Description
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Since I've moved into this house, I've noticed that the first shower of the day is colder than the any subsequent shower. More specifically, if no hot water has been used for a while, the water becomes more *luke warm* than hot. However, the maximum temperature of the water after it *recovers* from this issue is plenty hot.

To work around this problem, my wife and I use my dishwasher's timer feature and run it over night. This usage of hot water in the middle of the night helps the water be a more reasonable temperature in the morning. But we don't always have a load of dishes to do each night...

I've got some theories, and they're mostly related to the thermostat and/or it's position. I keep the temperature between the A and B settings (I'm in the US, if that matters). The water heater is only about 7 years old and I have flushed it recently. There was no difference after flushing it.

My Theories
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1. The thermostat is malfunctioning. The thermostat most likely has a high temperature shutoff (where it turns the heat off once it achieves the desired temperature) and a low temperature turn on (where it turns the heat on once the water gets too cold). One theory is that the slow cooling of the water over time isn't enough to trip the lower threshold on the thermostat. However, the cold water coming in from the main is enough to *shock* the thermostat and turn it on.
2. My second theory is that the thermostat is at the bottom of the hot water heater (I'm pretty sure it is) and the pilot light is keeping the water in that area hot enough that it doesn't kick on the heat. Since I keep the temperature at a moderate setting (I have young children), I'm thinking that the water at the bottom of the tank might never get cold enough to kick on the heat because of the proximity to the pilot light. However, when the tank starts to drain and the cold water from the main comes in, it does trip the thermostat.

Any idea if either of these theories might be right? How can I fix it? Do I just need a new water heater? Will that even help?