2001 Hyundai Tiburon by pass hose

The water pump runs all the time the engine is running. When it's still cold the thermostat is closed so the coolant will heat up. Since it's closed there is nowhere for that coolant to be pumped to by the water pump. The bypass hose allows some coolant to go around the thermostat, and in some cases it can be pumped through the heater core in the dash. That means the heater can start warming you up before the thermostat opens and allows hot coolant to flow through the radiator.

Of all the things that can suddenly spring a leak, the bypass hose has to be one of the least expensive. Some are hard to get to but they typically don't cost very much.