I read in sci.electronics.design that abekas67 wrote (in ) about 'How would I build a circuit to generate 4VAC?', on Sat, 27 Aug 2005:
The filament resistance varies with temperature and it's the 1 A that matters, not the 4 V. At switch-on, you need a lot less than 4 V, as you know.
The traditional method of doing what you want is to use a negative temperature coefficient (NTC) thermistor in series with the filament. This is extremely simple and works with an AC supply. Other methods of producing a time-varying AC supply can be complicated.
If you can't find an NTC thermistor that is suitable (cold resistance just under 4 ohms and hot resistance around 0.1 ohm or less, rated current 1 A or a little more), consider using an audio power IC, fed with a 50 Hz signal through a simple voltage-controlled amplifier (even a simple shunt FET attenuator) with a ramp control signal of the duration you require. Although not elegant (not Bloggsian), this is simple and uses no exotic parts.