Harmonic Analysis and the Prediction of Tides Dr. Russell Herman Mathematic and Statistics UNCW “THE SUBJECT on which I have to speak this evening is the.

Harmonic Analysis and Harmonic Analysis and the Prediction of Tidesthe Prediction of Tides

Dr. Russell HermanDr. Russell Herman

Mathematic and StatisticsMathematic and Statistics

UNCWUNCW

““THE SUBJECT on which I have to speak this evening is the THE SUBJECT on which I have to speak this evening is the tides, and at the outset I feel in a curiously difficult position. If I tides, and at the outset I feel in a curiously difficult position. If I were asked to tell what I mean by the Tides I should feel it were asked to tell what I mean by the Tides I should feel it exceedingly difficult to answer the question. The tides have exceedingly difficult to answer the question. The tides have something to do with motion of the sea.”something to do with motion of the sea.”

Lord Kelvin, 1882

OutlineOutline• What Are Tides?What Are Tides?

• Tidal ConstituentsTidal Constituents

• Fourier AnalysisFourier Analysis

• Harmonic AnalysisHarmonic Analysis

• Ellipse ParametersEllipse Parameters

AbstractAbstract

In this talk we will describe classical tidal harmonic In this talk we will describe classical tidal harmonic analysis. We begin with the history of the prediction of analysis. We begin with the history of the prediction of tides. We then describe spectral analysis and its relation tides. We then describe spectral analysis and its relation to harmonic analysis. We end by describing current to harmonic analysis. We end by describing current ellipses. ellipses.

The Importance of TidesThe Importance of Tides

Important for commerce and science for thousands of years Important for commerce and science for thousands of years

• Tides produce strong currents Tides produce strong currents • Tidal currents have speeds up to 5m/s in coastal watersTidal currents have speeds up to 5m/s in coastal waters• Tidal currents generate internal waves over various Tidal currents generate internal waves over various

topographies. topographies. • The Earth's crust “bends” under tidal forces. The Earth's crust “bends” under tidal forces. • Tides influence the orbits of satellites. Tides influence the orbits of satellites. • Tidal forces are important in solar and galactic dynamics. Tidal forces are important in solar and galactic dynamics.

Tidal Analysis – Long HistoryTidal Analysis – Long History

• Mariners know tides are related to the moon’s Mariners know tides are related to the moon’s phasesphases

• The exact relationship is complicated The exact relationship is complicated • Many contributors:Many contributors:

– Galileo, Descartes, Kepler, Newton, Euler, Bernoulli, Kant, Galileo, Descartes, Kepler, Newton, Euler, Bernoulli, Kant, Laplace, Airy, Lord Kelvin, Jeffreys, Munk and many othersLaplace, Airy, Lord Kelvin, Jeffreys, Munk and many others

• Some of the first computers were developed to Some of the first computers were developed to predict tides. predict tides.

• Tide-predicting machines were developed and used Tide-predicting machines were developed and used to predict tidal constituents. to predict tidal constituents.

““Rise and fall of the sea is sometimes called a tide; … Now, Rise and fall of the sea is sometimes called a tide; … Now, we find there a good ten feet rise and fall, and yet we are we find there a good ten feet rise and fall, and yet we are authoritatively told there is very little tide.”authoritatively told there is very little tide.”

““The truth is, the word "tide" as used by sailors at sea The truth is, the word "tide" as used by sailors at sea means means horizontalhorizontal motion of the water; but when used by motion of the water; but when used by landsmen or sailors in port, it means landsmen or sailors in port, it means verticalvertical motion of motion of the water.” the water.”

““One of the most interesting points of tidal theory is the One of the most interesting points of tidal theory is the determination of the currents by which the rise and fall is determination of the currents by which the rise and fall is produced, and so far the sailor's idea of what is most produced, and so far the sailor's idea of what is most noteworthy as to tidal motion is correct: because before noteworthy as to tidal motion is correct: because before there can be a rise and fall of the water anywhere it must there can be a rise and fall of the water anywhere it must come from some other place, and the water cannot pass come from some other place, and the water cannot pass from place to place without moving horizontally, or from place to place without moving horizontally, or nearly horizontally, through a great distance. Thus the nearly horizontally, through a great distance. Thus the primary phenomenon of the tides is after all the tidal primary phenomenon of the tides is after all the tidal current; …”current; …”

The TidesThe Tides, Sir William Thomson (Lord Kelvin) – 1882, , Sir William Thomson (Lord Kelvin) – 1882, Evening Lecture To The British Association Evening Lecture To The British Association

Tidal Analysis – Hard Tidal Analysis – Hard Problem!Problem!• Important questions remained: Important questions remained:

– What is the amplitude and phase of the What is the amplitude and phase of the tides? tides?

– What is the speed and direction of currents? What is the speed and direction of currents? – What is the shape of the tides? What is the shape of the tides?

• First, accurate, global maps of deep-sea tides First, accurate, global maps of deep-sea tides were published in 1994. were published in 1994.

• Predicting tides along coasts and at ports is Predicting tides along coasts and at ports is much simpler. much simpler.

Tidal PotentialTidal PotentialTides - found from the hydrodynamic equations for a Tides - found from the hydrodynamic equations for a

self-gravitating ocean on a rotating, elastic Earth.self-gravitating ocean on a rotating, elastic Earth.

The driving force - small change in gravity due to The driving force - small change in gravity due to relative motion of the moon and sun. relative motion of the moon and sun.

Main Forces:Main Forces:• Centripetal accelerationCentripetal acceleration at Earth's surface drives at Earth's surface drives

water toward the side of Earth opposite the moon. water toward the side of Earth opposite the moon. • Gravitational attractionGravitational attraction causes water to be attracted causes water to be attracted

toward the moon. toward the moon.

If the Earth were an ocean planet with deep oceans: If the Earth were an ocean planet with deep oceans: – There would be two bulges of water on Earth, There would be two bulges of water on Earth,

one on the side facing the moon, one on the opposite side. one on the side facing the moon, one on the opposite side.

Gravitational Gravitational PotentialPotential

Terms: Force = gradient of potentialTerms: Force = gradient of potential1. No force1. No force2. Constant Force – orbital motion2. Constant Force – orbital motion3. Tidal Potential3. Tidal Potential

2 2 21

1

, 2 cosM

GMV r r R rR

r

221

1 cos (cos 1)2M

GM r rV

R R R

Tidal BuldgesTidal BuldgesThe tidal potential is symmetric about the Earth-The tidal potential is symmetric about the Earth-

moon line, and it produces symmetric bulges. moon line, and it produces symmetric bulges. vertical forces produces very small changes in the vertical forces produces very small changes in the weight of the oceans. It is very small compared to weight of the oceans. It is very small compared to gravity, and it can be ignored. gravity, and it can be ignored.

High TidesHigh TidesAllow the Earth to rotate, Allow the Earth to rotate,

• An observer in space sees two bulges fixed relative An observer in space sees two bulges fixed relative to the Earth-moon line as Earth rotates. to the Earth-moon line as Earth rotates.

• An observer on Earth sees the two tidal bulges An observer on Earth sees the two tidal bulges rotate around Earth as moon moves one cycle per rotate around Earth as moon moves one cycle per day. day.

• The moon produces high tides every 12 hours and The moon produces high tides every 12 hours and 25.23 minutes on the equator if it is above the 25.23 minutes on the equator if it is above the equator. equator.

• High tides are not exactly twice per dayHigh tides are not exactly twice per day– the moon rotates around Earth. the moon rotates around Earth. – the moon is above the equator only twice per the moon is above the equator only twice per

lunar month, complicating the simple picture of lunar month, complicating the simple picture of the tides on an ideal ocean-covered Earth. the tides on an ideal ocean-covered Earth.

– the moon's distance from Earth varies since the the moon's distance from Earth varies since the moon's orbit is elliptical and changing moon's orbit is elliptical and changing

Lunar and Solar Tidal ForcesLunar and Solar Tidal Forces

• Solar tidal forces are similarSolar tidal forces are similar

• Horizontal Components – KHorizontal Components – KSS/K/KMM = 0.46051 = 0.46051

• Thus, need to know relative positions of sun Thus, need to know relative positions of sun and moon!and moon!

2

3

2 3sin 2 ,

4

K GM rH K

r R

Locating the Sun and the Locating the Sun and the MoonMoonTerminology – Celestial MechanicsTerminology – Celestial Mechanics

• DeclinationDeclination

• Vernal EquinoxVernal Equinox

• Right AscensionRight Ascension

Tidal FrequenciesTidal Frequencies

cos sin sin cos cos cos( 180 )p p

p is latitude at which the tidal potential is calculated,

is declination of moon (or sun) north of the equator,

is the hour angle of moon (or sun).

Solar MotionSolar Motion• The periods of hour angle:

solar day of 24hr 0min or lunar day of 24hr 50.47min.

• Earth's axis of rotation is inclined 23.45° with respect to the plane of Earth's orbit about the sun. Sun’s declination varies between = ± 23.45° with a period of one solar year.

• Earth's rotation axis precesses with period of 26,000 yrs.

• The rotation of the ecliptic plane causes and the vernal equinox to change slowly

• Earth's orbit about the sun is elliptical causing perigee to rotate with a period of 20,900 years. Therefore RS varies with this period.

Lunar MotionLunar Motion• The moon's orbit lies in a plane inclined at a mean

angle of 5.15° relative to the plane of the ecliptic. The lunar declination varies between = 23.45 ± 5.15° with a period of one tropical month of 27.32 solar days.

• The inclination of moon's orbit: 4.97° to 5.32°.

• The perigee rotates with a period of 8.85 years. The eccentricity has a mean value of 0.0549, and it varies between 0.044 and 0.067.

• The plane of moon's orbit rotates around Earth's axis of with a period of 17.613 years. These processes cause variations in RM

Tidal Potential PeriodsTidal Potential Periods

2 2

2

3

2 2

(3sin 1)(3sin 1)

3sin 2 sin 2 cos4

3cos cos cos 2

p

p

p

GMrV

R

Lunar Tidal Potential - periods near 14 days, 24 hours, and 12 hours

Solar Tidal Potential - periods near 180 days, 24 hours, and 12 hours

Doodson (1922) - Fourier Series Expansion using 6 frequencies

Doodson’s FrequenciesDoodson’s FrequenciesFrequency(°/hour)

Period Source

f1 14.49205211 1 lunar dayLocal mean lunar time

f2 0.54901653 1 monthMoon's mean longitude

f3 0.04106864 1 yearSun's mean longitude

f4 >0.00464184 8.847 yearsLongitude of Moon's perigee

f5 -0.00220641 18.613 yearsLongitude of Moon's ascending node

f6 0.00000196 20,940 yearsLongitude of sun's perigee

1 1 2 2 3 3 4 4 5 5 6 6f n f n f n f n f n f n f

Tidal Species Name n1 n2 n3 n4 n5Equilibrium Amplitude*

(m)Period

(hr)

Semidiurnal n1 = 2 Principal lunar M2 2 0 0 0 0 0.242334 12.4206

Principal solar S2 2 2 -2 0 0 0.112841 12.0000

Lunar elliptic N2 2 -1 0 1 0 0.046398 12.6584

Lunisolar K2 2 2 0 0 0 0.030704 11.9673

Diurnal n1 =1  

Lunisolar K1 1 1 0 0 0 0.141565 23.9344

Principal lunar O1 1 -1 0 0 0 0.100514 25.8194

Principal solar P1 1 1 -2 0 0 0.046843 24.0659

Elliptic lunar >Q1 1 -2 0 1 0 0.019256 26.8684

Long Period n1 = 0  

Fortnightly Mf 0 2 0 0 0 0.041742 327.85

Monthly Mm 0 1 0 -1 0 0.022026 661.31

Semiannual Ssa 0 0 2 0 0 0.019446 4383.05

The Tidal ConstituentsThe Tidal Constituents

Constituent SplittingConstituent Splitting

Doodson's expansion:399 constituents, 100 are long period, 160 are daily, 115 are twice per day, and 14 are thrice per day. Most have very small amplitudes. Sir George Darwin named the largest tides.

How to Obtain ConstituentsHow to Obtain Constituents

• Fourier (Spectral) AnalysisFourier (Spectral) Analysis

• Harmonic AnalysisHarmonic Analysis

Fourier Analysis … In the Fourier Analysis … In the beginning …beginning …

• 1742 – d’Alembert – solved wave equation1742 – d’Alembert – solved wave equation

• 1749 – Leonhard Euler – plucked string1749 – Leonhard Euler – plucked string

• 1753 – Daniel Bernoulli – solutions are 1753 – Daniel Bernoulli – solutions are superpositions of harmonicssuperpositions of harmonics

• 1807 - Joseph Fourier solved heat equation1807 - Joseph Fourier solved heat equation

Problems – lead to modern analysis!Problems – lead to modern analysis!

1

( ,0) sinkk

k xy x a

L

1

( , ) sin coskk

k x k cty x t a

L L

Adding Sine WavesAdding Sine Waves

Spectral TheorySpectral Theory• Fourier SeriesFourier Series

– Sum of Sinusoidal FunctionsSum of Sinusoidal Functions

• Fourier AnalysisFourier Analysis– Spectrum AnalysisSpectrum Analysis– Harmonic AnalysisHarmonic Analysis

+ =

Fourier SeriesFourier SeriesFourier Series

Eigenfunctions:

L 2 c x n( ) cosn x

L

s x n( ) sinn x

L

Function: f x( ) sin x( ) 2 cos 2x( ) x 0 .1 L

Fourier Coefficients:

a01

2L L

L

xf x( )

dN 8 n 1 N

a n( )1

L L

L

xf x( ) c x n( )

d

b n( )1

L L

L

xf x( ) s x n( )

d

ReconstructionReconstruction

2

0

a n( )

81 n

2

0

b n( )

81 n

2

0

a n( )2 b n( )2

81 n

F x( ) a 0n

a n( ) c x n( ) b n( ) s x n( )( )Fourier Expansion:

3

3

f x( )

F x( )

2 0 x

Comparison between f(x) and F(x)

Power Spectrum

Analog SignalsAnalog Signals

• Analog SignalsAnalog Signals– Continuous in time and frequencyContinuous in time and frequency– Infinite time and frequency domainsInfinite time and frequency domains– Described by Fourier TransformDescribed by Fourier Transform

• Real SignalsReal Signals– Sampled at discrete timesSampled at discrete times– Finite length recordsFinite length records– Leads to discrete frequencies on finite Leads to discrete frequencies on finite

intervalinterval– Described by Discrete Fourier TransformDescribed by Discrete Fourier Transform

Analog to DiscreteAnalog to Discrete

DFT – Discrete Fourier DFT – Discrete Fourier TransformTransform

Sampled Signal:Sampled Signal:

22 .p pf p

T

01

1( ) [ cos( ) sin( )],

2

M

n p p p pp

f t A A t B t

,nt n t ,T

tN

2

.p

pnt

T

and

DFT – Discrete Fourier DFT – Discrete Fourier TransformTransform

2

1

2( )cos( ), 1, 12

Npn

p n Nn

NA y t pN

N

nNpn

npNpty

NB

1

2 12,2,1),sin()(2

N

n

ntyNA

1

0 ),(1

N

n

nN ntyN

A1

2),cos()(

1

02

0 NBB

Matlab ImplementationMatlab Implementationy=[7.6 7.4 8.2 9.2 10.2 11.5 12.4 13.4 13.7 11.8 10.1 ...y=[7.6 7.4 8.2 9.2 10.2 11.5 12.4 13.4 13.7 11.8 10.1 ... 9.0 8.9 9.5 10.6 11.4 12.9 12.7 13.9 14.2 13.5 11.4 10.9 8.1];9.0 8.9 9.5 10.6 11.4 12.9 12.7 13.9 14.2 13.5 11.4 10.9 8.1];N=length(y);N=length(y);% Compute the matrices of trigonometric functions% Compute the matrices of trigonometric functionsp=1:N/2+1;p=1:N/2+1;n=1:N;n=1:N;C=cos(2*pi*n'*(p-1)/N);C=cos(2*pi*n'*(p-1)/N);S=sin(2*pi*n'*(p-1)/N);S=sin(2*pi*n'*(p-1)/N);% Compute Fourier Coefficients% Compute Fourier CoefficientsA=2/N*y*C;A=2/N*y*C;B=2/N*y*S;B=2/N*y*S;A(N/2+1)=A(N/2+1)/2;A(N/2+1)=A(N/2+1)/2;% Reconstruct Signal - pmax is number of frequencies used in % Reconstruct Signal - pmax is number of frequencies used in

increasing orderincreasing orderpmax=13;pmax=13;ynew=A(1)/2+C(:,2:pmax)*A(2:pmax)'+S(:,2:pmax)*B(2:pmax)';ynew=A(1)/2+C(:,2:pmax)*A(2:pmax)'+S(:,2:pmax)*B(2:pmax)';% Plot Data% Plot Dataplot(y,'o')plot(y,'o')% Plot reconstruction over data% Plot reconstruction over datahold onhold onplot(ynew,'r')plot(ynew,'r')hold off hold off

DFT ExampleDFT Example

Monthly mean surface temperature (Monthly mean surface temperature (ooC) on the west C) on the west coast of Canada January 1982-December 1983 (Emery coast of Canada January 1982-December 1983 (Emery and Thompson)and Thompson)

Fourier CoefficientsFourier Coefficients

Periodogram – Power Periodogram – Power SpectrumSpectrum

ReconstructionReconstruction

Reconstruction with 3 Reconstruction with 3 FrequenciesFrequencies

Harmonic AnalysisHarmonic Analysis

• Consider a set of data consisting of Consider a set of data consisting of NN values at equally spaced times, values at equally spaced times,

• We seek the best approximation We seek the best approximation using using MM given frequencies. given frequencies.

• The unknown parameters in this The unknown parameters in this case are the A’s and B’s. case are the A’s and B’s.

01

( ) [ cos(2 ) sin(2 )]M

k k k kk

y t A A f t B f t

Linear RegressionLinear Regression

• MinimizeMinimize

• Normal EquationsNormal Equations

2

2

01 1

[ ( ) ( [ cos(2 ) sin(2 )])]N M

n k k n k k nn k

e y t A A f t B f t

2

01 1

0 2 [ ( ) ( [ cos(2 ) sin(2 )])]( cos(2 )), 1, ,N M

n k k k k qn kq

e n n ny t A A f B f f q M

A N N N

2

01 1

0 2 [ ( ) ( [ cos(2 ) sin(2 )])]( sin(2 )), 1, ,N M

n k k k k qn kq

e n n ny t A A f B f f k M

B N N N

System of Equations – System of Equations – DZ=YDZ=Y

1( )

( )N

y t

y

y t

, ,

yA

Y Cy ZB

Sy

1 1

,N N

q qn q qnn n

c C s S

T TN c s

D c CC CS

s CS SS

cos(2 ), 1, , , 1, ,qn k nC f t q M n N

sin(2 ), 1, , , 1, ,qn k nS f t q M n N

1

cos(2 )cos(2 )N

T

k n q nqkn

CC f t f t

1

sin(2 )cos(2 )N

T

k n q nqkn

CS f t f t

1

sin(2 )sin(2 ) .N

T

k n q nqkn

SS f t f t

Matlab Implementation – Matlab Implementation – DZ=YDZ=Yy=[7.6 7.4 8.2 9.2 10.2 11.5 12.4 13.4 13.7 11.8 10.1 ...y=[7.6 7.4 8.2 9.2 10.2 11.5 12.4 13.4 13.7 11.8 10.1 ... 9.0 8.9 9.5 10.6 11.4 12.9 12.7 13.9 14.2 13.5 11.4 10.9 8.1];9.0 8.9 9.5 10.6 11.4 12.9 12.7 13.9 14.2 13.5 11.4 10.9 8.1];N=length(y);N=length(y);% Number of Harmonics Desired and frequency dt% Number of Harmonics Desired and frequency dtM=2; f=1/12*(1:M); T=24; alpha=f*T;M=2; f=1/12*(1:M); T=24; alpha=f*T;% Compute the matrices of trigonometric functions% Compute the matrices of trigonometric functionsn=1:N;n=1:N;C=cos(2*pi*alpha'*n/N); S=sin(2*pi*alpha'*n/N);C=cos(2*pi*alpha'*n/N); S=sin(2*pi*alpha'*n/N);c_row=ones(1,N)*C'; s_row=ones(1,N)*S';c_row=ones(1,N)*C'; s_row=ones(1,N)*S';D(1,1)=N;D(1,1)=N;D(1,2:M+1)=c_row;D(1,2:M+1)=c_row;D(1,M+2:2*M+1)=s_row;D(1,M+2:2*M+1)=s_row;D(2:M+1,1)=c_row';D(2:M+1,1)=c_row';D(M+2:2*M+1,1)=s_row';D(M+2:2*M+1,1)=s_row';D(2:M+1,2:M+1)=C*C';D(2:M+1,2:M+1)=C*C';D(M+2:2*M+1,2:M+1)=S*C';D(M+2:2*M+1,2:M+1)=S*C';D(2:M+1,M+2:2*M+1)=C*S';D(2:M+1,M+2:2*M+1)=C*S';D(M+2:2*M+1,M+2:2*M+1)=S*S';D(M+2:2*M+1,M+2:2*M+1)=S*S';yy(1,1)=sum(y);yy(1,1)=sum(y);yy(2:M+1)=y*C';yy(2:M+1)=y*C';yy(M+2:2*M+1)=y*S';yy(M+2:2*M+1)=y*S';z=D^(-1)*yy';z=D^(-1)*yy';

Harmonic Analysis ExampleHarmonic Analysis Example

Frequencies 0.0183 cpmo, 0.167 cpmoFrequencies 0.0183 cpmo, 0.167 cpmo

ReconstructionReconstruction

Example 2Example 2data = DLMREAD('tidedat1.txt');data = DLMREAD('tidedat1.txt');N=length(data);N=length(data);t=data(1:N,1); % timet=data(1:N,1); % timer=data(1:N,2); % heightr=data(1:N,2); % heightymean=mean(r); % calculate averageymean=mean(r); % calculate averageynorm=r-ymean; % subtract out averageynorm=r-ymean; % subtract out averagey=ynorm'; % height'y=ynorm'; % height'dt=t(2)-t(1);dt=t(2)-t(1);T=t(N);T=t(N);

% Number of Harmonics Desired and frequency dt% Number of Harmonics Desired and frequency dtM=8;M=8;TideNames=['M2','N2','K1','S2','O1','P1','K2','Q1'];TideNames=['M2','N2','K1','S2','O1','P1','K2','Q1'];TidePeriods=[12.42 12.66 23.93 12 25.82 24.07 TidePeriods=[12.42 12.66 23.93 12 25.82 24.07

11.97 26.87];11.97 26.87];f=1./TidePeriods;f=1./TidePeriods;

DataData

Harmonic AmplitudesHarmonic Amplitudes

Power Spectrum – Power Spectrum – FrequencyFrequency

Periodogram - PeriodPeriodogram - Period

Names =['M2', 'N2', 'K1', 'S2', 'O1', 'P1', 'K2', 'Q1'];Names =['M2', 'N2', 'K1', 'S2', 'O1', 'P1', 'K2', 'Q1'];Periods=[12.42 12.66 23.93 12 25.82 24.07 11.97 26.87];Periods=[12.42 12.66 23.93 12 25.82 24.07 11.97 26.87];

Current AnalysisCurrent Analysis• Horizontal Currents are two dimensionalHorizontal Currents are two dimensional

• One performs the harmonic analysis on One performs the harmonic analysis on vectorsvectors

• The results for each constituent are The results for each constituent are combined and reported using ellipse combined and reported using ellipse parametersparameters

cos 'sinu A t A t cos 'sinv B t B t

F. Bingham, 2005

C. Canady, 2005

General ConicGeneral Coniccos 'sinu A t A t cos 'sinv B t B t

cos( )

cos( )

u U t

v V t

2 2( ) ( ) ( )( )u v u v

fU V U V

2 cos( ) 2sin ( ).f

Coordinate TransformationCoordinate Transformationcos sinu p q sin cos .v p q

2 2

2 cos( )tan(2 ) ,

UV

U V

2 2

2 21,

p q

a b

22

2 2 2 2

( sin( ))

cos sin 2 cos sin cos( )

UVa

V U UV

2

22 2 2 2

( sin( )).

sin cos 2 cos sin cos( )

UVb

V U UV

GoalsGoals

• Maximum Current Velocity – Semi-major Maximum Current Velocity – Semi-major axisaxis

• Eccentricity – Ratio of semi-minor axis to Eccentricity – Ratio of semi-minor axis to semimajor axissemimajor axis

• Inclination – Angle semi-major axis Inclination – Angle semi-major axis makes to Eastmakes to East

• Phase Angle – Time of maximum Phase Angle – Time of maximum velocity with respect to Greenwich timevelocity with respect to Greenwich time

Ellipses and PhasorsEllipses and Phasors

cos , sin .x a t y b t

1 2

1 2

cos( ) cos( ),

sin( ) sin( ).

x r t r t

y r t r t

Any ellipse centered at the origin can be found Any ellipse centered at the origin can be found from the sum of two counter rotating phasors.from the sum of two counter rotating phasors.

Rotated EllipseRotated Ellipsecos sin cos cos sin sin

sin cos sin cos cos sin

u x a t b t

v y a t b t

1 2

1 2

cos( ) cos( ) cos( ) cos( )2 2 .sin( ) sin( )

sin( ) sin( )2 2

a b a bt t r t r tu

r t r tv a b a bt t

Changing the Initial PhasorsChanging the Initial Phasors

cos 'sinu A t A t

1 1 2 2

1 1 2 2

cos( ) cos( )

sin( ) sin( ).

u r t r t

v r t r t

1 1 2 2 1 1 2 2

1 1 2 2 1 1 2 2

( cos( ) cos( )) cos ( sin( ) sin( ))sin ,

( sin( ) sin( )) cos ( cos( ) cos( ))sin .

u r r t r r t

v r r t r r t

cos 'sinv B t B t

Relation to Current EllipsesRelation to Current Ellipses

1 1 2 2

1 1 2 2

1 1 2 2

1 1 2 2

cos cos( ) cos( ),

' sin sin( ) sin( ),

cos sin( ) sin( ),

' sin cos( ) cos( ).

A U r r

A U r r

B V r r

B V r r

cos( )

cos( )

u U t

v V t

Rotated EllipseRotated Ellipse2 2 2

1

2 2 22

1( ') ( ') ,

41

( ') ( ' ) .4

r A B B A

r A B A B

'tan ,

''

tan .'

p

m

B A

A BA B

A B

2 2

tan tan 2 cos( )tan( )

1 tan tanp m

p mp m

UV

U V

1 2

1( )

2 2p mINC

1p 2m

.21 mp

1( )

2 p mPHA max maxp mt t

SummarySummary

• History of TidesHistory of Tides

• Fourier Analysis – DFTFourier Analysis – DFT

• Harmonic Analysis – Wave HeightsHarmonic Analysis – Wave Heights

• Harmonic Analysis – CurrentsHarmonic Analysis – Currents

• Ellipse ParametersEllipse Parameters

BibliographyBibliography

• W.J. Emery and R.E. Thompson, W.J. Emery and R.E. Thompson, Data Analysis Methods in Physical Data Analysis Methods in Physical OceanographyOceanography, 2001., 2001.

• G. Godin, G. Godin, The Analysis of TidesThe Analysis of Tides, 1972., 1972.• R. H. Stewart, R. H. Stewart, Introduction to Physical OceanographyIntroduction to Physical Oceanography, 1997, Open , 1997, Open

Source TextbookSource Textbook• R. L. Herman, R. L. Herman, Fourier and Complex AnalysisFourier and Complex Analysis, Course Notes, 2005., Course Notes, 2005.• W.H. Munk and D.E. Cartwright, W.H. Munk and D.E. Cartwright, Tidal Spectroscopy and PredictionTidal Spectroscopy and Prediction, ,

Transactions of the Royal Society of London, A 259, 533-581.Transactions of the Royal Society of London, A 259, 533-581.• R. Paulowicz, B. Beardsley, and S. Lentz, R. Paulowicz, B. Beardsley, and S. Lentz, Classical Tidal Harmonic Classical Tidal Harmonic

Analysis Including Error Estimates in MATLAB Using T_TIDEAnalysis Including Error Estimates in MATLAB Using T_TIDE, , Computers and Geosciences, 2002.Computers and Geosciences, 2002.

• Sir William Thomson, Sir William Thomson, The TidesThe Tides, 1882., 1882.• Z. Xu, Z. Xu, Ellipse Parameters Conversion and Vertical Velocity Profiles Ellipse Parameters Conversion and Vertical Velocity Profiles

for Tidal Currentsfor Tidal Currents, 2000., 2000.

EpicycloidEpicycloid

( ) ( ) cos cos( )

( ) ( )sin sin( )

a bx t a b t b t

ba b

y t a b t b tb