Why does one cycle of tides take 24 hours and 50 minutes?
As Earth rotates through its 24-hour day, the Moon makes progress in its 27.3-day journey around Earth. So in 24 hours, the Moon has moved enough so that it takes another 50 minutes for any specific point on Earth to catch up so that the Moon will again be directly "overhead."
Why do the highest tides occur when the Moon is new and full?
The Moon appears new (dark) when it is directly between Earth and the Sun (with the Sun lighting up its backside), and full when Earth is between the Moon and the Sun. In both these cases, the gravitational pull of the Sun on Earth is added to the gravitational pull of the Moon on Earth. The Sun's pull is almost half as strong as the Moon's, so working together, the oceans bulge even higher.
Why do high tides sometimes occur either before or after the Moon is is highest in the sky?
If the Moon were the only object affecting the tides, high tide at a particular point on Earth would always occur when the Moon was at its highest point in the sky (i.e. the budge of the tide would be directed right toward the Moon). However, the Sun's gravitational pull, which is almost half that of the Moon, also gets into the picture. For example, when the Moon is in its crescent phase (when we see one-fourth of its face) or gibbous phase (when we see three-fourths of its face), the Sun's gravity is pulling at a 45-degree angle from the direction of the Moon's gravitational pull. So the highest tide will occur where the two forces combine to create the greatest pull, which will be somewhere in between the direction of the Sun and the direction of the Moon. The time when Earth has rotated to that exact position with respect to the Moon and Sun may be several hours earlier or later than when the Moon is at its highest point in the sky.
What causes neap tides?
Neap tides are the weakest tides, when high tide isn't very high at all. These occur when the Moon is in its first or last quarter (when we see half of its face), and the gravitational forces of the Moon and the Sun are acting at a 90-degree angle, thus nearly cancelling each other out.
Why do some places have only one high tide and one low tide in a day?
This is where predicting the tides gets more complicated. The Moon does not orbit Earth directly around the equator. For one thing, remember that Earth's axis of rotation is tilted by about 23.5 degrees from the plane of its orbit around the Sun. This tilt is what causes the seasons. Also, the Moon's orbit around the Earth is tilted by about 5 degrees from the plane of Earth's orbit around the Sun.
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Thus, the maximum tidal bulge will usually be either above or below the equator. At times some places on Earth experience only one of the two tidal bulges in a day, producing only one high tide.