

2 / /

15

Fig. 7. Graphical illustration of the effect of widening (a) andconstriction (b) of the basin on the sign of the integral

where n is the number of sections. The zero section is situated at

point x= 0(Fig. 7, b). The area of the section, calculated according to /161

the formula s = bh, diminishes to the apex. Let the points xo and x;,

correspond to the, area of the section s,) (x;, > b/4) .

Let us examine each term separately. In parts 1' and 5'

(Fig. 7, b) As < 0 and cos '> 0 (negative terms). In part 3' ,.&S >0 and

cos < 0 ( negative terms ) • Only in the very small parts 2' and G' do

the signs of the cofactors coincide (positive terms). The corrections

for bays of any shape are calculated by these formulas. A mouth

correction is also introduced to the period calculated with corrections

for cross section.

The dependence of the values of the mouth corrections on the

ratio of bay width to length is given by Defant [5] and the Japanese

investigators [6].

It should be noted that for bays, in which the ratio of the bay

width to length is greater than unity, this function does not apply.

, kJ = sulk> ,

--I

1)„ b il2

16

Defant [5], notes that when the shape of the bay differs

considerably from rectangular, this relation of the mouth correction to

the bay width/length ratio is incorrect.

In the 'majority of cases, the period calculated according to

Merian's formula, that is, TrzArdse, agrees sufficiently well with the

observed period. This is probably because for many bays, the correction

for cross section is close to the mouth correction, but has a negative

sign. This is explained by the fact that many bays and harbours

gradually constrict and shallow towards the apex, and thus the

corrections for cross section and the change in volume of the

oscillating liquid will be negative. When the correction for change in

cross section and volume is close in absolute value to the mouth

correction, then they cancel each other out and the period can be

determined from the formula . liowver, this formula can

[only] be used after preliminary studies of the bays for the indicated

corrections. For the example of three simple harbours on the Kuril-

Kamchatka coast (Fig. 8-10) the calculation of the periods of the

natural oscillations in level of 39 harbours is shown.

For the calculation, one must have a sufficiently detailed

bathymetric map. The length of the bay, the width of the cross section

and the depth are determined from the map. We note that bejacp ,,

they seksep and 14yW4 and ate calculated as follows:

where bo..% are the width and area of cross sections.

Kasatka Bay (Burevestnik settlement). Four positions of the

To 29,4 min., /I 2 - 5 , 8 en •

17

node have been selected for the bay (Fig. 8). Let Us examine the node

at positions ab and ac. For these positions, nine sections each have

been made with a step of 0.72 and 0.66 km respectively.

The carculated b 0 , s 0, h 0 are equal to 7, 9 and 9 km, 0.27 and

0.22 km 2 , 27.5 and 24.5 m. The measured length is 7, 25 and 68 km.

Considering these data we have:

The maximal mouth correction, established by the Japanese investigators,

is 0.34 for 14,1 1 -1 . For Kasatka Bay,b0ne>1 , though nevertheless for

this ratio we take the maximal mouth correction. The mouth correction

for Kasatka Bay is aT).„ ,.r0 .0,34=10 min.

Thus, with the node at ab we have:

1 ) 7. 0 + AT 14= 31 min.

With the node at ac, T o = 29.3 min, H I = -1.2 min, 11 2 = -5 min. The

mouth correction ATI ,,ct-1 10 min;

2) T = 33 min.

With the node on the boundary of the shelf, 20 cross sections were

selected with a step of 0.64 km with a lenght 1= 18.9 km. The results

were: bo = 20 km, so = 1.58 km 2 , ho = 78.8 m, T o = 45.3 min,

H I = -6.3 min, 112 = -8 min. The mouth correction AT ), = 15.4 min;

3) T = 4.6 min.

In the calculation of a transverse seiche, 20 sections were

taken with a step of 0.58 km with a length 1 = 11.6 km. The calculated

values are: bo = 5.34 km, so = 0.13 km 2 , ho = 24.2 m, To = 25 min, HI =-2.5

min, H2 = -5.2 min;

4) T = 17 min.

/162

18

Fig. 8. Diagram of Kasatka Bay (Burevestnik settlement)

Fig. 9. Diagram of Yuzhno-Kuril'sk Bay (Yuzhno-Kuril'sk).

Fig. 10. Diagram. of Kurillsk Bay (Kuril'sk).

In addition, for Kasatka Bay, we calculated the periods of

longitudinal seiches with the node line at ab. As a result the period T

was: 32 minutes by Merian's method, 25.5 minutes by Proudman's method,

.27 minutes by Defant's method, 25 minutes by Ertel's method, 31 by the

method of Honda and Terada, and 31 minutes by the method of

impedances.

Comparison of the calculated values with the periods found by

spectral analysis, allows us to state categorically that the

periods of natural oscillations lie in the range 30-40 minutes.

Yuzhno-Kuril'sk Harbour (Fig. 9). For the calculations, six cross

T o = 11.4 min H I = -1.3 min, H2 = -3.2 min. The mouth correction I

19

sections were drawn with a step of 0.32 km with a length of the

harbour of about 2 km. The computed values of bo, so, ho were 3.96 km,

0.025 km2 and 6.4 m respectively, T o = 16.5 min, H I =- 1.7 min, R2 =

-3.3 min.

The mouth correction was Airy,,z 5.6 min, T = 17 min.

Spectral analysis of observations of the oscillations in level

of Yuzhno-Kuril'sk Harbour revealed periods which vary in a wide range

(see paragraph 1). It is likely that the recorder of oscillations in

level is recording a change in the level of the Yuzhno-Kuril'sk strait.

For Yuzhno-Kuril l sk strait periods of about 50m and 2h were calculated

by the Defant method. The data obtained are in accord with the spectrum

of frequencies for Yuzhno-Kuril'sk Harbour.

Kuril'sk Bay. Ten sections were selected with spacing of 0.17

km with a length of the bay of 1.7 km (Fig. 10). The values of /163

1) 0 , s o , h o are equal to 2.8 km, 0.028 km2 and 10 m respectively,

3.9 min, T = 11 min.

A small part of the energy occurs at T = 11 min in the spectrum

of oscillations in level of the bay. In calculating natural

oscillations in level, one should probably examine a wider water area.

Kuril'sk Bay is part of a bay Which is situated between Cape

Breskens and Cape Ksana. For this water area, ten sections were

selected with a spacing of 1.15 km with a length of 11.5 km.

The results were: b 0 = 23.8 km s = 4.8 km 2 h0 = 200 m,

T o = 17.3 min, H I = -2.6 min, R2 = -3.2 min. The mouth correction

. 1171r„. 6 min, T = 14 min.

The values of the calculated periods lie in the high frequency

part of the spectrum, at which the least energy occurs. In order to

20

obtain a set of periods close to nature, it is necessary to calculate

with allowance for the shelf zone with various positions of the nodes.

Conclusions

In this paper, the natural periods of oscillations in level were

investigated by the method of spectral analysis and were calculated from

analytical formulas. Spectral analysis makes it possible to establish

the whole range of frequencies which is characteristic for a particular

water area. The spectra for the same point are close, which cannot be

said for the spectra at different points. In distant tsunamis, a long-

period component, which is absent in the spectrum of close tsunamis,

appears in the spectra. The nature of the spectrum primarily depends on

bottom topography.

There is no standard method for calculating the natural

oscillations in level. If the body of water is close to rectangular in

shape with a smooth bottom, Merian's formula is used. For sufficiently

simple conditions, the periods of seiches are calculated by Ertel's or

Proudman's method, for complex conditions, by Defant's method. One must

select the optimal number of cross sections when using the methods of

Defant, Proudman and Ertel, in order to avoid cumbersome calculations.

The method of Ronda, Terada, Yoshida and Isitani and the method

of impedances give reliable results in the calculation of periods for

basins which are close to rectangular, with uneven bottom.

Satisfactory results are obtained by calculation with allowance

for adjacent water areas with various positions of the nodes.

21

ABSTRACT

Free oscillations of harbour levels of the Kurile-Kamchatka coast bave been studied the method of spectral analysis of mareograms. With the help of special procedure the tide component of level oscillations have been removed from records. In calculated spectra sente maxima are clearly revealed, related to free harbour level oscillations. For tsunami waves from (listant sources low-frequencies appear in spectra, which are absent in spectra of near tsunamis. For 39 harbours of the Kurile-Kamchatka coast the analytical calculation of the period s . of free level oscillations have been made.. The results of calcula-Lions are in good agreement: with data of the spectral analysis of mareograms.

References

1 • Arsen'eva, N.M. et al. Seiches in lakes. Leningrad University Publishing House, 1963.

2. Davidan, I.N. et al. Probability characteristics of waves, methods of their analysis and calculation. "Trudy GOIN", 1971, no. 97.

3. Proudman, J. Dynamic oceanography, Moscow, IL (Foreign Literature), 1957.

11. Neumann, G. Impedance of mechanical oscillation systems and their application to the theory of seiches. "Ann. Hydrol. und Marin. Meteorol.", 1944.

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