running without alternator?

As long as it's a short amount of time running with no water pump, no. Not a problem. It's normal for the thermostat to be closed at cold startup, which is pretty much the same effect as having no water pump at all until the engine warms up enough to open the thermostat. For some engines, that can be several minutes - Plenty of time to decide if it's the alternator, pump, crank, or something else making the noise.

William R. Watt wrote in rec.autos.tech

Sure, you can do that. The battery will, if decent, provide plenty of electricity to run the engine for a while. However, if run too long on the battery you will not have enough juice to restart the engine. Before you go to that much effort though, just watch the pump run while the engine is running. Is the pulley running smooth and straight? It should be. If the pulley is wobbling then the pump is going out. When you pull the belt, check to make sure the alternator turns freely and smoothly. Then rotate the water pump by hand to see if it does the same. I would be willing to bet the problem is the pump, but it won't hurt to sure.

problem with that is you can't hold a stick against a rotating pulley. I've tried with a length of hose but still can't tell.

thanks for the all the feedback

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You can probe the bodies of the water pump, alternator, p/s pump, A/C compressor, idler/tensioner bulley bracket. If any of these is the culprit, you will hear a loud difference. I use a long screwdriver.