That never made a whole lot of sense to me.
Actually the bulge on the opposite side of the earth is due to the centripetal accelleration of the earth as it orbits the moon. Yes, the earth orbits the moon. Similarly, the sun orbits the earth. While quite small, I expect we could measure the earth's tidal influence on the sun.
I had to major in astronomy at Caltech before I was able to get anyone to tell me why the tides bulge on both sides. Even at Tech, no one seemed to grasp that the earth orbits the moon &c.
Imagine that only the Earth and the International Space Station exist. The ISS is doomed because it has stopped orbiting, and is falling straight down. Inside the ISS, an astronaut has placed 5 marbles in the shape of a plus sign (center, left/right/up/down). For convenience, the "center" marble is also at the center of mass of the whole ISS. There's no air or anything to mess up the results. What happens to the marbles?
Look at the problem from the frame of the Earth. The Earth is exerting a certain force on the center marble, call it F. The "down" marble is slightly closer to the Earth, so it experiences a greater force, F + Something. The "up" marble is slightly further from the Earth, so it experiences a lesser force, F - Blah. The "left" and "right" marbles experience forces very very close to F (slightly less, if they're in a straight line, but much closer to F than F - Blah). However, those forces are directed towards the center of the Earth, so those vectors have small components directed towards the "center" marble.
Now that we've figured out the forces, look at the problem from the frame of the marbles. We're in free fall, so the "center" marble floats motionless. The "down" marble is going to float further away from the center marble (F + Something pulls "down" harder), and the "up" marble will also float further away from the center marble (F - Blah pulls "up" less than everything else). The "left" and "right" marbles float towards the center.
Using only Newton's laws, this reveals that gravity's behavior over a volume has a stretching effect in the up-down direction, and a squeezing effect in the left-right direction. GR confirms this. This effect is the tidal force.
When applied to the Earth and the Moon, this explains the two bulges. The bulge closer to the Moon is due to F + Something, because that water is being pulled stronger. The bulge away from the Moon is due to F - Blah, because that water is being pulled less than everything else.
It can also be described via a conservation law. If the Earth were frozen solid, and the Moon+Earth system started up to something like we have now, then they would be in a stable orbit.
Next, thaw the water. If only one lobe forms then there's a violation of conservation of angular momentum.
To really see the problem, use the planet "H2O", made entirely of ice, instead of Earth. Once the ice thaws, which water decides to form the single lobe? If all of it flows towards the Moon then it wouldn't be a stable orbit.
There can't be zero lobes because the system is asymmetric around the barycenter. If everything is in equilibrium then the result needs to be axially symmetric, giving two asymmetric lobes; one towards and one away from the Moon. (The asymmetry is due to the size of the Earth vs. the distance to the Moon. It's about 60 radii away, so not that asymmetric, but the near-side tides will be higher. As another example, a micro black hole orbiting very close to the water surface will only have noticeable local effects.)
Of course, in real life the system isn't in equilibrium. The Earth isn't tidally locked to the Moon, the orbits aren't perfectly circular, the bodies wobble and aren't spherical, etc. and the sea isn't deep enough for this lobe wave to travel, so it isn't true that there are two bulges. http://physics.stackexchange.com/a/121858 gives a lovely explanation and http://www.lhup.edu/~DSimanek/scenario/tides.htm works through many "Tidal Misconceptions".
As a result, some places have more than two tides, like http://www.southamptonvts.co.uk/live_information/tides_and_w... , with High Water, Second Tide, and Low Water.
P.S. I hope I got that correct! The last time I did this sort of analysis was 20 years ago. And personally, my intuition is based on the stretching force interpretation. One of the mechanics problems I worked out then was to explain why tidal forces radially align the long axis of a spacecraft.
This also leads to the problem, were an astronaut to dive into a black hole to see what's inside, he would be stretched out in the direction of his fall.
