BackWe have seen that the tidal bulges move round the Earth's surface due to [primarily] the Earth's rotation and also the orbital movement of the Moon. This is not just a simple mechanical process though because, as seen from a point on the Earth's surface, the periodic rise and fall of sea-level that the movement of the bulges causes constitutes a standing-wave. Hydrodynamics tells us that the maximum speed a standing-wave can travel in a fluid is determined by the depth of the fluid. The ocean with the greatest average depth is the Pacific, at around 14,000ft: this corresponds to a free-propagation speed of about 455mph. It would thus take 54.7 hours for a wave crest to circle the equator, much longer than the 24 hours a single tidal bulge would take. We thus have a system whose natural motion is slower than the external force driving it: a "slow" standing-wave being driven by a "fast" tidal wave. As we move to higher latitudes the time taken for the standing-wave to get round the Earth decreases, as it has less far to go. Eventually, at above 64deg latitude, the standing-wave will be able to keep up with the forcing wave, and even go faster than it if it had to. We now have the reverse condition - a system whose natural motion is faster than the external force driving it. To find out what the result of these situations is we must turn to the physics of what is known as Forced Harmonic Motion.
Harmonic Motion is the physicists' name for the motion of, for example, a weight on the end of a coil spring. If the system is set in motion by extending the spring a little and letting it go, the weight will move up and down at a rate determined by the stiffness of the spring: this is its natural frequency. If, however, we start things off by moving the attachment point of the top of the spring up and down and then keep on moving it, the behaviour of the spring will depend on how the rate at which we are moving the attachment point (the driving frequency) relates to the natural frequency of the system. This is Forced Harmonic Motion, and there are basically three different modes the system can adopt.
Working on the principle that a picture is worth a thousand words, I searched the Internet for a "working model" of Forced Harmonic Motion and found a very good one. It does have two downsides though - it's written in the Java programming language, which means it might not run on your system unless you have the Java Run-time Environment loaded, and the terms it uses are rather "physics-based": I have thus added explanations where necessary. If you'd like to give it a try, click here. If all you see is a page title and a set of links round a blank area then you'll just have to click on the "Back" link as it would seem your system is unable to run Java. However, if your system offers to load Java for you then you might like to accept, as it will be useful for other webpages on the Internet. If you see a nice diagram with a spring on the left as well as the links, then you're off and running! Read the Explanations, set the model up by reading the Initial Conditions and then try the Examples.
The three Examples are exactly those mentioned above for a "slow" standing-wave, a standing-wave of the same speed as the forcing tidal wave, and a standing-wave that could go faster than the forcing wave. When transferring the results from the spring model to the tides we have to be a little careful what we mean by "out of phase" though, as there are actually two tidal bulges not one. In the case of the tides, "out of phase" means that there is low water where we would have expected it to be high, or high water where we would have expected it to be low i.e. the tide will be low when the Moon is overhead and high when it is rising or setting. In other words, there is a difference of 90deg between the position of the Moon and the actual tide.
We thus see that near the equator (the "slow wave" region) there is a 90deg difference between Moon and tides and at higher latitudes (the "fast wave" region) the tides correspond to our expected picture: Moon overhead means high tide. The physics tells us that this switch from 90deg shift to no shift is quite abrupt: it doesn't slowly decrease from 90deg to zero as the latitude increases. There is therefore an obvious question to ask - what happens at the changeover point? Not only will the phase relationship be changing rapidly but, because this is also the resonance point, the amplitude of the tides will be as well. Fortunately, two factors come to our aid and prevent turmoil in the oceans.
The first factor arises from the observation that the tidal currents which build up the tidal bulges flow from north to south (and vice-versa) as well as parallel to the equator. These currents are affected by the Coriolis force [mentioned in the earlier explanation of centrifugal force], which makes north-to-south flows turn to the west (and south-to-north flows turn to the east): the overall result is thus a very complex pattern of tides and eddies rather than a simple picture of standing-waves running round lines of latitude.
The most important factor however is the energy loss as the tidal bulges interact with the surface of the Earth, which tends to damp out any changes in flow. In terms of our spring model, one can imagine that the weight, instead of moving in free space, was placed in a vessel containing a fluid. The more viscous this fluid, the greater would be the resistance to the motion of the weight: this would clearly greatly affect the results of the Forced Harmonic Motion examples given above. In particular, the wild movements at resonance would be drastically reduced. This effect combines with the production of eddies mentioned above to "calm down" the situation near resonance.
Note that when I said "equatorial regions" and "higher latitudes" above I was really referring to the idealised case of an Earth whose rotational axis is "straight up". As we know from the previous page, it is actually leaning over at about 23.4deg. The position of the "tidal equator" can thus vary by +/- 23.4deg from the true equator (plus another +/- 5deg from the tilt of the Moon's orbit). This consideration will also affect what we mean by "higher latitudes", of course. In terms of real latitudes on the Earth "high" can thus be as little as 64-(23.4+5) or just 35.6deg, when the tidal equator is at -28.4deg.