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Investigation of the 1960 Chilean tsunami on the Pacific Coast of Canada Loucks, Ronald Harold 1962

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INVESTIGATION OF THE I960 CHILEAN TSUNAMI ON THE PACIFIC COAST OF CANADA by RONALD HAROLD LOUCKS B. Sc., University of British Columbia, 1961 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE RETIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE in the Department of PHYSICS We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA August, 1962 Txi presenting this thesis in p a r t i a l fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make i t freely-available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the Head of my Department or by his representatives. It i s understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of PrWVSKS  The University of British Columbia, Vancouver 8, Canada. Date flC60Sr 17, IKJZ i ABSTRACT A set of closely-spaced tide-gauge records of the I960 Chilean tsunami w^ <3i obtained on the Pacific Coast of Canada. The object of this study was to glean as much as possible of the information contained in these records. The investigation was carried on by power spectrum analysis, cross-spectrum analysis and visual inspection of the tide-gauge records. An interpretation is offered which invokes the mechanisms of wave build-up due to shoaling, clapotis effect, resonance, viscous dissipation, and shift of energy between wave and current by Reynolds stresses to explain the form of the power spectra. In the appendices are given a formula for the response characteristics of stilling wells, an application of an electrical engineering result for the response characteristics of pressure gauges, an elucidation of the conver-sion of power estimates to the positive frequency range, an interpretation of the phase difference from cross-spectra, and a formula for prewhitening the covariances before performing the Fourier transform in the spectral an-alysis. 20 ACKNOWLEDGEMENTS The author wishes to express his appreciation to Drs. R. W. Burling and G. L. Pickard of the Institute of Oceanography of the University of British Columbia for their valuable advice, to the Canadian Hydrographic Service, Vic-toria, British Columbia, for their most helpful cooperation, to the Computing Center of the University of British Columbia for assistance in carrying out the digital computations, and to Mr* V. K. Jain for the stimuli from our many conversations. TABLE OF CONTENTS ABSTRACT LIST OF ILLUSTRATIONS JaNTRODUCTION PROCEDURE RESULTS DISCUSSION A. Arrival Times B. Wave Amplitudes C. Power Spectra D. Those Records Not Studied in Detail E. Cross-Spectra CONCLUSIONS APPENDICES A. Response Characteristics of a S t i l -ling Well B. Response Characteristics of a Fox-boro Diaphragm-Box Level Recording System. C. Measurement of Spectra 1. Power Spectra 2. Cross-Spectra D. Derivation of a Formula for Prewhit-ening Covariances ACKNOWLEDGEMENTS BIBLIOGRAPHY CITED i i i LIST OF ILLUSTRATIONS Figure 1: Map of the Pacific Coast of Canada with tide-gauge positions c i r c -led. Figure 2: Some tide-gauge records of the tsunami. Figure 3: More tide-gauge records of the tsunami. Figure 4: The power spectra. Figure 5: Coherence and phase difference between the Tofino and Barkley Sound records. Figures 1 , 2 and 3 are copied, with permission, from Wigen. In figure 1, the locations of Nakwakto Rapids, Slingsby Channel, Outer Narrows, Cape Scott, Mayne Bay, Cape Flattery and the Gulf Islands were labelled by me. 1 INTRODUCTION On May 22, i960, there was an earthquake w i t h epicenter i n C h i l e at * f l * south l a t i t u d e and 73.5* west l o n g i t u d e . This earthquake generated an ocean wave or tsunami which propagated" away from the source r e g i o n w i t h Here, g i s the a c c e l e r a t i o n due to g r a v i t y , k i s the wave number, and D i s the depth of the ocean. I t happened t h a t , at t h i s time, the Canadian Hydrographic S e r v i c e had i n o p e r a t i o n , along the P a c i f i c Coast of Canada, twenty-one cl o s e l y - s p a c e d tide-gauges. Seventeen hours a f t e r the earthquake, the tsunami a r r i v e d on t h i s coast and was subsequently recorded by seventeen of the gauges. These tide-gauge records appear i n a manuscript r e p o r t by Wigen and are reproduc-ed, w i t h permission, as f i g u r e s Z 3. The one minute d i f f e r e n c e s between a r r i v a l times of the f i r s t wave of the tsunami at those p o i n t s on the outer coast i n d i c a t e that i t approach-es from the southwest with the f i r s t c r e s t about p a r a l l e l to the coast. The a r r i v a l times of the f i r s t wave at p o i n t s on the inner coast agree, w i t h the exception of the set of records up i n l e t from Nakwakto Rapids, with the times r e q u i r e d f o r t r a v e l from the outer coast p r e d i c t e d using distances and depths from the charts and the c e l e r i t y formula f o r long waves. (Long waves are those f o r which the depth i s l e s s than about one tenth of a wavelength. For long waves, c=gD.) The i n i t i a l i n f l u e n c e of the tsunami on t h i s coast was an e l e v a t i o n of the sea-surface. The wave of l a r g e s t amplitude u s u a l l y occurred some f i v e hours a f t e r the i n i t i a l disturbance. A feature most apparent i n the records; from the outer coast i s the s e r i e s of from one to three long p e r i o d waves which occurred during the two hours immediately f o l l o w i n g the a r r i v a l of." the tsunami. A f t e r two hours, the records show characters which are d i f f e r e n t f o r d i f f e r e n t records and which p e r s i s t f o r a few days. The Cape S t . James rgtsord shows innumerable waves of about f i v e min-utes p e r i o d . The A l e r t Bay record has higher waves of longer p e r i o d and i s smooth-ed compared to Cape S t . James. c e l e r i t y , c, given by 2: The Tofino and Barkley Sound records appear to be s i m i l a r i n that there are l a r g e r waves of very long (15 to 150 minutes) p e r i o d and smaller waves of long (2 to 15 minutes) p e r i o d than there are at Cape S t . James. The V i c t o r i a r ecord shows waves decreased i n amplitude and smoothed compared to the Bark l e y Sound r e c o r d . F u l f o r d Harbour shows l e s s tsunami a c t i o n than V i c t o r i a . Two gauges showed almost no tsunami i n f l u e n c e . P r i n c e Rupert was e f -f e c t i v e l y s h e l t e r e d by the Queen C h a r l o t t e I s l a n d s ; C a u l f i e l d , by the Gu l f I s l a n d s . The McKenney I s l a n d Record resembles the Cape S t . James r e c o r d except that the very long waves are more apparent and the long waves, l e s s apparent. The Klemtu r e c o r d i s smoothed compared to the McKenney r e c o r d . The G r i f f i n Passage record shows evidence of a 25 minute seiche w i t h resonant a m p l i f i -c a t i o n . Johnson P o i n t , Nugent Sound and Mereworth Sound show a s u r p r i s i n g amount of tsunami a c t i o n . Seymour and B e l i z e I n l e t s were i n f l u e n c e d only s l i g h t l y by the Tsunami. (The l o c a t i o n of a l l place names are shown on the map i n f i g u r e 1.) This study i s an attempt to e x p l a i n these f e a t u r e s . 3> PROCEDURE I t was decided e a r l y i n the course of the i n v e s t i g a t i o n to attempt power spectrum a n a l y s i s of some of the tide-gauge records. This y i e l d s an estimate, i n each of many frequency bands, of the mean square value of the water surface e l e v a t i o n s which i s r e f e r r e d to here as the power of the t s u -nami. With twelve hours of tsunami r e c o r d sampled at one minute i n t e r v a l s , i t was p o s s i b l e to c a l c u l a t e the power spe c t r a to w i t h i n c l o s e confidence l i m i t s . Thus, the s t a t i s t i c a l nature of the waves could be used to o b t a i n a more concrete r e p r e s e n t a t i o n of the character of the tsunami. . Before power spectrum a n a l y s i s could be performed, two questions had to be r e s o l v e d . The f i r s t question was the f o l l o w i n g . What are the d i s t o r t -i n g i n f l u e n c e s of the tide-gauges, i f any, and how do we measure and c o r r e c t f o r them? The answer i s that the d i s t o r t i n g i n f l u e n c e s can be measured and compensated f o r , f o r the two types of tide-gauges used, using the techniques described i n Appendices A and B. The second question was t h i s . Are the tsunami records a c c u r a t e l y r e -produced? The o r i g i n a l tide-gauge records are on l i n e a r or c i r c u l a r p l o t s , depending on the type of gauge, and have been marked f o r other purposes. The records presented i n Wigen are copies of the o r i g i n a l s drawn c l e a r l y on uniform format. These records were compared w i t h the o r i g i n a l s and seem to be very c a r e f u l l y done. They were chosen to form the b a s i s of t h i s study. Seven s t a t i o n s were chosen f o r power spectrum a n a l y s i s . These were, from north to south, Cape S t . James, Johnson P o i n t , Nugent Sound, A l e r t Bay, T o f i n o , B a r k l e y Soun<2, and V i c t o r i a . I t can be seen from f i g u r e 1 that these s t a t i o n s might provide i n f o r m a t i o n on the development of the tsunami along the coast and shoreward. The power s p e c t r a were c a l c u l a t e d by the method given i n Appendix C I . The f i r s t spectrum completed, f o r V i c t o r i a , c o n s i s t e d of some power estimates, f o r components of p e r i o d l e s s than three minutes, which were negative ( -31 .2 cms. per J-^cycles per minute). This was d i s t u r b i n g s i n c e negative power i s d i f f i c u l t to i n t e r p r e t . At l e n g t h , the f o l l o w i n g reasoning was developed. Since a power spectrum a n a l y s i s i s known to be unstable when the spectrum has high power and steep s l o p e s , and, s i n c e the V i c t o r i a spectrum i s seen to have these, perhaps the negative power i s the r e s u l t of i n s t a b i l i t y i n the a n a l y s i s . A need f o r prewhitening i s i n d i c a t e d . Prewhitening reduces the power 4; i n the high-power bands and amplifies the power in the low-power bands so that the Fourier transform process, referred to i n Appendix C 1 . , i s perform-ed on a more nearly smooth spectrum with correspondingly more s t a b i l i t y . The spectrum i s later restored to i t s actual shape. The derivation given i n Appendix D yields a method for prehwitening the autocovariances rather than the individual data. This was convenient because I had already calcu-lated the autocovariances with unprewhitened data. The cosine transform for Victoria was recalculated with prewhitened autocovariances. The power est-imates were found to be positive. Barkley Sound and Tofino have more power and steeper slopes than Vic-tor i a . Prewhitening was used on these records. The Alert Bay spectrum was calculated using prewhitening. Negative power estimates occurred at the component of 6 . 5 minutes period ( - 2 . 1 9 cms. per ~- cycles per minute) and at the component of k niinutes period ( - 1 . 1 1 cms. per cycles per minute). Calculations were not carried out for per-iods shorter than this. Recalculation without prewhitening yielded positive estimates. My interpretation of this i s that, with the decreased total power at Alert Bay compared to Victoria, prewhitening reduced the power i n the 6 . 5 minute band to such an extent that, with the small degree of uncertainty i n -herent i n the analysis, the value went slightly negative. Correction for pre-whitening just amplified this negative value. Prewhitening amplified the pow-er i n the k minute band by 3 6 per cent. I can't explain why i t was s t i l l neg-ative. Cape St. James, Nugent Sound, and Johnson Point spectra were calculat-ed without prewhitening and found to be everywhere positive. Therefore re-calculation with prewhitening was inadvisable and was not carried out. The records from Tofino and Barkley Sound seemed quite similar from visual comparison and therefor their cross-spectra were calculated. These yield estimates of the coherence and phase difference between frequency com-ponents of the two records. (Coherence i s a measure of the extent to which components of one frequency on the two records appear to have emanated from the same source.) The technique for calculating cross-spectra i s given i n Appendix C 2. 5 RESULTS The results f a l l into two classes. The tide-gauge records (figures; 2 and3 ) show the results of the tsunami incident on this coast. The power and cross-spectra (figures 4 and 5 ) are the results of my analysis of the tide-gauge records. The tide-gauge records are described i n the introduction since they constitute the basis of this study. The spectra are described below. The seven power spectra are similar i n that they a l l indicate more power in the very long periods (15 to 150 minutes) than i n the long periods (2 to 15 minutes). However, they are, at the same time, quite different from each other. At any given period, the power estimates have a wide range of values and the spectra, a wide range of slopes. Moreover, the ratio of pow-er i n long wave periods to power i n very long wave periods differs greatly between spectra. The cross-spectra calculations show that the coherence between Bark-ley Sound and Tofino i s , almost everywhere, quite low (0.2) and that the phase difference i s a rather smooth function of the period. \ DISCUSSION A. Arrival Times. The arrival times at those gauges up inlet from Nakwakto Rapids arc two hours later than one would predict using the distances and depths from the charts and the long wave celerity formula. The f i r s t wave took about 200 minutes to travel from Cape Scott to Johnson Point. The predicted tra-vel time i s about 80 minutes. The explanation of the discrepancy l i e s , I think, i n the fact that there was a strong t i d a l current flowing seaward through Nakwakto Rapids, Slingsby Channel, and Outer Narrows at the same time that the f i r s t wave was moving inward through these waters. Available estimates:' of the speed of the current i n Nakwakto Rapids place i t at about 15 knots at the ebb. This i s comparable with the speed of the wave i n the depths there. I suggest that the current held the wave almost to a standstill for an extended period of time. B. Wave Amplitude With the restricted access to Johnson Point from Queen Charlotte Strait and, from the submarine topography, l i t t l e p o s s i b i l i t y of tsunami energy being focussed strongly on the entrance to Slingsby Channel, i t i s amazing that the tsunami did progress through the narrows, channels and rapids to reach Johnson Point with appreciable amplitude. It has been shown, though, (Long-uet-Higgins and Stewart ( i 9 6 0 and 1961) that the waves can extract energy from an opposing current and lose energy to a following current via Reynolds stresses. Throughout the approaches to Johnson Point, the Reynolds stresses would be acting so as to increase the energy of the wave. On the basis of the results i n the two papers referred above, I suggest that, without the Reynolds stress effect, dissipation would have more strongly dominated. C. Power Spectra The power (and cross-) spectra were calculated from twelve hours of record starting, usually, two hours after the i n i t i a l disturbance. The pow-er i n this segment of record probably consisted largely of power i n response modes — the local configurations forming f i l t e r s or traps and determining these modes — and, to a lesser extent, of power i n incident tsunami stim-ulus. I suggest that, of my seven spectra (figure 4 ), the Cape St. James spectrum i s l i k e l y to represent most closely that of the tsunami i n the open ocean. Cape St. James i s directly exposed therefore l i t t l e dissipation i s expected to have occurred. Moreover, the continental shelf i s narrow here 7 so that minimal increase i n wave amplitude due to shoaling has taken place* (Shoaling, without dissipation, causes reduction of wave speed, concentra-tion of wave energy and, therefore, build-up of wave amplitude.) Strong re-flection seaward from Cape St. James i s hindered by the shunt afforded by Queen Charlotte Sound. Therefore the doubling of amplitude inherent i n re-flection at an abrupt obstruction (the clapotis effect) i s of diminished importance. Finally, Cape St. James i s at a node for the very long period resonance nodes which are l i k e l y to exist across Queen Charlotte Sound — the sound acting as an open-resonator. Thus there i s l i t t l e amplifi-cation of the very long wave amplitudes via resonance. Tofino i s the second most seaward station. Compared with that at Cape St. James, the tsunami spectrum here has a high broad peak i n the very long wave region. The continental shelf i s 20 kms. wide off Tofino. It i s possible that build-up of wave amplitude due to shoaling over this shelf accounts for the high power i n the very long waves. This build-up due to shoaling varies with the amplitude of the wave far offshore. Thus from the evidence of the Cape St. James spectrum, we would expect the very long waves to build-up more than the long waves. Since we are dealing with wavelengths as long as 5 0 0 kms., the continental slope, shelf and shore appear to be an abrupt obstruction. Therefore the amplitudes of the very long waves may be enhanced by the clapotis effect. I attribute the broadness of the very long wave peak to the fact that no very long period seiches are l i k e l y to exist around Tof-ino so that energy i s accumulated similarly by a l l very long periods without any resonance effects. There i s only half as much power, for waves of period less than seven minutes, at Tofino as there i s at St. James. This may seem strange since there i s opportunity for these waves to build up due to shoaling. I suggest that the energy i s being transferred from the long waves to the d r i f t current associated with the waves via Reynolds stresses. It i s suggested that four effects act to determine the characteristics of the Tofino spectrum. These are the energy build-up due to shoaling, the energy build-up due to reflection,' the shift of energy via Reynolds stresses associated with wave motion and viscous dissipation. It appears, from the spectra, that the energy of the long wave periods i s transferred or d i s s i -pated before the energy of the very long periods, Thus, i n the long wave re-gion at Tofino, Reynolds stress energy shift and dissipation dominate the shoaling build-up. In the very long wave region, the shoaling build-up dom-inates. 8 A numerical check that energy shifts due to viscous dissipation and Reynolds stress interaction can account for the energy losses observed (for instance i n the long wave periods at Tofino compared to Cape St. James)) was attempted. The average rate of dissipation of energy per unit cross-section-a l area via viscous stress i s ^ TJU/) where T i s the stress and JJ i s the long wave horizontal particle velocity. Energy i s dissipated with particle velocities i n either direction — hence the JT/j . NowTVCe£yVwh.ere C> i s the drag coefficient ( C o ~ ' ° 3 ) and { i s the density of the water. The to-t a l energy dissipated per unit breadth throughout the depth of the water and the length of the shelf via viscous stress i s It; For long waves offshore, the wave energy per unit surface area i s ap-proximately _L_C(\*a/xdep1"k. . Therefore the total energy per unit breadth for an undamped wave train on the shelf i s XC AlCL>x*«J«.p-tf\ * k^tU. opskdl. The ratio, R, of energy dissipated over total input energy i s C B e VZK A 3 co 3 x de-pfl * k^rk/twove spe^i _ j 8 _ C p f l j ^ , . Let B be tt^e. CL.Y*s^l\|wjoe. o£ a. wave cov^poKewt of vs/<u*e w^bev-j /s ki\ow>v "Prows, pewev &p«ci>u>v D There i s , associated with these waves, a mean d r i f t current i n the direction of propagation of the waves. The wave3 lose energy via Reynolds stresses to this current. The only numerical i l l u s t r a t i o n of this effect which I can offer i s the following. Longuet-Higgins and Stewart (I960) have shown that, when Reynolds stresses are taken into account, where c? and are the wave §aergy and group velocity respectively at a point where the mean current i s zero and Q' and are these same quant-i t i e s evaluated at a point where the mean current,, here the mean d r i f t cur-rent, i s Z(_ • The mean d r i f t current i s given by 9 Suppose that c^^c, . Then € - & ' ~. J , 8* cr» (O (£>v 7ofcv» 3 6 * ~ i n I consider Barkley Sound to be the third most seaward site of obser-vation. The character of the Barkley spectrum i s similar to that of the Tof-ino spectrum. The same effects dominate and their domination i s more pronounc-ed because they have had more time i n which to act. One additional effect apparent at Barkley i s resonance. The fundament-a l mode of a seiche through the length of Barkley Sound and up into Mayne Bay has a period of 60 minutes. This was calculated from the one-endopen where T i s the period, L i s the length of the channel and D i s the average depth of the channel. The second and third harmonics have periods of 20 and 12 minutes respectively. The Barkley spectrum has higher power than Tofino i n a peak around 72 minutes period which probably corresponds to the fund-amental lengthwise resonance. For periods from 15 to 3 0 minutes, the Barkley spectrum shows less power than the Tofino spectrum. At 12 minutes period, the Barkley spectrum shows a strong peak which i s interpreted to correspond to the third harmonic. Thus i t appears that only the fundamental and third harmonics are excited. Since the power i n the very long wave periods of the Victoria spectrum i s greatly reduced from that of Barkley, i t i s concluded that dissipation becomes comparable with shoaling build-up by the time the tsunami has tra-velled about lOOkms. i n shoal water. There are probably resonances along the length of the Strait of Juan de Fuca at periods of 80 or 50 minutes and seiches across the s t r a i t of per-iods 40, 2 0 , 13 or 10 minutes. The Alert Bay spectrum shows 75 per cent of the total power that Vic-toria does. The distance from Cape Flattery to Victoria i s 80 per cent of the distance from Cape Scott to Alert Bay. Thus once dissipation begins to prevail over the shoaling build-up, the energy of the wave decreases as some function of the distance travelled through shoal water. Strong currents com-plicate this relation of course. With the Cape St. James spectrum as reference, the Johnson Point spec-trum shows dissipation predominant. For waves of period less than 10 minutes at Johnson Point, dissipation and Reynolds stress energy shift have overwhelm-resonator formula 10 ed the shoaling build-up and energy input due to Reynolds stresses acting against the opposing t i d a l current. In the very long wave region, the shad-ing build-up i s more effective. The Reynolds stress energy input prevents the tsunami from being much more severely damped. I think of the Nugent Sound record as showing the response of Nugent Sound to the input shown i n the Johnson Point record. The long wave region of the Nugent Sound spectrum shows about the same power as that of Johnson Point. The very long wave region shows a surprising peak with twice the pow-er of Johnson from 150 to 72 minutes period.,and power in excess of Johnson down to 18 minutes period. Using the soundings given i n figure 2 of Thomp-son and Barkfey;, ( 1 9 3 8 ), I estimated the mean depth along the path of the wave i n Nugent Sound to be about ^5 meters. % e n I was able to calculate the one-end-open resonator nodes. These turned out to have periods of approximately 7 0 , 2 3 , 1^ and 10 minutes. Now the Nugent gauge was some distance from the head of the Sound. Therefore only 75 per cent of the power of the second harmonic (23 minutes) would be apparent at the gauge. Virtually no power of the third (1^ minutes) or fourth ( 1 0 minutes) harmonics would be apparent since the gauge was very near the respective nodes. I suggest then that high power i n the very long wave region i s caused by lengthwise resonance and that the low power from 18 to 8 minutes i s due largely to the fact that the gauge was near the nodes. D. Those Records Not Studied i n Detail Clapotis effect and shoaling build-up probably account for the McKen-ney Is. record being smoothed and heightened compared to the Cape St. James record. Klemtu i s further smoothed and heightened by shoaling build-up. The reason that McKenney and Klemtu show less power than Tofino i s probably that, i n the former, dissipation has had more time to act. The G r i f f i n Passage record shows resonant amplification. The regular 25 minutes period seiche of the Mereworth Sound record suggests that resonance i s responsible for i t s power being so much larger than that of Belize-Inlet. E. Cross-spectra The coherence calculated from the cross-spectra between Tofino and Barkley Sound was always so low ( 0 . 2 ) as to be hardly significant. (The sharp peak to 0 . 6 at 2 . 3 minutes period i s thought to be a manifestation of the s t a t i s t i c a l uncertainty of the analysis. No significance i s attached to i t . ) 11 This implies that the phase difference between Tofino and Barkley, which is also given by the cross-spectra, does not have much significance either. Thus, visual inspection of records for coherence cannot be trusted. The results of the cross-spectra calculations indicate that the distur-bance observed in a given region, following the i n i t i a l stimuli, is to a very great extent dependent on the local response system. 12 CONCLUSIONS ~ "My calculations show that tsunami records from locations close togeth-are characteristic of the individual locations. The f i r s t wave at each station was usually an elevation. The wave of largest amplitude usually arrived some five hours after the i n i t i a l distur-bance. The highest recorded waves of long period (2 to 15 minutes) occurr-ed in exposed places such as Cape St. James. The highest recorded waves of very long period (15 to 150 minutes) occurred in shallow embayments leading in from the open ocean such as Tofino and Barkley Sound. Calculations indicate that dissipation is greater for waves of higher amplitude. I speculate that the shoaling effect causes broad-band amplitude build-up and that the resonance effect causes narrow-band amplitude build-up. The clapotis effect is likely important at Tofino and Barkley Sound. In the presence of an opposing current, Reynolds stresses extract en-ergy from the current and add i t to the tsunami wave motion. From the spectra, i t appears that Reynolds stress energy-transfer from the waves into the drift current associated with the waves is significant. Calculations indicate that i t is more effective for higher amplitude waves. 13 APPENDICES A. Response Characteristics of a Stilling Well The Lege tide-guage consists of a float, in a stilling well, connect-ed to a pen which traces the level of the water in the sti l l i n g well on a chart wrapped round ^  clock-driven cylindrical drum. (A stilling well is a hollow upright cylinder with a small hole in the bottom. It acts as a low-pass f i l t e r to eliminate swell from the tide record.) The major part of the distortion in the response of the gauge to a high frequency input is due to the damping effect of the stilling well. I will now derive the response characteristics of a stil l i n g well. I present, f i r s t , a derivation of the governing equation from Bell and Boston ( 1 9 6 2 ) , Let x. be the height above mean level of the water in the stilling well* Let the water outside the sti l l i n g well execute simple harmonic motion of amplitude, B, and angular frequency, OJ> , about the mean level. My exper-imental observations show that, in this case, accelerations are relatively small and can be neglected. Therefore Bernoulli's equation cna be applied between points in the flow entering the inlet hole and the flow at the free surface outside the well. Assuming a homogeneous, incompressible fluid, we obtain ^°/2 + ^ — ^ " 3 x ( 1 ) where T-T0 is the vertical velocity of the water surface outside the well and is the velocity of the flow at the inlet hole. From continuity, K D d = D- ( 2 ) where d and D are the areas of the inlet hole and the well cylinder respect-ively. K D is a factor allowing for friction and hole effects found by Bell and Boston ( 1962) to be a constant with value, 0 . 7 7 . Since T/. a n d a r e approx-imately equal, i t is seen that V°AL/ ~ ^/b • ( 3 ) Therefore the V0 term can be dropped from equation 1 provided d«D. Then combining equations 1 and 2 yields = K* J L [i-^&s^c^t - X ) J ^  . ( 4 ) To account for flow in both directions through the hole, equation 4 must be modified slightly to <^>c - s ^ f g s ^ ^ t -;c) « o 2.5 / G w ^ - ^ J ^ ( 5 ) The following is strictly my own';work. Without loss of generality, let 8 s w v J be greater than x . TJU.^, (p)^= ~*]'L ( 6 ) p/re-s t U u t a*~x_ =" O 0 . 6 X = c? o d , /Vo 'A w B c o s U - gs^c^i J / l - ( 7 ) L e t the. •steady stoute. y-«-spoevse of Ike, s t t fUn^ we'/ tc 3C " b Scvv cj C©m.l . (8) No w jcjorva'wlex- tke sC4uja.£tovv. out jj!i.=o. Fror*>. (8 ) , o = bco cos c o ( t - r ) . (9) . CO ("t -T) = oeW r\ • T£. (/o) CL^ vri t = oJJ K. -f- T* . (if). Tkere-rVe x - b s ^ (>«W »v E.) » (-/)^ b £ = o . (*) ^ 8 ( - 1 ) ^ c o i l o t » b ( . - O 5 ^ 04) cn. B co& . o r - b (?s) Now lc± t = " f < ^ (7) a-~d SiAistctoute |r.«v G s ) . cub = W cor 04) In Appendix C. l . i t i s shown that £/*/teii.)Pr |e»£|where i s the mean square elevation recorded xn the f frequency band, r> i s the power est-imate for the band, and m i s the number of lagged correlations. The mean square amplitude for the s t i l l i n g well response i s • Longuet-Higgins (1952) gives ^ pr=0.88fewhere b i s the mean amplitude. We take b above as an estimate of b • Then b^o.88t * VT* ' 0.878\/T" M • ( 1 8 ) One can now solve for the phase lag, c O t , i n (17) and (15) gives the amp-litude of response. The power estimate,; corrected for attenuation due to the s t i l l i n g well, i s p / r 5 « - * » r • ( i 9 ) The largest phase lag imposed by the s t i l l i n g well at Victoria was 17 for a wave component of 2 . 1 minutes period. B. Response Characteristics of a Foxboro Diaphragm-Box Level Recording Syste The Cape St. James and Barkley Sound gauges were Foxboro Diaphragm-Box Level-Recording Systems. These consist of a diaphragm-box on the sea floor connected by pneumatic tubing to a pressure bellows actuating a pen on a circular clock-driven chart on the shore. I quote from Loucks ( 1 9 6 2 ) . An experiment was planned to determine the phase lag and amplitude-attenuation characteristics — distortion being due to inertia and f r i c t i o n i n the pen mechanism. Briefly, the procedure was as follows: the diaphragm box was tied to a long pole and plunged or hoisted rapidly through a certain distance ( 4 , 6 , 8 feet). The response of the system to this step-function input was recorded — the steps being taken i n 15 both directions at various amplitudes, ...The step function response curve was visually differentiated to yield an ap-proximation to the S -function response curve. This curve was analysed to give the response characteristics using the method i n V.V. Solodovnikov , i 9 6 0 , pp. 3 5 - 3 7 . This method i s based on the fact that the phase characteristic and the amplitude characteristic of a linear system can be determin-ed i f the resonse of the system to an impulsive or J-func-tion input i s known. In our case, the assumption was made that the response of the level-recording system i s linear. Also i t was argued that, since the derivative of a step-func-tion i s c a :>cfr-function, then the derivative of the step- func-tion response i s the 6-function response. The maximum phase lag at Cape St. James was 2 2 ° with 2 per cent atten-uation while at Barkley, i t was 5 ° with no measurable attenuation — a l l these for waves of 2 minutes period. C. Measurement of Spectra 1. Power Spectra So avoid having t i d a l power swamp the very long wave range, I subtract-ed a mean tide curve from the record. This difference was read at intervals of one minute for twelve hours. From this data, autocovariances were calcu-lated on the Alwac III-E d i g i t a l computer at the University of British Col-umbia using a program written by Froese and Duffus ( 1 9 5 9 )• Autocovariances, Cr « are defined as follows. Q « _ U £ (xC - <Xc>XXrtr - v W > i [*~? 1 V O, » . . . where N i s the total number of data points (N=720), r i s the lag, m i s the maximum lag (m=36), i s the value of the i***" data point and 4,X<> = 4,^c^z O H i n our case, (<^>s12^- .) The autocovariances, }C<-} , give the degree of correlation between a record, , and that same record displaced by r data, points. These autocovariances were prewhitened where necessary (see PEOCED-URE and APPENDIX D.). Next, the f i n i t e series cosine transform, V< , was computed on a desk calculator according to the formula ( C i s the separation of data points ~ one minute i n this case.) The set, \V¥J , are raw power estimates and can be seen to arise through f i t t i n g co-sine curves to the set, \CY^ . The spectra for Victoria, Tofino and Barkley were smoothed usingTj^o.SoCYo+V.J.Tj^o.isK--! + 1£+o.zsVr4, l^v^v*v,L^=.,vC-,+ V*J) (ka^w-in^). The spectra for Alert Bay, Cape St. James, Johnson Point and Nu-gent Sound were smoothed using^ j*0S4Vroto.4^ v;^ ur=o.23Vv..,+-o.54»C.<-fliijVr+l I^VAV**-I, t^ .x«.S'VV(Kto.4tv/^ ( K»».-»^ |). It was f e l t that W»*w.mtv\j preserved more detail while |«*.«*v.tivg preserved phases (see Holloway ( 1 9 5 8 ) ) . This might have been im-16 portant i n the cross-spectra. The estimates were then corrected for prewhit-ening where appropriate (see APPENDIX D.). The correction to remove zero frequency power from the x~*-o estimate given i n Blackman and Tukey ( 1 9 5 8 ) , p. 53 was applied. The analysis up to this point applies to both positive and negative frequencies. Thus the power estimate for r=0 applies to the band from - o.s/71 to+O'^cycles per minute. The power estimate for r=20 i s typical for Kr&v»-i and applies to the band from i ^ s ^ t o cycles per minute and to the band from-fl-S^to-io.S/^cycles per minute. The power estimate for r=m=36 applies to the half band from3*.^ to cycles per minute and to the half band from -yss/^to - y!o/ cycles per minute. V/e now convert into the positive frequency range because negative frequencies are d i f f i c u l t to interpret. Clearly the conversion involves multiplying the r=0 and r=m estimates by one and the other estimates,by two. The resulting quantities are the smoothed one-sid-ed power spectrum estimates and are denoted by ?<\a9&/, t.nc4es- ] • Finally, the spectra were corrected for attenuation due to tide-gauges. In this analysis of 720 data points, one can be 80 per cent confident that a single observed power estimate w i l l be within + 3 0 per cent of the true value. Since, from examination of the records and spectra, there i s very l i t -t l e power i n waves of period less than two minutes, the danger of aliasing i s judged to be negligible. Blackman and Tukey (1958) i s an important^reference for this section. (There i s a b u i l t - i n check on the arithmetic i n the power spectrum analysis. It can be seen that C 0 i s just the mean square elevation of the tsunami. Now (3 i s the mean square elevation per frequency band of width J^YA cycles per minute of the component of the tsunami at the frequency. Therefore C e= 7Z Pr X V tc~~"3 • ( 1 ) r - o AlSO C : !/ ? F ( 2 ) Equation (l) was satisfied to within 1 per cent i n my spectra.) 2 . Cross-spectra The cross-covariances, S *" aad S r were computed on the d i g i t a l com-puter with the data adjusted to zero mean. 17 Sy. and were prewhitened since Barkley and Tofino were. The Fourier transforms were calculated. For the co-spectrum, fV S T + cosf-nr -»- Z IS S*co5<rrit : ^ = C i ^ . (~ v ^ ^ s P - e w l v i ^ d j . For the quadrature spectrum, X . " V V V The estimates were smoothed using hawvxing, corrected for prewhitening and converted to positive frequency exactly as for power spectra. I w i l l denote the completed co-spectrum estimate by ^ and the quad-rature spectrum estimate by,£f,_ • Then the coherence i s given where Py- i s the power estimate of the x -record (Tofino) and ^ r i s the power estimate of the y -record (Barkley). The angle, © r , which i s the phase lead of the X -record over the (j, -record i s given b y ^ ' S r 5 ^ . Ifj£>o and > Q , theno&0 r6'^ • I f j £ > o a n d ^ - ^ $ thenar ^  <9Y £ rr . I f ^ . ^ 0 and ^ f r > o » then /*-£©v^V?r • I f X ^ ° and <£o , then £ 2,?r . In some situations, such as point A i n figure 5 lb), i t seems more li k e l y that the phase slips i n -to the next cycle rather than making an abrupt change and staying i n the same cycle. The main reference for this section i s Ward and Shapiro ( 1 9 6 2 ) . D. Derivation of a formula for Prewhitening Covariances From Blackman and Tukey ( 1 9 5 8 ) , p. 52, the formula for prewhitening the data points i s x* * Xi -OJQ-, jt'ij-jiK . This formula i s equivalent to :Split:1vi"nj>p3 the raw data mVo a differentiated and an undifferentiated part — 5(1 = A(Xi-*i - i ) + 6~*-)*f "*~ which in effect multiplies the spectrum by the fre-quency thus reducing power in the low frequency bands and increasing power in the high frequency bands. I now substitute this formula into the defin-i t i o n of covariance. K/-f IV-r / \ i - v A / - T /v-v 18 CoCz+a-j - 2o_C, These are the relations used to prewhiten the covariances. I now substitute the prewhitened covariances in the finite series cosine transform to verify that the transfer function i s , as given by Blackman and Tu-key, 1958, p. 53, -ZA - C ° s j £ £ . Let \£ be the prewhitened finite series cosine transform. Vr - C 0( |tCL*) -2.CX.C, + C—(i-tcJ- )cos rK - CO* r-T^ + = \£ (i-tcx*-) -Zcv^c^co^,r-?r -t-C,(itco-52^2f j f Cv(c©*xjr -t-cos^r)^---+ CV^M ( c o s ( y ^ - 0 r 7 c ^ C * S ^ K ) + C ^ ( O D S co-5 3v7C - 2-cos v-?r. cx»S 2^ r7t"\+- . . . + C (co-b (vyv-x-)vTT -t-coS — >~ ' v^ C J ^ ^ vv^  /J I" now prove b«« ivvefouciSiot^ /^vo-t cos Ofc-tifft! -f 00s (.1+0 r ft* ~2» c o s rTT c-g»5 C>y"7r = 0 "/ov »*vvj Cvcfc^.^cv jCjJ^O , AJoio. t U t f i e reJo-^'ow is &ojfcls#«*l -fe-r 0= i . /(ssu-wve. tU_t c o s C&.-I)YTT +c*sj&+)rlr " 2 cos rTT cosA-rK - O Q>rA = 2,3,... o-/. /1/oM/ OL-' )rTT 4. cos (qj-Qyir - 2. «osxlc 0 3 54xJZL, -WN. VW YW. 19 — cos T7T I c * s Cq , -^ ) rT r + c o s «LVTT — 2 c o s r j £ c o s 6 j r Q r r r j *• -St^jrTT N - s ' ^ ^(V r ° ° s ^~J>)-rTV -e cos(g- i ) r r _ 2 o o s r Z t <^>s(c»-z)y-7r1 ^ L *~ *~ J) It can be seen that, by continuing the trigonometry, we will be left with We have also shown that s ^ ^ O j L E + Cj+Oljr - z e«»s v 7r qr?f - © . V v » r*v Thus the transfer function for prewhitened cosine transformed covariances is \/^ s [lV0^-2o.cosx&j\£. This formula can be shown, via an analogous proof, to hold for the finite series sine transform. Prewhitening the covariances involves far fewer operations than prewhit-ening the raw data and is most convenient where digital computer techniques are available for covariances but not for spectral estimates. It seems that the pre-sent formula has not been pointed out before. 21 BIBLIOGRAPHY CITED Bell, W. H. and Boston, N. E. J. ( 1 962) Stilling Well Design for the Hecate Model Tide Gauges. Fisheries Research Board of Canada. Pacific Oceano-graphic Group, Nanaimo, B. C . Circular 1 9 6 2 - 1 . Blackman, R. B. and Tukey, J. W. (1958) The Measurement of Power Spectra. New York, Dover. Froese, C. and Duffus, H. J. (1959) Programming the Alwac III-E Digital Com-puter for Geomagnetic Data, Defense Research Board. Pacific Naval Lab-oratory, Esquimalt, B. C . Technical Memorandum 5 9 - 4 * Holloway, J. Leith Jr. (1958) Smoothing and Filtering of Time Series and Space Fields, Advances in Geophysics, vol. 4 , pp. 3 5 1 - 3 8 9 , Longuet-Higgins, M. S. (1952) On the Statistical Distribution of the Heights of Sea Waves, Journal of Marine Research, vol. XI, no. 3t pp. 2 4 5 - 2 6 6 , Longuet-Higgins, M. S. and Stewart, R. W. ( i 9 6 0 ) Changes in the form of short gravity waves on long waves and tidal currents. Journal of Fluid Mechan-ics, vol. 8 , pp. 5 6 5 - 5 8 3 . Longuet-Higgins, M. S. and Stewart, R. W. (1961) Short gravity waves on non-uniform currents. Journal of Fluid Mechanics, vol. 1 0 , pp. 5 2 9 - 5 4 9 . Loucks, R. H. (1962) Response Characteristics of a Foxboro Diaphragm-Box Level Recording System, Institute of Oceanography, Uviversity of British Columbia. Manuscript Report No. 1 2 . Solodovnikov, V. V. (I960) Introduction to the Statistical Dynamics of Auto-matic Control Systems. New York, Dover. Thompson, T. G. and Barkey, K. T. (1938) Observations on Fjord-Waters. Trans-actions of the American Geophysical Union, vol. 1 9 , pp, 2 5 4 - 2 6 0 , Ward, F. and Shapiro, R. ( 1962) Decomposition and Comparison of Time Series of Indices of Solar Activity. Journal of Geophysical Research, vol, 6 7 , no, 2 , pp. 5 4 1 - 5 5 4 . Wigen, S. 0 . Tsunami of May 2 2 , I 9 6 0 , West Coast of Canada. Canadian Hydro-graphic Service (Victoria, B. C ) . Manuscript Report. 


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