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What causes neap and spring tides
Did You Know?
The 'spring' in 'spring waves' refers not to the season, but the action of 'springing forth'.
Everybody knows that the emergence and schedule of tides has 'something to do' with the Moon's gravity. Well, it does, but there is much more to it.

Tides are actually formed by the gravitational interplay between the Earth, the Sun, and the Moon. While the Earth's superior gravity keeps the Moon in check, the latter also influences proceedings on Earth with its significant gravity. The effect of lunar gravity is not felt on land, which doesn't yield easily, but water, much more fluid and flexible than solid land, responds to it in a much more noticeable fashion.

The terms 'spring tide' and 'neap tide' refer to particular points during the lunar cycle, when the tide is the highest and the lowest, respectively. This doesn't refer to 'high tide' and 'low tide', but rather, the average size of the waves during that period. Waves are the largest during a spring tide, and smallest during a neap tide. The deviation from average is about 20% either way. Let's take a look at what causes these phenomena.
Causes
Spring Tide
Neap Tide
Spring tides are caused by the Moon coming in line with the Sun and Earth. Due to this arrangement, the gravity of both Sun and the Moon act on the water in the same region. Hence, waves during this period are higher than average.

When the Moon forms a right angle with the Sun and Earth, the gravitational effects of the Moon and the Sun, combined with the effect of the Earth's rotation, act in opposite directions. Thus, waves during this period are smaller than the average. These waves are known as the neap tide. This phenomenon occurs just after the first quarter and the third quarter moon, or when the Moon is half visible.

These phenomena are observed just after the new moon and just after the full moon, since the effect of the Moon's gravity begins a few days after the exact phase in the lunar cycle. It is for this reason that, in the adjoining illustrations, the moon is slightly ahead of the described situation.
Spring neap tides cycle
The effect of the Sun on tides is constant, and less intense than the effect of the Moon, because the latter is much closer to Earth. It's the Moon's revolution around the Earth that causes the variation in tides. In its revolutionary cycles around the Earth, it causes sea water to rise a small (in astronomical terms) amount due to its gravity (think of a global Mexican wave of the oceans, if that helps). The high tide in the region opposite to the Moon is called the antipodal tide. It is caused due to Earth's rotation.