This would normally provide a geodesic great arc, but when applied to the Gleeson map, it would yield a straight linear distance. **S = rθ** **S = Distance between two addresses** **r = Radius of Earth** **θ = Angle introduced at the center of the Earth by the two addresses** But if you have **GPS** coordinates of two places, you need to determine from the **Haversine Formula**. Look at the **Haversine Formula** shown below. ![[../Attachments/Attachments 1/Pasted image 20240301145005 1 1 1.png]] ![[../Attachments/Attachments 1/Pasted image 20240301145012 1 1 1.png]] Let’s introduce you to the parameters of the **Haversine Formula**. **φ1** **= Latitude of the first place** **φ2 = Latitude of the second place** **ℽ****1** **= Longitude of the first place** **ℽ****2** **= Latitude of the second place** Now, I’ll be showing you how to apply this formula in Excel step by step. **Steps:** - First, make a cell to store the distance value and type the following formula in cell **C8**. =2*6400*ASIN(SQRT((SIN(RADIANS((C6-C5)/2)))^2+COS(RADIANS(C5))*COS(RADIANS(C6))*(SIN(RADIANS((D6-D5)/2)))^2)) The formula uses **ASIN**, **RADIANS**, [**SQRT**](https://www.exceldemy.com/excel-sqrt-function/), **SIN**, and **COS** functions. It’s pretty simple if you just look at the **Haversine Formula**. We measure the distance in kilometers, so we put the radius of the earth in kilometers which is **6400 km**. **ASIN** refers to the **inverse Sine** or the **ArcSine**. If we compare the parameter angles of the **Haversine Formula** with our Excel formula, we get, **φ1 = Latitude of Ohio (C5)** **φ2 = Latitude of Alaska (C6)** **ℽ****1** **= Longitude of Ohio (D5)** **ℽ****2** **= Latitude of Alaska (D6)** - After that, press the **ENTER** button to see the distance between **Ohio** and **Alaska** in **Kilometers**. - Thereafter, if you want to [measure the distance in miles](https://www.exceldemy.com/excel-calculate-miles-between-two-addresses/), use the following formula in cell **C8**. **`=2*3959*ASIN(SQRT((SIN(RADIANS((C6-C5)/2)))^2+COS(RADIANS(C5))*COS(RADIANS(C6))*(SIN(RADIANS((D6-D5)/2)))^2))`** ### **1. Using Latitude and Longitude to Calculate Miles between Two Addresses** In our first method, we’ll use the latitude and longitude within a formula. The formula will use some trigonometric functions- **ACOS**, **SIN**, **COS**, and **RADIANS** functions to determine distance as miles. Let’s follow the instructions below to [calculate the distance between two addresses](https://www.exceldemy.com/calculate-distance-between-two-addresses-in-excel/) in miles. **Steps:** - Activate **Cell D8**. - Then type the following formula in it- **`=ACOS(COS(RADIANS(90-C6)) *COS(RADIANS(90-C5)) +SIN(RADIANS(90-C6)) *SIN(RADIANS(90-C5)) *COS(RADIANS(D6-D5))) *3959`** - Finally, just press the **ENTER** button to get the distance in the **Miles unit**. **Formula Breakdown:** - **COS(RADIANS(90-C6)) *COS(RADIANS(90-C5))** – the **RADIANS**  - functions convert the values into radians form and **COS** provides the cosine of the values, the cosines for latitude are multiplied then. Output – 0.365377540842758 - **COS(RADIANS(D6-D5))** – it provides the cosine value for the longitude difference between two locations. Output – 0.716476936499882 - **SIN(RADIANS(90-C6)) *SIN(RADIANS(90-C5))** – calculates the diversion of longitudes from 90 in radians form and multiplied the sine values Output – 0.627884682513118 - **SIN(RADIANS(90-C6)) *SIN(RADIANS(90-C5)) *COS(RADIANS(D6-D5))** - – becomes 0.627884682513118 * 0.716476936499882 Output – 0.449864893802199 - **COS(RADIANS(90-C6)) *COS(RADIANS(90-C5)) +SIN(RADIANS(90-C6)) *SIN(RADIANS(90-C5)) *COS(RADIANS(D6-D5))** – becomes 0.365377540842758 * 0.449864893802199 Output – 0.815242434644958 - Then the **ACOS** function arccosine the value Output – 0.617648629071256 - Finally, multiplying the value by 3959 – 0.617648629071256 *3959 provides the result in miles Output – 2445.270922 Related: [[Tides pdf]] [[Tides - Waves, lunar and Solar Tides, Tidal periods and Amplitudes, Tidal Forces, and Tides in Ocean Basins]] [[Geocentric Tides]] [[Tide Notes - Waves, Tides, Tidal Nodes, and Amphidromic Points]] Heliocentric explanation basics Sites: https://courses.lumenlearning.com/suny-osuniversityphysics/chapter/13-7-tidal-forces/ - [NOAA Tide Predictions](https://tidesandcurrents.noaa.gov/tide_predictions.html) https://www.vims.edu/research/units/labgroups/tc_tutorial/tide_analysis.php " Because the tides are essentially waves with extremely long [wavelengths](https://opencontent.ccbcmd.edu/ccardona2023oceanography/chapter/5-2-dynamic-theory-of-tides/#term_138_831) extending halfway across the Earth, they behave as shallow water waves, and they are influenced and refracted by the bottom contours, leading to regional tidal variation" "The result of all of this is that instead of a simple standing wave moving back and forth across the ocean, the tidal crest follows a circular pattern around the ocean basin, counterclockwise in the Northern Hemisphere and clockwise in the Southern Hemisphere." Dynamic tidal theory and equilibrium tidal theory are two different approaches to understanding and predicting the movements of ocean tides. ### Equilibrium Tidal Theory: 1. **Origin**: Developed by Sir Isaac Newton, this theory is based on the gravitational pull of the moon and the sun on the Earth's oceans. 2. **Simplicity**: It assumes a simplified Earth covered entirely by water that responds instantaneously and uniformly to gravitational forces. 3. **Tide Generation**: Tides are explained as the result of the Earth's waters trying to align with the gravitational forces of these celestial bodies, creating two bulges: one facing the moon (or sun) and one on the opposite side of the Earth. 4. **Predictions**: It considers the positions of the moon and the sun relative to the Earth and predicts the timing of high and low tides based on these positions. 5. **Limitations**: The equilibrium theory doesn't account for the Earth's rotation, the continents' presence, ocean basins' depth and shape, or the inertia of water. Therefore, while it provides a broad understanding of how tides should work in an idealized situation, it is not accurate for predicting real-world tides. ### Dynamic Tidal Theory: 1. **Origin**: This theory began to develop in the 19th century with the work of Pierre-Simon Laplace and others who recognized the need for a more complex model. 2. **Complexity**: It incorporates the Earth's rotation, the real shape of ocean basins, and the water's inertia. 3. **Tide Generation**: Tides are seen as shallow water waves created by gravitational forces and modified significantly by Earth's rotation, landmasses, and basin geometry. 4. **Predictions**: Dynamic tidal theory uses real-world oceanographic and astronomical data to predict tides by solving complex mathematical equations that simulate the actual motion of tidal waves around the globe. 5. **Accuracy**: It is much more accurate and is used for practical tide predictions. This theory can explain phenomena such as amphidromic points (where tidal range is minimal) and the formation of tidal streams and currents. In summary, the equilibrium theory gives us a conceptual model for how tides work in a simple, idealized Earth, while the dynamic tidal theory provides a detailed, practical model used to predict actual tides in the complex, real-world ocean environment. Script from AI Intro:   📍 📍 ​ The traditional equilibrium tidal theory and the dynamic tidal theory are two models used to understand and explain the behavior of tides. Here's a detailed comparison of the two: Traditional Equilibrium Tidal Theory: Assumptions: This theory is based on the assumption that the ocean responds instantaneously to the gravitational forces of the moon and the sun. It assumes that the ocean reaches equilibrium with the tidal forces, resulting in a predictable and stable tidal pattern. Simplified Model: The traditional equilibrium tidal theory simplifies the complex interactions between the Earth, moon, and sun into a static mod el. It considers the Earth as a rigid body with a uniform ocean depth, neglecting factors such as the Earth's rotation, ocean basin shapes, and frictional effects. Predictability: This model provides a straightforward way to predict tidal behavior based on the positions of the moon and the sun. It allows for the calculation of tidal heights and times using harmonic analysis, which involves decomposing the tidal signal into a sum of sinusoidal components with specific frequencies and amplitudes. Limitations: While the traditional equilibrium tidal theory is useful for making tidal predictions over long time scales, it does not account for certain dynamic phenomena such as tidal currents, the effects of coastal geography, and the influence of atmospheric pressure on tides. Dynamic Tidal Theory: Consideration of Dynamic Factors: The dynamic tidal theory takes into account the dynamic nature of tidal phenomena, considering factors such as the Earth's rotation, the shape of ocean basins, frictional effects, and the influence of atmospheric and oceanic circulation patterns. Time-Dependent Model: Unlike the equilibrium tidal theory, the dynamic tidal theory recognizes that the ocean's response to tidal forces is time-dependent and can exhibit complex variations over different time scales. Incorporation of Nonlinear Effects: This model incorporates nonlinear effects, such as resonance phenomena in coastal areas and the amplification of tides in certain ocean basins, which are not accounted for in the traditional equilibrium tidal theory. Applications: The dynamic tidal theory is particularly valuable for understanding short-term variations in tides, predicting tidal currents, and studying the impact of tides on coastal and estuarine environments. In summary, the traditional equilibrium tidal theory provides a simplified and predictable model for understanding long-term tidal behavior, while the dynamic tidal theory offers a more comprehensive and nuanced approach that considers the dynamic and nonlinear aspects of tidal phenomena, making it particularly useful for studying short-term variations and complex tidal interactions in specific geographical settings. Or, In other words, equilibrium tides are kinematic descriptions of the forces involved in the celestial procession that pay no attention to the actual factors controlling tides on earth, and dynamical tides detail how the water is actually pushed around on earth without incorporating the mathamagical descriptions of the gravitational forces supposedly causing all of this. ![[Attachments/ezgif.com-animated-gif-maker 1.gif]] To understand and predict harmonic tides, or convert tides into frequencies, you would need to use the method of harmonic analysis. This technique involves the decomposition of the complex, observed tidal signal into basic sine and cosine components, each associated with a specific tidal frequency. These frequencies correspond to the gravitational effects of the moon, sun, and other astronomical bodies. The principal lunar semidiurnal constituent (M2), the principal solar semidiurnal constituent (S2), and other identified components like the solar annual constituent (SA) are among those used to predict tidal conditions. The prediction is done by summing the effects of cosine curves representing each constituent to produce the predicted tides. The calculation of harmonic constants, which represent the amplitude and phase of each tidal component, requires the analysis of a location's tidal data. Typically, a minimum of 30 days of data is needed to observe most lunar and solar cycles, while a year of data allows for direct observation of all the major tidal constituents. The formula used for the calculation of tidal predictions from harmonic constants is: ℎ=�0+∑(��cos⁡[��+(�0+�)−�])h=H0​+∑(fHcos[at+(V0​+u)−K]) Here, ℎh is the height of the tide at any time �t, �0H0​ is the mean height of water level above the datum used for prediction, and �H is the mean amplitude of any constituent �A. The rest of the terms represent factors for amplitude reduction, speeds of individual harmonic constants, and corrections for equilibrium arguments. For more detailed explanations and the mathematical process behind this analysis, resources such as NOAA's Special Publication NOS CO-OPS 3 - Tidal Analysis and Predictions and Special Publication No. 98: Manual of Harmonic Analysis and Prediction of Tides can be extremely helpful. These documents offer insights into the harmonic analysis method in greater detail than a brief overview can provide​ ([NOAA Tides and Currents](https://tidesandcurrents.noaa.gov/about_harmonic_constituents.html))​. If you're looking for more academic resources, the Geosciences LibreTexts website offers an explanation of the method used for tidal analysis and prediction, including the formulas and factors involved in the calculation of harmonic constituents​ ([LibreTexts](https://geo.libretexts.org/Bookshelves/Oceanography/Coastal_Dynamics_(Bosboom_and_Stive)/03%3A_Ocean_waves/3.09%3A_Tidal_analysis_and_prediction))​. For a practical understanding of how harmonic analysis is applied to real-world tidal data, the Virginia Institute of Marine Science provides accessible explanations and even an analysis program​ ([Virginia Institute of Marine Science](https://www.vims.edu/research/units/labgroups/tc_tutorial/tide_analysis.php))​. # Tide Analysis Looking at water level records in coastal waterways, the most obvious clue confirming the presence of the tide is a characteristic, sinusoidal oscillation containing either two main cycles per day (_semidiurnal tides_), one cycle per day (_diurnal tides_), or a combination of the two (_mixed tides_). The underlying principle of tide analysis is that, no matter how complex they may appear, tidal oscillations can be broken down into a collection of simple sinusoids (sinusoids usually represented by the cosine function from trigonometry). Each “cosine” wave will have the same period of oscillation as the celestial forcing that gives rise to it (see Tide Model – _Static_ or _Dynamic_?). As it turns out, there are quite a few of these. The purpose of tide analysis is to determine the _amplitude_ and _phase_ (the so-called _tidal harmonic constants_) of the individual cosine waves, each of which represents a _tidal constituent_ identified by its _period_ in mean solar hours or, alternately, its _speed_ in degrees per mean solar hour (speed = 360°/T where T = period). Finding the tidal harmonic constants at a place allows one to predict tides at that place. Tidal constituent amplitudes are usually given in feet or meters, and phase is usually expressed in degrees. Putting these parts together, the _partial tide_ corresponding to a single tidal constituent is represented by the following equation,  In this equation, _h_(_t_) is the height of the partial tide calculated for time _t_, and _R_ is the constituent amplitude (equal to one-half the constituent _range_). The argument for the cosine function includes two terms: _Tt ,_ the constituent speed multiplied by time and _N_, the constituent phase. Notice that the argument could be written as _T( t-__N )_ if we wanted to express the phase in hours instead of degrees. Calculating _h_ for a series of times ranging from 0 to 36 hours yields the following tide graph using _R_ = 1 meter and _N_ = 5 hours:  The term _phase lag_ is sometimes used when the phase is expressed in hours as it is in the above example. If the phase had been zero, the cosine wave shown in red would “peak” at the zero hour; Instead, it peaks 5 hours later since _N_ = 5. Although the amplitude and the phase are arbitrary numbers picked for this example, the period of 12 solar hours uniquely identifies this wave as the _principal solar semidiurnal constituent_ with a speed of 30 degrees per mean solar hour. In the jargon of tide analysis, this constituent is represented by the symbol S2  (the subscript “2” means that two of these cycles occur each day). Suppose the red curve shown above was a record of the actual water level measured at a certain tide station. We wouldn’t really need analysis to come up with the amplitude and phase for a single cosine wave. But what if the curve happened to be the result of several cosine waves added together, all with different amplitude, phase and period. How could we find the amplitude and phase for S2 as well as the other constituents? The answer depends on the length of the record, measurement error, and the analytical technique used but an _estimate_ of the amplitude and phase for any set of tidal constituents whose period we know can be obtained by _harmonic analysis, method of least_ _squares_ (HAMELS). In this method, a set of cosine terms is used as a model. The complete set is made to “fit” the data according to the least squares criterion – simply picking the combination of R and _N_ that causes the sum of the squared differences between observed and model-predicted water levels (calculated, say, at half-hour intervals) to be as small as possible. An easy-to-use HAMELS analysis program is offered to the reader at the end of this tutorial. **BUILDING BLOCKS OF THE TIDE** The following are among the **_major tidal constituents_** contributing to the astronomical tide: M2 - Principal lunar semidiurnal constituent (speed: 28.984 degrees per mean solar hour) S2 – Principal solar semidiurnal constituent (speed: 30.000 degrees per mean solar hour) N2  - Larger Lunar elliptic semidiurnal constituent (speed: 28.440 degrees per mean solar hour) K1 - Luni-solar declinational diurnal constituent (speed: 15.041 degrees per mean solar hour) O1 - Lunar declinational diurnal constituent (speed: 13.943 degrees per mean solar hour) M4 - First overtide of M2 constituent (speed: 2 x M2 speed) M6 - Second overtide of M2 constituent (speed: 3 x M2 speed) S4 - First overtide of S2 constituent (speed: 2 x S2 speed) MS4 - A compound tide of M2 and S2 (speed: M2 + S2 speed) What’s so special about these symbols representing a bunch of squiggly lines? They are literally the building blocks of the tide. The first five constituents are the main players that determine the type of tide that a region experiences. If the amplitudes for M2, S2, and N2 are large compared to the amplitudes for K1 and O1, then tides in the region will be of the semidiurnal type (two highs and two lows each day); if K1 and O1 amplitudes are large compared to the others, then the tides will be of the diurnal type (one high and one low tide each day). While tidal type stems from amplitude differences among the major tidal constituents, cycles in _tidal range_ (height difference between successive high and low tides) depend on differences in speed. The spring-neap cycle, for example, is due to the speed difference between M2 and S2. S2 is going to complete each 360° cycle a little sooner than M2 because it completes 30° of that cycle in an hour while M2 completes only 28.984°, a speed difference of a little more than a degree per hour. At that rate, S2 will gain on M2 by a full 360° cycle – a spring-neap cycle – every 14 and ¾ days (two cycles every 29 ½ days, a lunar month). As the M2 wave continues to lag behind the S2 wave, the two waveforms pass in and out of phase. We get spring tides when M2 and S2 are in phase so that both waves peak at the same time causing tides of greater range. Neap tides occur when M2 and S2 are out of phase and tend to cancel one another, reducing tidal range. But if you are interested in tides of maximum range, consider what happens when M2, S2, and N2 all peak at about the same time. This results in the so-called _perigean-spring tides_ of maximal range that occur several times a year. Because the tide at any given time is the result of adding a number of different waveforms together, there is always a lot of variation – we shouldn’t expect one spring tide to look exactly like another! The last four tidal constituents shown above are called _shallow-water tides_. Tides entering waters where the tidal range is no longer insignificant compared to the depth undergo a transformation that yields additional waves called _overtides_. Like the “overtones” produced by a musical instrument, the frequency (speed) of an overtide is always an exact multiple of the fundamental frequency – the frequency of the parent wave that underwent transformation. Since their speeds are exact multiples of the parent wave speed, overtides appear fixed or “phase-locked” in time series plots– their waveforms do not move relative to the parent waveform. Instead they “deform” the parent wave and give rise to permanent tidal asymmetries; e.g., differences in the duration of a rising tide versus a falling tide (note there is no difference in duration for the simple cosine wave shown above). In addition to overtides, other tides called _compound tides_ also arise in shallow water. A compound tide (e.g., MS4) results from the shallow-water interaction of its two parent waves (M2 and S2). There are many more shallow-water tidal constituents but the four listed above do a good job of reconstructing the tidal asymmetries and other fine-scaled behavior seen at most locations.  **Two Examples from the Middle East**  The above graphic depicts a 29-day analysis of a tide record from Ras Tanura, Saudi Arabia. The red curve is the measured water level, the blue curve is the predicted tide based on a tide model using the harmonic constants obtained from the analysis, and the green curve is the residual or the difference between the two. Since the green curve shows some pretty strong deviations from zero, does this mean the predictive model is flawed? Not really. We should expect to see some non-tidal variation in water level showing up in the green curve; in fact, the strongest oscillations that appear there have a period of about 5 days – way longer than tidal. Still, since the analysis used only nine tidal constituents to fit the data, it’s possible that the green curve also contains some minor tidal variation not accounted for in the predictive model. In statistical terms, the tide model appearing above accounts for more than 90 percent of the total variation in water level (r2 = 0.91). The spring-neap cycle is very apparent at Ras Tanura, a big oil terminal on the west side of the Persian Gulf. The tidal type there is semidiurnal with two highs and two lows each day. But the Persian Gulf (Arabian Gulf to the Saudi Arabs) is a strange place in terms of tidal dynamics. The tide wave entering the Straits of Hormuz generate two large rotary waves for the semidiurnal tide and a single large rotary wave for the diurnal tide inside the Gulf. As you might expect, the amphidromic points for these waves are spaced far apart. To see the consequence of this arrangement on tidal type, look at the next graphic for Safaniya, a coastal town less than two hundred kilometers to the northwest of Ras Tanura.  At Safaniya, the tide is mixed, mainly diurnal. In contrast to Ras Tanura, which has a large spring tide on Julian day 97, the tide range is near a minimum on the same day at Safaniya. That’s because the tropic-equatorial cycle takes precedence over the spring-neap cycle at places where the tide type is mainly diurnal. Just as the phasing in-and-out of the principal semidiurnal constituents M2 and S2 produces the spring-neap cycle, the in-and-out phasing of the principal diurnal constituents, K1 and O1, results in the tropic-equatorial cycle. A quick glance at the residual curve for Safaniya shows that it is almost identical to the one at Ras Tanura. Thus we see that two tide stations with completely different tidal types can experience the same _meteorological tide_. In fact, the oscillations shown in the green curve coincided with a _Shamal_, a desert wind that can reach 60 miles per hour and blow for several days, causing periodic water level oscillations in the Northern Persian Gulf basin that last for many more days, like the ringing of a bell. Note that the range of the astronomical tide at Safaniya is smaller than that at Ras Tanura while the meteorological tide range is about the same at both stations. This is reflected in the percent of total variance accounted for by the tide model, which is only 76 percent (r2 = 0.76) at Safaniya as opposed to 90 percent at Ras Tanura. Clearly the meteorological tide has to be taken into account before evaluating the success of the astronomical tide model! Finally, a place with mixed tides can produce some strange looking curves as the tide transitions from semidiurnal to mixed to diurnal. The following series of daily tides from Safaniya provide a good example: Now it’s semidiurnal  Now it’s mixed  Now it’s diurnal with some strange looking highs  DO YOUR OWN TIDE ANALYSIS - IT’S NOT THAT HARD! In its infancy, the theory of tides occupied the minds of Europe’s greatest physical scientists in the seventeenth and eighteenth centuries – Newton, LaPlace, Maclaurin, Euler, Bernoulli, among others. Long after the basic principles were put in place by the geniuses of that age, the practical matter of analyzing and predicting sea tides remained something of an art in the hands of a very few. In the days before computers, calculations that we consider simple today represented a huge effort and consequently the task fell to government agencies to perform. Now, at the start of the twenty-first century, a typical desktop computer program requires only five lines of code to derive a set of tidal harmonic constants and very little more to generate tide predictions from these numbers. Yet few of us ever look inside the “black box” since tide prediction tables have always been easy to obtain – the government does these things for us at nominal cost. But if you’re the curious type and even a novice user of MATLAB software by the MathWorks, Inc., a very simple set of programs for tide and tidal current analysis is available at VIMS. You can use these programs to analyze a huge database of water levels and incorporate other data available for free downloading on the NOAA/NOS website: [http://co-ops.nos.noaa.gov/](http://co-ops.nos.noaa.gov/). To obtain a copy of the analysis and prediction programs at no cost, please [click here](https://www.vims.edu/research/units/labgroups/tc_tutorial/yourown.php) and refer to the BOTTOM of that page for information. ![[Attachments/Pasted image 20240416185714.png]]