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Looking at the tidal calendar for the German North Sea, I wonder why the interval between times is so irregular. Internet searches lead to pages that say the tides should be regular 24 hours 50 minutes cycles, but when looking at each column, the changes vary between 16 and 86 minutes. Why is that?

Tidal Calendar

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    $\begingroup$ 24 hr 50 min is the average length of the lunar day. There is a small variation due to the elliptical orbit of the Moon. But I suspect the larger difference is topography. The way the water sloshes around the Earth affects the timing. timeanddate.com/astronomy/moon/tides.html has more detailed information without going into details about why the time interval changes so greatly. My guess is the topography adds or subtracts to the time, possibly depending on the phase of the Moon and how the water moves globally (or at least over continental distances). $\endgroup$
    – JohnHoltz
    Aug 22, 2022 at 3:44
  • $\begingroup$ Whenever one comes across a website that still uses the archaic notion of two tidal bulges, one should run away. There is no tidal bulge. I'll expand this comment into an answer. $\endgroup$ Aug 22, 2022 at 6:31

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As David Hammen's answer illustrates, the dominant tide in different locations vary and the magnitude of each of the tidal constituents will change. The tide in the North Sea is relatively well understood and a map of the dominant constituent ($M_2$) is available from many sources. For instance, Reynaud and Dalrymple (2012) M2 North Sea map

Map showing the $M_2$ (principal lunar semidiurnal) tide amphidromic systems in the North Sea. Cotidal lines are perpendicular to the coast, which means that the tidal wave travels parallel to the coast, creating tidal currents that are also coast-parallel. The tidal range increases outward from each amphidromic point (point with zero amplitude for that tidal constituent; https://en.wikipedia.org/wiki/Amphidromic_point), with the highest tidal ranges within embayments such as the German Bight.

Now, if we choose the water level time series (provided by the German Federal Maritime and Hydrographic Agency) at a specific point in the North Sea (Borkum, Germany) for the last couple of weeks, we can see the varying water levels and also the changing magnitude of the peaks depending on whether the main tidal constituents ($M_2$ and $S_2$) are in phase or out of phase. water level at Borkum

We can go one step further and conduct a harmonic analysis of the water level time series to quantify the magnitude of each of the tidal constituents that significantly (larger than 5 cm) contribute to the water level oscillations.

Tide  Description       Period (h)   Amplitude (m)   Phase (degrees GMT)
 
O1  Lunar diurnal           25.82       0.0930        233.46    
K1  Lunar/solar diurnal     23.93       0.0875         51.20   
M2  Lunar semidiurnal       12.42       1.1440        306.06    
S2  Solar semidiurnal       12.00       0.2978         49.81    
M4  1st Overtide of M2       6.21       0.0789         82.87   
M6  2nd Overtide of M2       3.10       0.0658        276.37   

The main constituent, as expected, is the $M_2$ with over 1 meter of amplitude (tidal range will be double that). The second largest constituent, with about 30 cm amplitude, is the $S_2$. So when high values of the $M_2$ and $S_2$ coincide in time, the total magnitude will be the addition of the two (~1.44m), while when the 2 are out of phase, then the total magnitude will be decreased. Now as they peak at different times you might have that the space of time between successive high tides will not match either 12.42 hours or 12.00 hours.

Additionally, the diurnal tides ($O_1$, $K_1$; periods around 24 hours) have a magnitude of about 10 cm and they will also shift the timing of the peak high tide.

Finally, the overtides ($M_4$, $M_6$; shallow water constituents caused by the interaction of the main components with terrain or other constituents) have magnitudes of about 7 cm in this location and they will also contribute to the intensification of some of the high tides, but should not alter their timing.

Reference: Reynaud, J.Y. and Dalrymple, R.W., 2012. Shallow-marine tidal deposits. In Principles of tidal sedimentology (pp. 335-369). Springer, Dordrecht.

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Whenever one comes across a website that still uses the archaic notion of two tidal bulges, one should go to a different website. There is no tidal bulge. Quoting from the linked answer at the physics StackExchange,

This was one of Newton's few mistakes. Newton did get the tidal forcing function correct, but the response to that forcing in the oceans: completely wrong.

Newton's equilibrium theory of the tides with its two tidal bulges is falsified by observation. If this hypothesis was correct, high tide would occur when the Moon is at zenith and at nadir. Most places on the Earth's oceans do have a high tide every 12.421 hours, but whether those high tides occur at zenith and nadir is sheer luck. In most places, there's a predictable offset from the Moon's zenith/nadir and the time of high tide, and that offset is not zero.

One of the most confounding places with regard to the tides is Newton's back yard. If Newton's equilibrium theory was correct, high tide would occur at more or less the same time across the North Sea. That is not what is observed. At any time of day, one can always find a place in the North Sea that is experiencing high tide, and another that is simultaneously experiencing low tide.

The mathematical tools needed to properly analyze the tides were not available in Newton's time. It was Laplace rather than Newton who properly explained the tides. Even Laplace didn't quite get it right. The harmonic analysis tools needed to properly model the tides arose about a century after Laplace, two centuries after Newton.

The tides at any one place result from multiple frequency components. Both the Moon and the Sun contribute to tides, with the Moon typically dominating over the Sun, roughly by a factor of two. Two of the many (over 30!) frequency components are

  • M2, the principal lunar semidiurnal component. This component of the tides typically is the dominant frequency with a period of 12.4206012 hours.
  • S2, the principal solar semidiurnal component. This component of the tides typically has a bit less than half of the amplitude of the M2 component and has a period of 12 hours.

When one adds multiple sinusoids (keep in mind that most places use over 30 frequency components to model the tides), the result is a complex shape with peaks that don't exactly match the 12.4206012 hours between the peaks suggested by using the M2 component only.

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