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UBC Theses and Dissertations

Seiche excitation in a coastal bay by edge waves travelling on the continental shelf Lemon, David Douglas 1975

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. SEICHE EXCITATION IN A COASTAL BAY BY EDGE ''WAVES TRAVELLING ON THE CONTINENTAL SHELF. b y DAVID DOUGLAS LEMON . B.. Sc. , University of British Columbia, 1972 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE. REQUIREMENTS'FOR THE DEGREE OF MASTER OF SCIENCE in the Department o-f PHYSICS and the Institute of Oceanography We accept, this, thesis as conforming to the required standard. THE UNIVERSITY OF BRITISH. COLUMBIA .AUGUST, 19 75 In present ing th is thes is in p a r t i a l fu l f i lment of the requirements for an advanced degree at the Un ivers i ty of B r i t i s h Columbia, I agree that the L ibrary sha l l make it f r ee ly ava i l ab le for reference and study. I fur ther agree that permission for extensive copying of th is thes is for s c h o l a r l y purposes may be granted by the Head of my Department or by h is representa t ives . It is understood that copying or p u b l i c a t i o n of th is thes is for f inanc ia l gain sha l l not be allowed without my wr i t ten permiss ion. Department of Physics  The Un ivers i ty of B r i t i s h Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 ABSTRACT »: Short p e r i o d h o u r ) o s c i l l a t i o n s are often seen superimposed on t i d a l curves at Port Renfrew, B.C. In order to determine the nature of these o s c i l l a t i o n s , time-series of sea-level v a r i a t i o n were obtained i n Port San Juan; thei r analysis revealed energy peaks at periods of 34.7 min. and 13.5 min. A theoretical model of wave excitation of a basin of variable depth f i t t e d to Port San Juan by edge waves t r a v e l l i n g on the shelf accounts reasonably for the observed frequencies. Bottom f r i c t i o n was taken into account to investigate the dependence of seiche amplitude on the amplitude of the shelf waves. i i i TABLE OF CONTENTS Page ABSTRACT . i i . TABLE OF CONTENTS • • • i i i LIST OF TABLES v LIST OF FIGURES v i ACKNOWLEDGEMENTS • ........ ix CHAPTER 1: INTRODUCTION v.. 1 CHAPTER 2: DATA.;... ... '. •••• 4 1) Prior Evidence of Seiches 4 ; 2) Method of Measurement ... 7 3) Data Collection 8 4) Time Series Processing . 10 a) Digitization and record patching 11 b) Calibration 13 . c) Detrending %1 5) Spectral Calculations 29 6) Discussion of the Spectra 3| 7) Damping Effects CHAPTER 3: THEORY 41 1) Solution of the Long-Wave Equations i n a Bay with Exponential Bottom %1 2) The Outside Edge Wave 49 3) Matching the Solutions 53 i v Page 4) F r i c t i o n a l E f f e c t s £5 CHAPTER 4: CONCLUSIONS . 70 BIBLIOGRAPHY • 72 APPENDIX A: Cross-Channel Modes 73 APPENDIX B: Green's Function f o r a Wave on a Sloping Bottom 75 APPENDIX C: The Instrument 79-LIST OF TABLES TABLE Page 2rt The d i g i t i z e r scale constants 12 2.Jl The s e n s i t i v i t y as a function of depth I4 2„iiii -The t i d a l constituents at Port Renfrew 22 2„iy Q as a function of the mode number ... 39 3oV The wavelength of the edge wave as a function of i t s period. 52 3oVI The observed and calculated seiche periods ... "68 LIST OF FIGURES v i Fig. ?age 1.1 The S t r a i t of Juan de Fuca and Port San Juan 2 1.2 Port San Juan. Measurements were taken at the government wharf i n Snuggery Cove 3 2.1 Seiche action on the government tide gauge, 29 January 197$; : 5 2.2 Seiche action on the government tide gauge, 30 January 197'1'i 6 2.3 The wraparound technique for scale expansion 9 2.4 The c a l i b r a t i o n curve for channel 4 of water depth vs. d i g i t i z e r count ..15 2.5 The c a l i b r a t i o n curve f o r channel 2. The plot shows counts for channel 2 vs. counts for channel 4 ...16 2.6 Water dep th y-S. time from Channel 4. .............. 18 2.7 The power spectrum of the raw time series i n Sjiig. 2.6. 19 2,8 2.8 The predicted tide at Port Renfrew for the period -.-of seiche measurement: 0600 - 1915 27 February-1974 23 2.9 The time series from Fig. 2.6, corr§cted_for the tide 24 2.10 The t i d a l l y c o r r e c t e d ^ ^ e % ^ e ^ c ^ f i t subtracted to remove the residual 26 2.11 The t i d a l l y corrected series with a spline f i t Subtracted to remove the residual 27 Page Fig. 2.12 The spline - corrected series Cfrom Fig. 2.11) with a polynomial f i t also subtracted. 28 2.13 The power spectrum for the polynomial —corrected time series in Fig. 2.10, smoothed oyer 4 points... 30 2.14 The powersspectrum for the spline-corrected time series in Fig. 2.12, smoothed over 4 points. ...... 31 2.15 The power spectrum for the spline-polynomial corrected time series in Fig. 2.13,ssmoothed over 4 points 22 2.16 The power spectrum for the polynomial-corrected time series from channel 2, logarithmically band averaged 34 3.1 The coordinate axes and the assumed geometry of Port San Juan for theory: L=6400m, w= 1000m,,$k>. = 16.97m, 0=4.95596 x l O ^ n A ' v 42 3.2 A comparison of the theoretical and actual bottom profiles for Port San Juan. The solid line is the fitted theoretical curve; the open circles are soundings taken from the chart. 43 3.3 A comparison of theoretical and actual bottom profiles for the shelf off Port San Juan. The solid line soliatelineaisdtheoas >.: are soundings taken from the chart. 50 3.4 The coordinate axes for the outside wave calculations 49 V l l l Page Fig. 3.5 The transport thro ugh the mouth, of the bay as a function of the frequency of the forcing wave 63 3.6 The surface elevation Csolid line) and phase r e l a t i v e to the forcing current (broken line) at the position of the wharf . 64 3.7 The absolute value of the surface elevation along the bay for the fundamental mode. (The mouth i s . a t x = - 6400m. and the head i s at x = O.Ki 66 3.8 The absolute value of the surface elevation along the bay for the second mode. (The mouth i s at x = - 6400m: the head i s at x = 0.) ....... 6/ ACKNOWLEDGEMENTS I would l i k e to thank my supervisor Dr. P. H. LeBlond, for his encouragement and assistance i n the w r i t i n g of t h i s thesis. I would also l i k e to thank'Dr.T.R. Osborn for h i s assistance, e s p e c i a l l y with the experimental part of t h i s work. Special thanks are due to Mr. J. L. Galloway f o r h i s assistance i n the design and construction of the wave gauge. I am g r a t e f u l f o r t, the cooperation from a l l the members of the I n s t i t u t e of Oceanography at U.B.C. F i n a l l y , I wish to thank the National Research. Council of Canada f o r supporting met"persM.^(liy.xduiiA;g",.t570ayears of1'this research. CHAPTER 1 INTRODUCTION The purpose of this work was to compare the results of analytic modelling of a seiche i n a coastal bay to measurements of the seiche. In order to do t h i s , a bay., amenable to analytic modelling was necessary. Port San Juan i s such a bay. I t i s situated on the west coast of Vancouver Island, near the entrance to the S t r a i t of Juan de Fuca (see Fig. 1.1). The bay i s extremely regular i n plan, being rectangular with steep side walls and a gently shelving beach at the head (see Fig. 1.2). The bottom slopes regularly and smoothly from the head of the bay* to the entrance.-V The bay i s approximately 6.5 km. long, 2 km. wide and about 16 metres deep at the entrance. A tide gauge operated by the Department of the ^nyi£pnmenti(Water Management) i s situated on the government wharf at the small v i l l a g e of Port Rerffcew,, about 2 tffe&S- from the head of the bay on the southeast shore. Two small r i v e r s , the San Juan River and the,,|^rHon;River entei- at the head of the bay. The entrance of the bay looks out in t o the open P a c i f i c past Cape Flattery and i s exposed to surf from the open ocean. Port San Juan was very w e l l suited to analytic modelling because of i t s extremely regular shape. The other requirement of this study was that seiches should e x i s t there. The government tide gauge"did" show evidence of seiche^ action. F i g . 1.2 P o r t San Juan. Measurements were taken a t the government wharf i n Snuggery Cove. Co CHAPTER 2-DATA 2.1) P r i o r Evidence of Seiches Evidence of seiche action i n Port San Juan bay exists on the records of the government tide gauge. (S.O. Wigen, Personal communication) Anr^xample of these motions as detected by the government gauge i s shown i n Fig. 2.1. The gauge i n use at Port Renfrew i s a very simple instrument, consisting of a clock-work-driven chart recorder which i s marked by a pen connected to a f l o a t and a counterweight. The f l o a t i s suspended inside a s t i l l i n g w e l l consistoingeof a long piece of pipe, about 12 inches i n diameter wfhich i s attached v e r t i c a l l y to the wharf p i l i n g s . The s t i l l i n g w e l l does not perform i t s f i l t e r i n g function very w e l l , with the result that the record i s contaminated by high frequency noise produced by surface waves. Other examples of the occurence of the seiche are shown i n Fig. 2.2. Visual examination of the record reveals two d i s t i n c t periods for the seiche: one of the order of 35 minutes, and one of the order of 15 minutes. Investigation of a l l these records for the year 1973 revealed no correlation between Ithe occurrence of seiches and the phase of t i d a l motions. The seiche may occur on any phase of the t i d e ; many t i d a l cycles may go by without any appearance of the seiche at a l l . Neither i s there any obvious correlation with storm a c t i v i t y Fig. 2.1 Seiche action on the government tide gauge, 29 January 1971. Fig. 2.2 Seiche action on government tide gauge, 30 January 1971. 7 (as revealed by the width of the noise on the record), although there tends to be more seiche a c t i v i t y i n the late f a l l , winter and early spring than i n the summer months. The seiche amplitude \\;',/ varies markedly as w e l l ; usually i t i s of the order of 5cm. The largest observed amplitude was 15 cm. The amplitude also shows no obvious correlation with tide or storm a c t i v i t y . 2.2) Method of Measurement. The government tide gauge records were not suitable for making precise determinations of the period and amplitude of the seiche. Therefore a more precise measurement using another instrument designed for the purpose was deemed necessary. A bottom mounted pressure gauge, based on a design by J. L. Galloway (Galloway, 1974) using a vibrotron transducer was b u i l t to measure the seiche. The instrument was o r i g i n a l l y designed as a shallow-water tide gauge; some modifications were necessary to enable i t to measure seiches, but the basic form of the instrument . remainedd the same. Details of the design and operation of the instrument may big; found i n Appendix C. The record from the instrument was a time-varying analogue voltage. In order to achieve the necessary precision of measurement over the large t i d a l range, a scale expansion technique was used. When the change i n water l e v e l exceeded the f u l l scale range-of "the t r c - i i . -./^z-iz : t i . '.iiqu instrument output, the signal would wrap around and return to zero, e f f e c t i v e l y keeping only the least s i g n i f i c a n t pcXrilion of the sig n a l . Thus a r i s i n g and then f a l l i n g tide would produce signals l i k e that i n Fig.lp. These wraparounds could then be removed after the record was d i g i t i z e d . The instrument had four output channels; two low-frequency channels incorporating a f i l t e r to remove the surface waves, and two channels with a high sampling frequency which recorded surface waves. Both groups incorporated a high and low resolution channel. The low-resolution channels were designed to operate without wraparounds. They were included i n case the wraparounds could not be removed l a t e r ; since wraparounds >p0^W^0^tssui^,^: the low resolution channels were not used i n the analysis. '$2g(3)Data Collection'^ I collected the f i e l d data f o r the study at Port Renfrew on 26, 27, and 28 February 1974, i n s t a l l i n g the instrument on the government wharf near the tide gauge, (see Fig. 1.2), where 110 v o l t AC l i n e power was available. The t r a i l e r containing the electronics was l e f t parked on the wharf, and the sensor head was lowered on a heavily weighted frame down to the bottom. The water depth over the instrument at high tide was 5.95 metres. The output from the instrument was recorded on a 2-track Brush recorder as w e l l as the magnetic tape, to g i v e f a - v i s u a l 9 Fig. 2.3 The wraparound technique for scale expansion check on the instrument's operation. After the pressure head had been lowered, inspection of the government tide gauge showed that no seiches were happening, so I did not immediately turn on the recorder. The recorder was turned on at 0600 27 February when seiche motions appeared on the government tide gauge. Recording continued , interrupted only by tape changes every 6 hours, u n t i l 1915 27 February when a general power f a i l u r e occured throughout Port Renfrew. A severe storm going on at the time had blown a tree across the power l i n e s . Power was restored at 2315 and recording continued. At some time between 0445 February 2:8 and 1030 February 28 the tape recorder f a i l e d and I was unable to repair i t with the ^ equipment at hand. Since I had 13 hours of good data with the seiche occurring, I returned to Vancouver. Data c o l l e c t i o n was f i n i s h e d . 2.4) Time Series Processing: The time series obtained from the above measurements- could not be d i r e c t l y used for the computation of a power spectrum of the series. As i t was stoned i n analogue form on several tapes i t was f i r s t necessary to (digitize i t , and then to mate the i n d i v i d u a l records into one long record. I t was then necessary to remove the wraparounds, calibrate the d i g i t i z e d record and remove the t i d a l trend. The details of these operations are 11 described below. a) D i g i t i z a t i o n and record patching: The records were d i g i t i z e d using the A-D system on the Ins t i t u t e of Oceanography PDP-12 computer. The system accepts analogue signals between +2 and -2 volts and produces a d i g i t a l output signal between -511 and +512 which_ i s written out on'9.T track computer tape. The sampling rate may be adjusted to any value up to 16000 s e c " 1 W&&%?X£SP$ :t^d'HS^^li%sSprf.©& separately and dig-Ja.^^. into >£s;ep;a-Datiea; dataf f i l e s . Only the two high-resolution channels were used i n the actual analysis. The playback was done at 16 times the recording speed<jland the d i g i t i z a t i o n rate was adjusted accordingly. On Channel 2 the sampling rate of 2 Hz., when replayed at the higher rate, produced a signal varying i n r e a l time at 32 Hz (=2 x 16 Hz.). Since the d i g i t i z a t i o n system works only at certain fixed frequencies, the d i g i t i z a t i o n was carried out at the closest available frequency, which was 50 Hz. S i m i l a r l y , Channel 4, which sampled at 1/21 Hz., was d i g i t i z e d at 1.5 Hz. Each channel had a + and - f u l l scale signal recorded at the beginning for c a l i b r a t i o n purpose. After adjustment of the tape recorder output amplifiers, these signals werei-digitized to give the scale for each channel which was as below i n table I ', , Table I D i g i t i z e r scale constants. Channel + FS -FS 4 429 -431 2 425 -427 These maxima were set somewhat below the f u l l range of the d i g i t i z e r i n order to avoid saturating the system. The next problem was to f i t the i n d i v i d u a l tapes together, f i l l the gaps between them, and then remove the wraparounds. Since the time between tapes was known I f i l l e d these intertape gaps by l i n e a r l y interpolating the correct number of values (n= At/dt; dt= d i g i t i z a t i o n period; At= gap time) between the l a s t reading from one tape and the f i r s t reading on the next. Then the wraparounds were removed numerically. A c r i t e r i o n of 70% of f u l l scale was taken to represent a wraparound, and the time series were then run through a computer program which automatically added or subtracted f u l l scale to a l l values following a wraparound. The result of these treatments was a r e l a t i v e l y clean time series with only minor data gaps (^minutes out of 755, or gaps of 1?.2% of the records.) Channel 4 had a t o t a l of .4225 points, 10.72 seconds apart, making a t o t a l record length of 45,292 seconds or 12.581 hours. Channel 2 produced 131,689 points, 0.32 seconds apart, baking a t o t a l record length of 42,141 seconds. Both records had to be truncated somewhat i n order to f a c i l i t a t e the use of the Fast Fourier Transform program. Channel 2 was cut back to 4096 points for a t o t a l record length of 43,909 sec. (12.20 hrs.) and channel 4 was reduced to 131,072 points M r a t o t a l record length of 41,943 sec. (11.65 hrs.) b) Calibration The vibrotron response i s not l i n e a r with frequency as equation C l i n Appendix C shows. The departure from l i n e a r i t y i s small, but nevertheless a ca l i b r a t i o n of the results seemed to be i n order. Instead of i n d i v i d u a l l y c a l i b r a t i n g each part of the instrument, I decided to treat the entire system (including the d i g i t i z e r ) as one black box, and calibrate the output trace i t s e l f . I decided to calibrate the instrument against the government tide gauge. The non-linearities i n the vibrotron responses are s u f f i c i e n t l y 'sTnaiLtli that they w i l l only show up over large depth ranges. Within smaller depth excursions, the response of the instrument should be l i n e a r . To remove the effect of surface waves, half-hour averages of the channel 4 output were plotted against the appropriate values from the government tide gauge trace. Fig. 2.4 shows a plot of the actuall water depth vs. the d i g i t i z e r counts for channel 4 over the period of measurement. The v e r t i c a l error bars arise from errors i n estimating the position of the government gauge trace and are equal to 1/2 the width of the trace. Fig 2.5 shows the r e l a t i o n between counts for channel 2 and channel 4. The curve i n Fig 2.4 was f i t t e d with a cubic, resulting i n the following c a l i b r a t i o n equation for channel 4 Z=18.78 + 1.97 x 10 - 2C + 2.9 x 10~ 7C 2 + 9.18 x IO" 1 1 C 3 where C i s the count and Z i s the water depth i n feet. The count for channel 2 i s related to the count for channel 4 by C2 = 0.403C4 +35.87 where C2 = channel 2 count C4 = channel 4 count These results show that the instrument s e n s i t i v i t y changes s l i g h t l y with the t o t a l depth of the water. Table i>I 2, l i s t s the s e n s i t i v i t y as a function of the t o t a l water depth: Table I I Se n s i t i v i t y as a function of depth. Depth (feet) S e n s i t i v i t y (mm./ count) 18.75 0.598 17.31 0.510 16.01 0.525 Fig . 2.4 The calibration curve for channel k of water depth vs. d igi t izer count. 5 0 0 Fig. 2.5 The calibration for channel 2. The plot shows counts for channel 2 vs. counts for channel k. Note that these counts are counts from the d i g i t i z e r output, not the i n t e r n a l instrument .counts mentioned i n Appendix C. Since these are smaller, the net effect i s that the d i g i t i z e r count always changes by at least three. c) De trending '  Fig. 2.6 shows the time series resulting from channel 4. The high-frequency o s c i l l a t i o n s are p a r t i a l l y f i l t e r e d surface waves (see Appendix C for the f i l t e r i n g process!)'. The seiches i s v i s i b l e as the longer period o s c i l l a t i o n s superimposed on the main t i d a l curve. The t i d a l curve presents a major problem i n analyzing the data. The purpose of the measurement was to obtain a more precise measurement of the seiche period than that possible using the government gauge© However a spectrum of the curve shown i n Fig. 2.6 would not y i e l d any information about the seiche beacuse of the t i d a l trend. The record i s f a r too short to resolve the t i d a l peaks properly, with the result that the low frequency end of the spectrum i s ^ bs^urgd| including the frequency band containing the seiche frequencies. This may be seen i n Fig. 2.7 which shows a power spectrum computed from the trace shown i n Fig. 2.6. Note that a l l d e t a i l i s obscured i n the low frequency region of the spectrum, i . e . _ 3 t T at frequencies less than 5.6x10 n z* 100.0. 120.0. 160.0 100.0 200.0 220.0 2«0.0 300.0 320.0 340.0 350.0 , 380.0 TIME (X10* ) IS1 g. 2.6 Water depth vs. time from channel k. 19 . Some method must therefore be found to remove the effects of the trend from the very low frequencies. This effect arises because the record i s not long enough to resolve the t i d a l Fourier c o e f f i c i e n t s , and then retransform the spectrum were not suitable, since this would have destroyed the very information I was attempting to -extract from the record. Any f i l t e r i n g had to be performed on the time series i n such a manner as to preserve mo-tions near the seiche period. There i s a class of d i g i t a l f i l t e r s described i n Godin 0-9.72) which work on the time series i t s e l f by taking successive d i f f e r e n -ces between values, ( S^(cr) , i n Godin's notation) .These however usually cut off everything below ha l f the Nyquist frequency, which i n this case meant that only the surface wind-waves would survive the f i l t e r i n g process. Therefore, some other method of removing the t i d a l trend had to be found. Since the trend i s t i d a l i n o r i g i n , the best method of remov-ing i t appeared to be" to/ "take .thepredicted Jtide> for Port;. Renfrew and subtract i t from the time series data. I f the amplitudes and phases of the various constituents of the tide at a port are known, then i t i s possible to predict the tide there from the formula motions. D i g i t a l f i l t e r s of the type which eliminate fedesired h = H + o (2.3) a l l constituents 21 (Schureman, 1958), where h= the tide height at time t , RQ= the mean water height abovedatum } H= the amplitude of a t i d a l constituent, f= a factor reducing the mean amplitude H to the year of prediction, a= the speed of the constituent of amlitude H ? t - the time measured from s t a r t i n g point of predictions, (VQ+U)= The value of equilibrium argument of the constituent of ampitude H when t=0, at Greenwich, k= the phase of the constituent with respect to Greenwich, ( i n Canadian System). The values of f, a and ( V , + u) are tabuliated (Schureman, 1958) o and may be found for any s t a r t i n g time. The amplitudes and phases for Port Renfrew are shown i n Table £M-, The seven largest constituents (and the only ones used are) M2,K1, 01,S2,N2,P1, and K2. The tide at Port Renfrew for the period of measurement was then calculated from (2.3) for the period of observations, producing a.the curve shown i n Fig. 2.8. This was subtracted from the o r i g i n a l time series to produce a t i d a l l y corrected curve, shown i n Fig. 2*9. A rather large residual was l e f t . There was s u f f i c i e n t energy contained i n this residual that i't' was necessary to remove i t as w e l l as the tide , before the peaks due to the seiche could be made clearly v i s i b l e . Therefore I decided to try to f i t a curve to the over a l l trend i n the data. I t r i e d two different methods, i n order to have some comparison between them. This might therefore help to keep a check on a r t i c i a l effects introduced by the f i t t i n g . Table III Tidal constituents at Port Renfrew. Name Amplitude (m) Phase (°) M2 2.351 9.3 Kl 1.492 134.4 01 0.918 122.9 S2 0.656 29.1 N2 0.488 354.0 PI 0.488 133.5 K2 0.174 6.3 ] « . 0 360,0 , WOO TIME (XiO a I IS) F i g . 2.8 The predicted t ide at Port Renfrew for the period of seiche measurement: 0600 - 1915 27 February 197A. Fig. 2,9 The time series from Fig. 2.6, corrected for the tide. The f i r s t detrending method chosen was a poly n o m i a l / f i t using the U.B.C. Computing Centre Program '0LQF ( B i r d , 1974).. The f i t t i n g had to be done to the large scale trend, without obscttrlngi I the seiche motions. There were 4096 data points i n the time s e r i e s . These points were averaged i n groups of 100, to produce 40 points representing the long-term trend i n the time s e r i e s . The polynomial f i t t i n g program was then run on this time s e r i e s . The program employs'' a l e a s t squares f i t t i n g technique to f i n d the best f i t polynomial of degree less than or equal to an exter n a l l y supplied maximum! degree. Since I d i d not want to destroy the seiche information, a polynomial of r e l a t i v e l y low degree was necessary. I ran the program with the maximum degree set at 5. The program chose a cubic as the best f i t . This polynomial was then used to generate a new time s e r i e s , one point f o r each i n the o r i g i n a l -s e r i e s . This new serie s was subtracted from the o l d and the r e s u l t was the corrected time s e r i e s showniin F i g . 2.10. The other method used was a s p l i n e f i t t i n g techniq,uetwitht;errcir (Bird, 1974 ). This program uses a s p l i n e f i t t i n g technique to f i t a curve to a series of-points with errors. Therefore I l e t the surface wave and seiche amplitude be errors i n the points (allowing an e r r o r of 7.0 cm i n each point) and then ran the program to f i t a correction to the trend i n the s e r i e s . The r e s u l t i n g time s e r i e s i s shown i n F i g . 2.11. As a further check, I "then ran the polynomial f i t t i n g program on the s p l i n e - c o r r e c t e d time s e r i e s . This time s e r i e s i s showniin F i g . 2.12. N3 ON Fi"g. 2 . 1 1 Trie t i d a l l y corrected ser ies with a spl ine f i t subtracted to remove the res idua l . N3 oo 29 5)Spectral Calculations Spectra were calculate' (for a l l these time series using the standard Fast Fourier Transform algorithm is the U.B.C Computing Centre program FOURT (Froese, 1970). Fig. 2.13 show the transform computed for the cubic-corrected 2 time series shown in 2.10. Log (f^ ) is plotted vs. log f. Each point on the plot is composed of a 4 point average of the Fourier coefficients, giving a somewhat smoothed plot. Figs. 2.14 and 2.15 show the spectra for the spline and spline-polynomial corrected series in Figs. 2i.ll j^anflji 2.12 respectively. It i s necessary to perform smoothing operations on the spectrum . in order to obtain consistent spectral estimates. For a Ga*uss^a% var/lable, a power spectrum computed by direct Fourier tansform methods results in a normalized standard error in each estimate of E =1. (BenHafe & r Piersol, 1971). There are two possible methods to reduce the error i n each spectral estimate. One may either perform an ensemble average on s p e c t r a resulting from several time series of the same phenomenon (i.e. break a longer record into short segments and perform a spect:ial analysis on each, short record), or one may average the spectral estimates in a given frequency band together to produce one estimate for that band. The two processes are entirely equivalent (Bendat & F i e r s o l 1971).v The error E referred to above i s a random error only; the r averaging processes above wil reduce i t by the formula F i g . 2.13 The power spectrum for the polynomial - corrected time ser ies in FTg. 2.10, smoothed over k po ints . 31 i n r—' in -5.0 -4 .5 -4.0 -3.5 -3.0 -2.5 -2.0 I - ] .5 -1.0 LOG f F i g . 2.14 The power spectrum for the spl ine-corrected time ser ies in F i g . 2.12, smoothed over k po ints . r <1> Cf) "§> = computed value <J> = -real value (2.4) where n i s the number of degrees of freedom. Since each Fourier coeffi c i e n t consists of an amplitude and a phase, averaging over d-values produces a result with 2d degrees of freedom, so that However, the average of the spectral estimates overaan i n t e r v a l (f'.; - Af/2, f 0 + A f/2 ) when assigned to the frequency f 0 may not truly represent the spectral value at f 0 . I t may be greater or smaller, depending upon the shape of the spectrum. Thus , while averaging w i l l reduce the effect of random errors i n the i n d i v i d u a l spectral estimates, i t does introduce a bias error. Nevertheless, averaging i s necessary i f the spectral estimates are to carry any significance at a l l . The spectra i n Figs. 2.13 to 2.15 have been averaged as described above, and the standard errors plotted were computed from 2.4. These operations of t i d a l correction and cubic f i t t i n g were also performed on the record from channel 2; theihi'gh^resolution, high-frequency record. A spectrum (Fig. 2.16) was calculated for this record as w e l l , i n order to compare i t with the channel 4 results. In this way, any aliasing effects from the p a r t i a l f i l t e r i n g performed on channel could be discovered. n = 2d LOG f F i g . 2 . 1 6 The power spectrum for the polynomial - corrected time ser ies from channel 2 , logar i thmica l ly band averaged. 35 6) D i scuss ion of the Spectra: The spectrum shown i n F i g . 2.14 i s the f o u r - p o i n t averaged spectrum of the s p l i n e - c o r r e c t e d time s e r i e s i n F i g . 2 .12 . Note the four prominent peaks i n the low frequency p o r t i o n of the spectrum. These have frequencies corresponding to per iods of 55+ 3 m i n . , 3 8 . 5 + 2 min. ,21+ 1 min. and 14.6 + 6 min. The peaks at 38.5 and 14.6 minutes c o r r e l a t e f a i r l y w e l l w i th the per iods o f motion observed i n the government t ide gauge t races . The other peaks do not . They do n o t , i n f a c t , c o r r e l a t e wi th anything observed on the government gauge. There are two p o s s i b l e explanat ions f o r t h i s . The f i r s t i s that these do not correspond to r e a l motions but are an a r t i f a c t of some s o r t , in troduced by the detrending procedures . I f t h i s were the case, then a spectrum produced by a d i f f e r e n t detrending procedure should a l t e r these anomalous peaks. With th i s i n mind, examine F i g . 2.13 which i s a spectrum of the same data, using a cubic po lynomial f i t i n s t e a d of a s p l i n e f i t . The same b a s i c s t r u c t u r e i s s t i l l pre sen t , although not as c l e a r l y . The peak at i 15 min. p e r i o d has s p l i t i n t o two, although the two peaks are not r e a l l y re so lved . The peak at "38.5 min. appears now as no more than a shoulder on the 55 min. peak which has become much broader . F i g . 2.15 shows the spectrum c a l c u l a t e d from the s p l i n e correc ted time s e r i e s a f t e r i t had again been f i t t e d by the polynomial method. There i s no s i g n i f i c a n t difference between this.spectrum and ^ the spectrum of the s p l i n e - f i t t e d ; C R E C G R D - ' cjhi'?- i s ''f','. because the program choose a l i n e a r f i t as best, and changed the time series only by a very small amount. The spectrum of channel 2 shown i n Fig. 2.16 was computed i n order to check on possible aj^asihgr^done on Channel 4 of the instrument. Although calculations showed that there should be no s i g n i f i c a n t effects from this process on the low-frequency end of the spectrum, comparison wwith the u n f i l t e r e d record serves as a valuable check. The sampling rate was s u f f i c i e n t l y high to resolve completely the surface wave spectrum which may be" seen centred about a frequency of 0.3 hz. The structure of the low-frequency region shows the same peaks found i n the f i l t e r e d spectra. The frequencies and r e l a t i v e magnitudes are the same, which leads to the conclusion that the f i l t e r i n s t a l l e d i n the low sampling frequency c i r c u i t had adequately suppressed the a l i a s i n g effects. Examination of these spectra resulting from the various de.trending methods leads to the conclusion that while some portions of the anomalous low-frequency energy may be a r t i f a c t s introduced by the detrending most of i t must be r e a l . I t i s possible only to speculate on possible sources of this energy. Some of i t may just be sp'uriojj.s^ values remaining because of imperfect detrending. The rest may be a response to long waves of various sorts which impinge upon the entrance of the bay. A wide spectrum of frequencies i s possible for edge waves along this coast (see chapter 3, section 2) and the bay w i l l respond to some extent to a l l of them, although not i n a resonant mode. The data set I obtained did not contain a very strong example of the seiche and the p o s s i b i l i t y e x i s t s , therefore, that the response seen at other frequencies^' was simply the off-resonance response of the system to forcing at those frequencies. Taken i n conjunction with the government tide gauge records, the results presented here support values of 38.5+2 min. and 14.6+.6 min. for the f i r s t and second longitudinal seiche modes, although the error involved i s rather large. The quoted errors are equal to one half the bandwidth over which the spectral values were averaged at those frequencies. •The difficulty^^identi^fy-in!g the seiche motions clearly i n the spectrum tfe'rms "from 3 . t n e nature of the seiche i n Port Renfrew i t s e l f . As the government tide gauge record showed, i t i s of small amplitude, short duration and random occurrence. The signal usually remains coherent over only a few periods, and thus i s always superimposed on a large t i d a l trend. The effect cannot be avoided by taking longer records to resolve the t i d e , since the seiche ( i f i t occurs at a l l ) w i l l not occur throughout the f u l l length of the record, but only for a r e l a t i v e l y short time. Because of the error effects explained above, an o s c i l l a t i o n l a s t i n g only a few periods w i l l unaviodably have a large error i n the Fourier c o e f f i c i e n t s ; thus an incoherent signal l i k e the seiche cannot be defined precisely. The effects of the large t i d a l trend only aggravate the problem, as they can never be removed perfectly. I t does not appear, therefore, that a very much more precise measurement of the seiche i s possible, although i f one were to be lucky enough to encounter a stronger, more long-lasting manifestation than the one I found, some improve-ment could be possible. For example, a 15 cm o s c i l l a t i o n , l a s t i n g 24 hours does occasionally occur C s e e Figs. 2.1 and 2.2)..* This might be expected to produce power spectral coefficients 10 times larger than those found here, with only h a l f the error i n frequency. 7) Damping Effects Description of the seiche i s not complete without an estimate of the damping of the motion as w e l l as i t s period. In an open bay such as this the damping w i l l arise from two sources; the f i r s t due to radiation of long wave energy ::6rom the mouth of the bay and the second from f r i c t i o n a l damping i n the shallow water of the bay i t s e l f . These two dissipative mechanisms combine to l i m i t the amplitude of the seiche and to damp i t out i f the force driving i t should cease to operate. I t i s not possible to separate these two effects observationally, but t h e i r combined ^effect may be estimated by finding the Q of the bay. Q may be estimated i n two different ways from the data at hand. The f i r s t estimate i s f rom the h a l f power bandwidthxAf of the resonance peaks i n the spectrum of the o s c i l l a t i o n s . In this Case Q = f Q /Af where f i s the resonant frequency. Because of the unavoidable uncertainty i n the spectra, i t i s only possible to obtain an order of magnitude estimate, which gives the following results. Table IV Q as a function of mode number. Mode # Q 1 8 2 7 These should be regarded as no more than order of magnitude r e s u l t s , meaning that Q for both modes i s on the order of 10 The other method to estimate Q i s to use the decay time of the seiche once i t has been excited. Q i s related to the logarithmic: decrement X by Inspection of the government tide gauge records i n Fig. 2.1 and 2.2 reveals that the seiche decays i n 4 or 5 periods, so that QiV 12 - 15 by this method, which agrees with the estimate derived from the bandwidth c a l c u l a t i o n s . CHAPTER 3 THEORY 3.1) Solution of the Long-Wave Equations f o r a Rectangular Bay with an Exponential Bottom. Port San Juan i s very close to rectangular i n surface plan, and the bottom p r o f i l e approximates an exponential very w e l l . I s h a l l the-refore take as a model a rectangular bay of width 2w and length L , as sketched i n F i g . 3.1. I t s h a l l have steep l a t e r a l w a l l s , a sloping beach at the head ( x = 0), and be open at the mouth (x = - L) . The bottom depth w i l l be allowed to vary with x as h = h Q (1 - e C X ) , C 3 . U where the constants h and c are taken to match: re a l i t y - as closely-o as p o s s i b l e . The depth does not change with y, across- the bay-. F i g . 3 shows a comparison between the exponential bottom p r o f i l e (.3.1) and soundings from the chart of Port Renfrew. The agreement i s quite good _A _ l with w = 1000 m, L = 6400 m, h = 16.97 m , c = 4.956 x 10 m ' o The analysis i s performed under the following assumptions: 1) No r o t a t i o n : the observed frequencies are s u f f i c i e n t l y high. (1^ 2 cycles/hour) that the C o r i o l i s force i s n e g l i g i b l e . 2) Shallow water baro'tropic conditions, applicable to the propaga-: tion of long waves, are used throughout. 3) L i n e a r i z e d equations are used, adequate for small amplitude s e i -ches . X=-L X=0 The c o o r d i n a t e axes and the assumed geometry o f P o r t San Juan f o r t h e o r y : L=64O0m, w=1000m, hi =16.97m, C=4.95596 x \o~% = 1 . BAY DEPTH PROFILE 20\ 2000 x(m) 4000 6000 F i g . 3.2 soundings taken from the chart. . open c i r c l es are Co 4) The depth depends on x only, as per (3.1). S t a r t i n g with the usual long-wave equations, with n the surface displacement from mean water l e v e l j u the current i n the x - d i r e c t i o n , and v the current i n the y - d i r e c t i o n : ^ + g - ^ = 0 9t g 9x u ' — + g 3^ = 0 , (3.2) 3t 6 3y 3 x + m 3y 3t • The v e l o c i t y components u and v are eliminated from the above to y i e l d an equation for n, alone: 0 +0) = - • , » . 3 , Let us look for harmonic solutions of the form ri(x,y;t) = F(x,y) e ± V t . ('. ' { > Substitution i n t o (3.3) gives h 2 V 2 F + 7-— F + \ F = 0 . (3.4) h x gh Note the presence of the F term, due to the i n c l u s i o n of bottom to-x pography. With the s p e c i f i c bottompprofile (3.1), the c o e f f i c i e n t & o f F x has the following form: h cx x c e h 1 - e Let's write F(x,y) = X'(x)?Y(y); then, (3.4) takes the following form, where primes indicate differentiation with respect to the relevant arguments: X Y .. cx X , /-, ex. 1 - e gh 0(l-e ) This equation must be solved in the bay subject to the relevant boundary conditions. At the sides, y = - w there are steep walls,through which no flow may pass; therefore, v = 0 , i.e. F =0 at y = t. w. (B.C.I) y At the head of the baye1! ( x = 0) there i s a sloping beach, so that the depth h -»• 0 at x = 0. (This produces a regular singular point at x = 0 in the equation for r|.)The wave.;.^runsri < up some distance on the beach: the appropriate boundary condition at x = 0 is then merely that both r) and u be f i n i t e : F , F x f i n i t e at x = 0 . (B.C.II) At the mouth, x = -L, both current and pressure must be continuous, which means that both r|(-L) and u(-L) must match the outside solution. This w i l l be dealt with later, after the solution for the exciting wave has been found. Equation (3.5) is separable; the solution for Y subject to (B.C.I) may easily be found to be Y(y) = A c o s^-H (y-w)J , (3.6) with n = 0, - 1, - 2 Substituting for (3.6) into (3.5), the X dependence is now given by X" - c e cx 1-e cx X*- + 2 2 n r g h o ( l - e C X ) 4,/ X = 0 (3.7) Let us introduce the new variables K = nTT/2wc a 2 = v 2 / g h 0 c 2 ; s = e c x X'(x) = G(s) Equation (3.7) for X(x) then becomes for G(s) s 2 ( l - s ) G " + s ( l - 2 s ) G ' + [ a 2 - k l ( l - s ) ] G = 0 . (3.8) This equation has regular s i n g u l a r points at s = 0 ( x = - ° ° ) , and s = 1 ( x = 0, the head of the bay); i t i s an equation of the general form (1 -s)G" + s ^ + b 1s)G' + ( a 2 +bb 2s)G 0. with a 1 = 1 , 2 2 a 2 = a - K b 1 = -2 , b 2 = ^ Now l e t G = s u , where k 2 + k( a n - 1) + a 0 = 0, or '1 "2 k 2 = K 2 - a 2. Replacing G by u, (3.8) becomes s ( l - s ) u " + (a- bs)u' + cu = 0, with a = b = 1 7.2 2k + a, = 1 + 2 ( K 2 -02fJv c = 2k - b 1 = 2[ it IK2 - a*) b 2 + (1+b^k - k 2 1 /2 = o2 I ( K * - az) 1/2 (3.9) With these s u b s t i t u t i o n s , (3.9) takes the form s ( l - s ) u M + [ l-'-+2 / K 2 -a 2 - 2( 1 t /K 2 -a 2)s]u' + (a 2 + / K 2 -a 2 ) u = 0. (3.10) This i s a form of Gauss' Hypergeometrie Equation (Erdelyi et a l . , 195 s ( l - s ) u " + [c -(l+a+b)s ]u' - abu = 0, with a = \ [1 + 2/ K Z -oz 1 / 1 +4KZ ] , b = -| [1 + 2/ K 2 - 0 2 ~ / 1 +4K2 ] , c ' = 1 + 2/ K 2 -a 2 , K -a . I s h a l l ? also choose the p o s i t i v e root for / 1 + 4K2, since the other sign merely interchanges a and b and the equation i s symmetric i n those qua n t i t i e s . Equation (3.10) i s that s p e c i a l form of Gauss' Hypergeometric Equation i n which a,b,c, s a t i s f y the r e l a t i o n c - a - b = - q ; q = 0,1,2, In t h i s case, q = 0. For th i s s p e c i a l form, the general s o l u t i o n consists of a sum of two l i n e a r l y independent solutions expanded about the regular singular point at x = 0 (Murphyj- 1960) ; these two solutions are y l = |F 1(a,b;l;z) y 2 = y 1 InCz) • + Y(a,b ;l;z) , where 2 ^ has the series expansion - F l ( a i b ; c ; , ) = -JSBL Y r c ^ ) r o ^ , i ) ^ 3 . 1 r ( a ) r ( b ) r(c+j) J : j=o and the exact form for Y need not be written, as the second solu-tion w i l l be rejected. The condition (B.C.II) requires that the solution be f i n i t e at the head of the bay. Thebh'ead of the bay (x=0) corresponds to z = 1-e^ = 0. The second solution, y^ » becomes negatively i n f i -n i t e there and must therefore be discarded. The general solution for the x-dependence (In the o r i g i n a l va-riables) i s therefore 0 0 j X(x) = A e x P [ c / ^ 7 x ] V R ( A + ' 1 ) R ( B + - L ) ( 1 " E C X > • ( 3 - U > « C J r ( a ) r c b ) j'. J : The constants /K 2 - 0 " 2 , a ,,and b are i n general complex. X(x) w i l l be complex and o s c i l l a t o r y i n character i f K 2 < a 2 ; this i s certain-ly the case when there i s no cross-channel v a r i a t i o n (n=0). , The complete solution for the amplitude inside the bay i s then \cosInfT(y-w) /2w]cosVt l)(x,y;t) = R e a l J A ^ x p I c ^ - a 2 ' x] 2F 1(a,b ; l ; l - e C X ) ; (3.12) for the cross-channel mode. This solution must be matched to the outside solution, which drives that inside the bay. 3.2) The Outside Edge Wave. To calculate the form of an edge wave travelling along the coast outside Port San Juan, I shall follow the treatment given by Ball (1967). The shelf profile off Port San Juan i s a good f i t to an expo-nential of the form h = h (.1 - e b x) , (3.13) with h = 112.22 m , o b = -3.545 x 10~4 rn"1. Fig. 3.3 shows a comparison between the f i t t e d function and sound-ings taken from the hydrographic chart. Let now x = 0 along a straight coastline, stretching to y = _ as in Fig. 3.4. + Fig. 3.4 The coordinate axes for the outside wave calculations. SHELF DEPTH PROFILE 3.3 o Starting from the equation for the surface elevation n (3.3), we look for t r a v e l l i n g wave solutions along the coast, of the form ri o(xvy;t) = F(x> expfi(vt -my)]. Under the same assumptions used for the analysis inside the bay, and now using the new variables bx 2 2 / i * 2 s = e , d z = v V g h Q b z , k = m/c , F(x) = G(s) (v Hhese transformations lead to the equation s 2 ( l - s ) G " + s ( l - 2s)G' + [ a 2 - k 2 ( l - s ) ] G = 0 . (.3.14) for the surface elevation, affoirm i d e n t i c a l to that found i n the previous analysis (.3.8). Looking for a series solution about x = -00, subject to the bound-ary conditions that F and F' must both Be f i n i t e at x= 0, and F"K) as x •+ - o o , we f i r s t define p 2 = k 2 - O 2 . (3.15) The requirement that F tend to zero as x tends to minus i n f i n i t y demands that the series for F terminate after a f i n i t e numb.er of terms, which means that k 2 = (p+n) (p+n+1) , C3.16) where n i s an integer. This gives for G(s). the solution G(s) = A sP 2 F 1 ( -n,n+2p+l;2p+l;s). . (3.17) (The boundary condition at the coast allows us to discard the second solution, which has a logarithmic s i n g u l a r i t y there). Thereieare therefore a whole hierarchy of modes possible, cor-reponding to different values of n. The zeroth-order modes are the most easily stimulated (Ball, 1967), and therefore the most likely to be present. I shall accordingly take the outside exciting wave to be a zeroth-order edge wave. In that case, n = 0 , and (3.14) and (3.15) reduce to P 2 = k 2 - a 2 , k 2 = p 2 + p , (3.18) which gives, eliminating p, a 2 = | [ -1 + 7 1 + 4k2 J . (3.19) Choosing the positive root gives real travelling wave solutions.We therefore use a 2 = | I / 1 + ,4k2' - 1 J. C3.19a) Substituting for periods comparable to those of the seiches obser-ved in the bay into (3.19a) yields zeroth-mode wavelengths as given in Table 3.1. For these short periods, the form of the edge waves is controlled by the topography, and not by the rotation; these wa-ves can therefore travel at equal speeds in either direction along the coast. Period (min.) Wavelength (km) 60 119.4 30 59.7 15 2 9 - 8 Table 3 V. The wavelength of zeroth-mode edge waves for a few selected periods. The zeroth order mode has the form T](x,y;t) = A Qe pbx 2F 1(0,2p+l;2p+l;e D X) e i(vt -my) (3.20) But(Abramowitz and Stegun, 1970, p556) , 2F 1(a,b;b;z) = (1-z) -a He re , a = 0 and the outside wave reduces to i(v t - my) (3.21) At the coast, we find for the real part of the outside wave displa-cement : 3.3) Matching the solutions. In order to find the complete motion inside the bay, we must now match the free inside solutions with, the outside forcing wave at the mouth of the bay. The necessary boundary conditions are conti-nuity of pressure and current across the mouth of the bay. Under the assumptions made at the beginning of the chapter, these requi-rements are satisfied by the continuity of current and surface ele-vation across the mouth. This balance cannot however be expressed simply as an equality between the inside free wave and the outside wave; this would overdetermine the system. The presence of the open mouth of the bay means that the induced oscillation inside w i l l radiate energy in the form of long waves out into the open sea. The appropriate matching is therefore between the outside, inside and scattered waves, as given by n o(0 ,y;t) = A cos(vt-my). C3.22) n. C -L ) = n Co) + n CO) , 1 o s (3.23) u, C-L) = u (0) + u (0) . i o s The meaning of the subscripts should be clear from what has been explained above. Different arguments apply to the inside and out-?) side coordinate systems, as they have been defined d i f f e r e n t l y Csee Fig. 3.1 and Fig. 3.4). The equations C3.23) must be solved for the unknown amplitude coefficients of the inside solution i n order to f i n d the resonant response. I t i s now necessary to f i n d a form for the scattered wave, i n i n terms of the existing elevation at the -mouth. Matching must be done for each of the hierarchy- of cross-channel modes. In order to do this correctly, the outside exciting wave must be decomposed into an eigenfunction expansion i n terms of the cross-channel modes of the inside system. From C3.6), the inside cross-channel eigenfunctions are c|> = cosInTTCy-wl/2w] . C3.24X Taking the r e a l part of (3.21) as the outside wave, i t i s necessa-ry to express (3.22) i n the form H = 5~ a <b cosVt + ^ 3 <$> sinvt , C3.25) o "*—«r n n n n n n Inside the bay, only the eigenf unctions (3.24). e x i s t , and therefo^-re, n must be continued as an even function i n [-w,3w], i n order to perform the expansion correctly. The result i s the same as far as the inside response i s concerned. Transform variables by l e t t i n g £ = y - w. In the equation for T| "simply replace y by £ to simplify the integrations. This merely introduces a change of phase i n X)Q , which was arbitrary i n the f i r s t place. 2w Jw I out Then . _ b L °n = 2w • A *- 6 P c o s ( m ? ) cos(n7r£/2w) dg -2w 2w 3 = h- j A ^ e sin(m^) cos (nTr£/2w) d£ n 2w / out -2w Evaluation of these integrals yields m A ^ e sin(2mw) cos(nfr) out a = n 2 _ 2„.2 I , 2 w [ mz - n^TT/Aw ] m A e P^ L r cos (2mw) cos (nir) - 1] B = e s t : :  n 2 2 „ 2 / , 2 (3.26) (3/27) w I mz - . n^ T T / t o * ] These expressions may then be used to evaluate the outside elevation i n trttermsof the cross channel modes o"f> the bay. Reference to }Ta.ble*3.>l: shows that fbr periods comparable to those observed, the wavelengh of n C-.- lis very much longer ;>;6ut than the 2Km entrance width of the bay. In this case we may neglect the e x c i t a t i o n of the higher modes and be concerned only with the case where n=0: the amplitude of the e x c i t i n g wave i s then taken as) constant across the mouth. (There i s Q due to the y t r a y ^ l l i n g n a t u r e of the wave, a change i n phase, which w i l l cause the modes i n s i d e the bay to be forced w i t h r i * " ? \'l d i f f e r e n t phases - the usual cos Vt term and another i n s i n Vt; t h i s i s of no consequence, however, i n computing the resonant response frequencies) . The problem to be dealt with now i s how best to account for the s c a t t e r i n g of waves from the entrance of the bay when attempting to match the solutions across the entrance. In order to do this I s h a l l follow the approximate equivalent -c i r c u i t analysis i^jieHbKodiJpf Miles (1971) as modified by Garrett (1975). In this formulation the edge wave produces an elev a t i o n n 0 at the mouth of the bay; the surface of the bay being taken as being at rest. The discrepancy i n elevation at the mouth of the bay then produces a mass f l u x F through the mouth of the bay, i n order to balance the pressure difference at the mouth. This r" i i n turn forces a response r\i ^inside the bay, and scatters waves i ' / put i n t o the open ocean. The rela t i o n s h i p s between these various quantities are analogous to those i n an e l e c t r i c a l c i r c u i t , and may be treated correspondingly. I s h a l l treat only the case where the cross-channel mode number i s zero, and leave the problem of-the-h-igher-^order cross-channel modes u n t i l l a t e r (see appendix A) I s h a l l follow the method of Garrett, and f i r s t define the impedance of the bay i t s e l f and that of the ocean outside. The elevation discrepancy at the mouth of the bay induces a flu x F through the mouth of the bay. Let a point source of transport i n the mouth h u*n = 6(s- a) produce the responses K (s,a) and K Q(s,a) on the bay and open ocean sides respectively. Then, equations (3.23) require that JKd(S,O--,?) F(a) da = x] (s) - IK (s,a) F(a) da , (3.28) » a o J o M M which defines an i n t e g r a l equation for F(a). I f we assume that there are no cross-channel modes and that the bay mouth i s small with respect to the wavelength of free waves, H n and r|i are independent of s (distance across the mouth) and we u ' i can define F(a) = I f(a) , C3.29) where J f(a) do = 1 , and I i s the mass flu x through, the H mouth, Then i f we define an equivalent exciting voltage V q by V = In "(s) f(s) ds , (3.30) o J o M (3.28) becomes Z T V - Z I , (3.3 1 ) B o o where the impedances!^,, and Z are defined by v 'B o J KgCsycr) f (s)i(ff) dads , M M IK (s,a) f ( s ) C3.32) f(a) dads M M These impedances define the r e l a t i o n between the ele v a t i o n of the surface at the mouth and the corresponding f l u x through-i t . They are stationary with respect to small changes of f ( s ) about the exact s o l u t i o n (Miles 1971) and therefore may be approximated quite weljiby t r i a l s o l u t i o n s . The s o l u t i o n of (3.31) gives the f l u x through the mouth I = , (3.33) O D and the corresponding elevation V..- B ° , (3.34) Z + Z„ ; N : O B which i s not very d i f f e r e n t froin V Q i f Z B » Z Q , a condition which can be obtained i f there i s e i t h e r a large change of depth from the bay to the deep ocean,(in this case a f a c t o r of ^ 7) and the bay i s r e l a t i v e l y small ( as this one i s ) . Since the solutions f o r the free waves i n s i d e the bay are known,lit i s a simple matter to f i n d Z since D n,(-L)/[2wh u.(-L)] , 1 m i (3.35) where V i s defined as i n (3.30), and h i s the depth at the mouth. o m Using (3.2) for u ±(x) and (3.11) for X(x) gives u ±(x) = - i S Real[X'(x)] , (3.36) •«where X' (x) = A exp[c/ K 2-o" 2 x]'« {clf^h^f^p^(asb ; 1; l-e°X) -abc e ^ 2F 1(a+l,b+l;2;l-e C X) using the form given by Abramowitz and Stegun (1970) for the derivative of 0 F (a,b;l;z) . In this case, the bay impedance Z becomes Zfi = i v Real . exp[-c/ K 2 o2' Lj 2F 1(a,b ; 1; l-e~ c I L) 1 '•• 2wh (1-e C L ) Real o - abc e ° L 2F1(a+l,b+l;2»l-e °L) -1 (3.37) 1 In order to solve the problem completely, an expression for the impedance of the outside ocean to scattered waves must be found. Given a unit current impulse i n the mouth of the bay, the outgoing waves produced i n an ocean of uniform depth are given (Mi'les?,J 19 71; Buchwald, 19 71) by the i n t e g r a l of a Green's function which i s a c y l i n d r i c a l Hankel function. At the bay mouth i t s e l f (x = - L), this takes the form K (y,n) = — H ( 2 ) ( k |y-n|) , (3.38) o o o out with k Q = v/ / g h Q u t , since we are dealing with frequencies where f becomes negl i g i b l e . The depth h o u t appearing i n (3.38) i s the asymptotic value of the actual offshore depth. The assumption has been made that the effect of the sloping bottom i s not too great, and that the Green's function (3.38) appropiate for a flat-bottomed, f r i c t i o n l e s s ocean may be used. There does not e x i s t i n the l i t e r a t u r e , to my knowledge, a Green's function appropriate for a sloping bottom. See appendix B for further comments on this problem. Using (3.38) the outside impedance.(from 3.32) i s given by wH _w z 8 2ghQ4w^ H ( 2 )(k- |y-n|)dndy (3.39) o o J -w -w • The function f(s) has been simply taken as-f^fS since i n the case of no cross-channel v a r i a t i o n this i s very nearly correct. 2k wv7 i s of the order of 1/30 and therefore the asymptotic form o of H v '(£) for small z may be used (Abramowitz & Stegun, 1970): H ( 2 ) ( z ) * 1 - | ^ [ l n ( z / 2 ) +vy], O IT where y = 0 . 5 7 7 2 i . . . i s Euler's constant. Evaluation of the i n t e g r a l gives Z = o 2gh . ° out lnlVw/VglTj + ^ -.3/2 (3.40) V must be evaluated next; from C3.22) and ( 3 . 2 0 ) , A O U t cos[vt -my] dy , (3.41) V o 1 -w 2w A cos(vt) sin(mw) out wm The fl u x through the mouth given by 03.32)- takes the form V A ^ cos(vt) sin(mw) / 1= ^ = - S H t T (3.42) Z + Z R Z + Z B o B o B Resonance w i l l occur wheriathere i s a minimum' i n Z + Z„ , 1 o B j This produces a large flux through the mouth, and a correspondingly large surface elevation at the head of the bay. Because of the complicated nature of the functions, involved, i t i s not possible to solve a n a l y t i c a l l y for the minima of | Z Q + Z^ The hypergeometrie functions are not tabulated and therefore must be evaluated from the series expansion i n (3.11). A Fortran program was written to sum up terms of the series. Convergence 6 to within 1 part i n 10 was found after 3001;terms were added. In order to find the resonant frequencies and the shape of the response curve for the bay, a program was written to evaluate I as a function of V and also to calculate n vs V at the position corresponding to the wharf where the measurements were taken. The value of n at the wharf i s given by v ri(x ) = — — Real nC-D exp ;I-c /K 2 -a 2' x ] F (a,b;l;l-e ^ ) (3.43) where x = - 2000 m i s the position of the wharf, and V i s given w v ' m 6 biy, (3.34). Fig. 3.5 shows I as a function of frequency. The forcing function was as i n (3.41), with A t = 0.5 cm. The graph shows the magnitude of the flux through the mouth as a function of frequency. _ 3 There are two peaks; the f i r s t at a frequency of 3.02 x 10 Hz corresponding to a period of 34.6$ minutes; the second at a frequency _ 3 of 7.78 x 10 Hz corresponding to a period of 13.46 min. The respnance peaks are of f i n i t e width and height due to the effect of the radiation of waves from the mouth of the bay. This effect i s due to the real part of Z , which l i m i t s the value of V /[Z +Z_J , since Z_. i s o purely imaginary. Reference to (3.40) demonstrates that Real(Z Q), and hence the radiative damping, increases l i n e a r l y with V. This effect i s recognizable i n the decreasing amplitude of the resonance peaks as V increases i n Fig. 3 .5. The value of the radiative Q , % Q = f / 'A'f o xir 'ea:cla'- cfse^is^a^^llOT.srhfn. cthe> "compufed' ~fesponse.: F i r s t mode: Q = 21.57, Second mode : Q = 32242 . Fig. 3.6 shows the surface amplitude andephase (dotted line) r e l a t i v e to the exciting current at the position of the wharf. The second re-sonance peak appears very much lower than the f i r s t because the wharf F i g . 3.6 The surface elevat ion (sol id l ine) and phase re la t i ve to the forc ing current (broken l ine) at the posi t ion of the wharf. happens to be close to a node of the surface amplitude at that frequency v.(Fig.-J3.'8) Resonance can e a s i l y be i d e n t i f i e d from the phase curve (dotted l i n e ) as occurring at l;that frequency where the phase difference between the surface response and the e x c i t i n g current i s zero. This corresponds to a phase difference of TT/2 between the •s:ur;fa;ce e l e v a t i o n at the wharf, and that of the outside e x i t i n g wave current. Figs. 3.7 and 3.8 show the surface configurations f o r the f i r s t and second modes, respectively. These pl o t s show the absolute value of surface displacement; points where the curve touches- the axis correspond to sign changes (phase changes of TT). . Note that i n F i g . 3.8 the p o s i t i o n of the wharf {(x=2S2'000Am)is near a node. The a m p l i f i c a t i o n r a t i o A i / A ^ ^ at the head of the bay i s 36 f o r the f i r s t mode and 20 for the second mode. 4) F r i c t i o n a l E f f e c t s The preceding calculationscdonudt tgaK ,e.l n t o account the e f f e c t s of f r i c t i o n a l damping ins i d e the bay. The resonance peaks i n F i g . 5.5 and 5.6 are l i m i t e d only by r a d i a t i o n a l damping due to the s c a t t e r i n g of waves from the mouth of the bay*/ The calculated periods are somewhat shorter than the observed periods as Table Iyi2 shows F i g . 3.7 The absolute value of the surface elevat ion along the bay for the fundamental mode. (The mouth is at x = -6400m. and the head is at x = 0 . ) . Fig 3.8 The absolute value of the surface elevation along the bay for the second mode. (The mouth is at x = -6A00m: the head Is at x = 0 . ) . Table VT The observed and calculated seiche periods. Mode Calc. Period (min) ; Observed Period (min) 1 34.7 - .5 38.5 ± 2 2 13.5 ±-.5 14.6 ±.6 The e f f e c t of f r i c t i o n a l damping may be estimated by adding an imaginary part to the frequency so that V •> V + i e where £ i s rel a t e d to Q by e = V/Q The i n c l u s i o n of £ introduces a r e a l part i n t o Z , thereby D increasing the dissipation'' of energy and lengthening the period. I f one adds an £ equivalent to Q=l, then the calculated period i s increased to 39.4 + .5 min. f o r the fundamental mode. The amp l i f i c a t i o n r a t i o i s reduced by a f a c t o r of 2. One may calculate the value of Q nnecessary to bring about an overlap of the possible values of the observed and calculated periods (taking the low value of 16.5 min. for the observed wave and the high value of 35.2 min. for the calculated period), using the approximate formula for the harmonic o s c i l l a t o r u v 2 = co2-co2/4Q2 (3.45) '• where i s the new resonant frequency (16.5 min.) and, w/ i s the >. f r i c t i o n l e s s resonant frequency C35.2 min.). The required Q i n this case i s approximately 2.5/ These values of Q are u n r e a l i s t i c a l l y low and compare unfavourably, with the values of 10 - 20 found i n Chapter '2<c$ section 7/ R e a l i s t i c amounts of f r i c t i o n cannot account for the discrepancy between the observed and calculated results. One must conclude that, while f r i c t i o n a l effects are importan^j there are other causes for this disagreement. There i s probably a greater degree of radiative damping than that calculated, due to the approximations made i n the calculation of the scattered waves. Geometrical effects may also be included, since the r e a l bay does not f i t the assumed model shape exactly. CHAPTER 4: CONCLUSIONS. An a n a l y t i c model of Port San Juan as a rectangular basin with an exponentially shelving bottom predicted the f i r s t two l o n g i t u d i n a l seiche modes to have periods of 34.7 min. and 13.5 min.. Measurements taken with a bottom mounted pressure gauge showed corresponding motions at periods of 38.5 anand 14.6 minutes. Other p e r i o d i c i t i e s v i s i b l e i n the data record did not match with the theory or with other observations from the government t i d e gauge at Port Renfrew. These other motions are probably due to long period waves on the shelf which put energy i n t o the bay, but do not e x c i t e i t to resonance. The t h e o r e t i c a l response was l i m i t e d only by r a d i a t i o n a l damping. Under these circumstances, the f i r s t and second l o n g i t u d i n a l modes had t h e o r e t i c a l Q's of 22 and 32 respectively. Measurement ind i c a t e d a Q value of 10 - 15 f o r both modes. Addition of f r i c t i o n a l damping to the model brought the calculated periods closer to the observed periods, however very large amounts of f r i c t i o n were needed to bring them i n t o exact agreement. Such large f r i c t i o n a l e f f e c t s seem rather u n r e a l i s t i c ; part of the discrepancy i s no doubt due to f r i c t i o n , but the balance is, more l i k e l y due to the approximations made i n jm'akings.tn^ model. The seiche motions are intermittent, short l i v e d and show no obvious c o r r e l a t i o n with tides or storm a c t i v i t y . The most l i k e l y generating mechanism seems to be long period waves impinging on the bay entrance from outside. In the region near Port San Juan these are most l i k e l y edge waves t r a v e l l i n g along the coast of Vancouver Island. I t appears possible that they maybe excited i n Imperial Eagle Channel whichysbecaueetbf.qits^]sizre.v.cwouid.. ebe ah\ e f f i c i e n t : s c a t t e r e r of larger period surf ace waves,, and t r a v e l south-along the coast i n t o the S t r a i t of Juan de Fuca, i n c i d e n t a l l y e x c i t i n g Port San Juan on t h e i r way. BIBLIOGRAPHY Abramowitz, M. and Stegun, I.E. (1970) Handbook of Mathematical Functions; 7th Ed., New York: Dover Publications: 1046 pps. B a l l , F.K. (1967) Edge waves i n an e^cean of f i n i t e depth. Deep-Sea Research ]4, 79-88. Bendat, J.S. and P i e r s o l , A.G. (1971) Ramdom Data: Analysis and  Measurement Procedures; New York; Wiley-3Mnte.rscien.ce;' 407 pps. Bir d , C. , Lee,CM. and Streat, J. (1974) U.B.C. Curve; Vancouver; University of B r i t i s h Columbia Computing Centre: 71 pps. Buchwald, V.T. (1971) The d i f f r a c t i o n of tides by a narrow channel. J . F l u i d Mechanics 46, 501-511. Defant, A. (1960), Physical Oceanography, Vol. I I ; New York; Pergamon Press: 598 pps. Erd e l y i , A., Magnus, W., Oberhettinger, F. and Tricomi, F.G. (1953) Higher Trans cendental Functions' j Vol. I ; New York; McGraw-Hill 302 pps Froese, E. (1970) U.B.C. FOURT: Vancouver; University of Bri t i s h . Columbia Computing Centre; 13 pps. Galloway. J.L. (1974) Prototype of a Continental Shelf Tide Gauge; M. Sc. Thesis, University of B r i t i s h Columbia: 73 pps. Garrett, C. (1975) Tides i n gulfs. Deep-Sea Research. 22, 23-35. Godin, G. (1972) The Analysis of Tides; Toronto; University of Toronto Press; '264 pps. Lath i , B.P. (1965) Signals, Systems and Communication; New York; Wileyj 607 pps. Lautenb^acher, C.C. (1970) Gravity wave refraction by islands. J. F l u i d Mechanics 4d, 241-265. Murphy, G.M. (1960) Ordinary Diffcerential Equations and Their  Soutions; New York; Van Nostrand Reinhold; 541 pps. Miles, J. W. (1971) Resonant response of harbours: a\n equivalent-c i r c u i t analysis. J. F l u i d Mechanics hb_, 241-265. Schureman, P. (1958) Manual of Harmonic Analysis and Prediction  of Tides; Special Publication 98; Washington,: U.S. Dept. of Commerce-Coast and Geodetic Survey j 3.175 .ppsl.7 ' APPENDIX A: CROSS CHANNEL MODES Cross-channel modes w i l l be excited whenever the denominator i n (3.27) becomes zero, i.^e. i f .. 2 = " vf_ = n 2 7f 2 ' „ . . . . . . — — gh 4wlj> ; >' (Al) The fundamental cross-channel mode has a period of 5.17 minutes. I f the outside e x c i t i n g wave has a period less than or equal to 5.17 minutes then one may expect cross-channel modes to be excited. However, i f t h i s i s the case, then the assumptions upon which the 'snatching jpyocedure i n Chapter 3 i s based are no longer v a l i d . The mouth of the bay i s no longer very much smaller than the wavelength of the e x c i t i n g wave, and the mouth of the bay can no longer be regarded as a point source of ra d i a t i o n . In c a l c u l a t i n g the edge wave outside the mouth, the mouth can no longer be regarded as a n e g l i g i b l e i n t e r r u p t i o n i n a long s t r a i g h t c o a s t - l i n e . The t r i a l function f - X s ^ u s e d i n Chapter 3 can no longer take the simple form used there; because of the cross-channel -variation 2w, -i t must take a form close to the s i n u s o i d a l dependence of the cross-channel modes. I f that i s true, then i t i s no longer p o s s i b l e to have yi (s) dS ="l k ' f (s)ds=l since the flow i s alt e r n a t e l y i n and out across the channel and there i s no longer any net fl u x . Under these circumstances, i t i s necessary to solve (3.23) exactly, using an exact form f o r the scattered wave. This necessitates solving the problem of a wave moving over a sloping bottom from an opening of f i n i t e width. This problem was beyond scope of this thesis ( f o r reasons described i n Appendix B) and therefore the problem of cross-channel modes must remain unsolved f o r the time being. • APPENDIX B': ,GREEN"S FUNCTIONS FOR A WAVE ON A SLOPING BOTTOM An exact form f o r the scattered wave moving out of the mouth of the bay requires the use of a Green's function f o r a wave moving from a point source out over a sloping bottom. This problem has not been solved i n the l i t e r a t u r e to the best of my knowledge. This appendix of f e r s an approach based on the treatment used by Buchwald (Buchwald, 1971) but derives only an i n t e g r a l form f o r the Green's function, not a closed expression. Assume topography and coordinate axes i d e n t i c a l tfo> those used f o r the edge waves i n Ch. 3. Let the free edge wave t r a v e l l i n g , „ i>(vt^my^i ,)-• along the coast be denoted by F (x)es.^. XC- - where i % stands f o r the wave number. Assume that the frequency involved i s s u f f i c i e n t l y high to neglect r o t a t i o n . Let there be a point source of current at (0,0) U (o,y) = 6(y) e i V t Then one may express the surface elevation at any point i n the ocean i n the following i n t e g r a l form (Buchwald, 1971) T1G = j*(m) e _ i m y F m(x) e i V t dm , (Bl) and the v e l o c i t y f i e l d as i v u = -g f$(m) e " i m y F'(x) e i V t dm . J m (B2) At the coast (x = 0 ) , these expressions take the form i v 6 ( y ) = - g $(m) e i m y F'(0) dm m (B3) hk 2 5(y) = iv$(m) e " l m y F'(0)dm J — C O m (B4) Applying Fourier's inversion theorem: iv$(m) F' (0) = hk 2/2fr m Then, the expression for r\ , which i s the Green's function desired, i s , 2 F (x) lhk m -imy . e J dm , 2TT(JJ — 0 0 F'CO) m - l 2Trg J F (x) V(m) ^ — e - l m y dm F' (0) m (B5) Equation (3.17) gives the form of F (x) for edge waves such as are consi-m dered here: F _ W = A e P b x T ( 2 P + 1 ) . r(-n)r(n+2p+l) n m r(1-n) r(2p+l+i+n) e j " The derivative, F'(x), has the form m n F m ( x ) p + j^Tcp-hOa. GKL) j j=o J where a i s an abbreviation for 3 rC2p-KU rC2p+n+i+l). . 7 7 aj rX2p+n+l) r ( 2 p +j+l) The expression for the Green's function TI then becomes G n„(x) = G w 2TTgb I n=0 -°° r -e r n p b X 1+ S l a . e " ^ (-l)J e _ l m y  L 1=1 J J dm (B6) I n P + 51 (p+n )C-l )a . This" Integral must be evaluated for every possible value of n and the results summed i n order to express completely . G This appears to be impossible to perform, since reference to 5.14 w i l l show that both the a^ and p contain m, the vari^blle coff-i n teggati on. In practice, i t may be possible to expand the integrand for the f i r s t few values of n, thereby finding an approximation to V For example, i f n=o then F (x) = A e 2 m and p = ey and the expression for ri reduces to G i u I expI-V 2x/gh b] e " l m y dm , 7 v I 2 ^ b ' W g h b > which i s very s i m i l a r to the result for the case of a f l a t bottom: 00 ... 2 | -imy -sx , , N xhk I e dm / T, O N " f w • ~2ir J — ~ s '  ( B 8 ) & / 2 2 where s = / m - k . Higher order terms become progressively more complicated. , However, the forms'of (B7) and (B8) are quite s i m i l a r , Sw.hich means - .'that the flat-bottom approximation i s not bad, e s p e c i a l l y as the fundamental (n=0) edge wave mode i s the most e a s i l y ^stimulated ( B a l l , 1971). 79 APPENDIX C: THE INSTRUMENT The general requirements that the instrument had to s a t i s f y were that i t should be portable, capable of measuring small surface elevations with high accuracy and that i t should be capable of resolving frequencies from surface waves to t i d e s : a period range of 1 second to 24 hours. A short summary of the operation of the instrument i s given here; those i n t e r e s t e d i n a d e t a i l e d d e s c r i p t i o n should r e f e r to Galloway (19 74). The instrument consisted of 3 stages: a pressure transducer, a d i g i t a l processing s e c t i o n , and an analogue output section. The pressure transducer was a v i b r o t r o n , mounted together with a preamplifier i n a pressure casing. The v i b r o t r o n was a shallow water model, capable of measuring pressures up to 25 p s i (17.3 KPa) . It.:s= output i s a s i n u s o i d a l o s c i l l a t i o n whose frequency depends on pressure i n the following manner F 2 = -aP + b < c e c 9 i ) with a s e n s i t i v i t y m = - 3 5 H z / f t (c.2) fdz -: . and a centre frequency of 14.5 KHz. The vibrotron s i g n a l i s transmitted up a cable to a waveshaper which converts the s i n u s o i d a l s i g n a l to a square wave of the same i£feqtier£cy; The s i g n a l i s then passed to 4 channels i n s i d e the instrument for measurement of frequency (with the aid of a c r y s t a l o s c i l l a t o r as a?reference clock). These 4 channels are divided into 2 groups: 2 high sampling rate channels and 2 low sampling rate channels. The high frequency channels were sampled every 0.5 sec. ( i . e . at a sampling frequency of 2 Hz.) i n order to resolve the surface wave spectrum; the low frequency channels were sampled every 21 sec. (0.048 Hz.) and a running mean was performed i n between i n order to f i l t e r l p a r t i a l l y the surface waves. (The effect of a 21 second average would reduce the eff e c t i v e surface wave amplitude by a factor of 10.) Each of these groups contained a high and low resolution channel; the high resolution channels u t i l i z e d the wraparound technique of scale expansion described i n Ch. 2 Section 2 while the low resolution ccb^ann^lsiididijnp.t. The low resolution channels were included i n case the wraparounds could not be removed from the data afterwards. I t did not prove necessary to use them as the wraparounds were removed successfully. These four signals are written as d i g i t a l numbers into four buffers on each sampling. The contents of the buffers are then written v i a four d i g i t a l to analogue (D/A) converters onto a four track F.M. tape recorder (Hewlett-Packard). When the data was taken at Port Renfrew the channels were set as follows: High Resolution: 0.43 metres f u l l scale Low Frequency 1.70 mm. s e n s i t i v i t y Low Resolutions: 6.97 metres f u l l scale 27.2 mm. s e n s i t i v i t y High Resolution: 0.83 metres f u l l scale „ . i -p 3.23 mm. s e n s i t i v i t y High Frequency 3 Low Resolution: 6.62 metres f u l l scale „ . 25.8 mm. s e n s i t i v i t y The D/f.A converters worked i n the range-5v. to +5v. with an error of - 5 mv. The major source of noise i n the instrument i s the tape recorder, which has a noise l e v e l of 10 mv. rms. This, together with the If/A converters contributed a l l the s i g n i f i c a n t noise. For the most sensitive channels, this means a random v a r i a t i o n of 0.43 mm. and 0.83 mm. for the low and high frequency channels respectively. In each case this represents about 30% of the smallest detectable v a r i a t i o n , and therefore i s not s i g n i f i c a n t . 

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