How can I design a circuit that will turn on when a wire is cut?
A simple circuit to do this would use a single transistor, one resistor and a buzzer. Connect two 1.5 volt batteries in series to get 3 volts.  Connect one end of a 10 kilohm resistor to the positive terminal of the batteries and the other end to the base of a general purpose NPN transistor (2N2222, 2N3904, etc.).  Connect the negative end of the battery to the emitter of the transistor.  Connect one wire of the buzzer to the positive end of the battery and the other wire to the collector of the transistor.  If the buzzer has polarity markings, follow them: positive to positive end of battery, negative to transistor collector.  Connect your sensing wire from the base to the emitter of the transistor.  As long as the wire is connected , it will short the base to the emitter and prevent the transistor from turning on.  When it is cut, the batteries will send current to the base of the transistor through the resistor.  This will turn on the transistor which means the collector to emitter voltage will be very small and most of the battery voltage will be across the buzzer which will then turn on.  With the sense wire connected, the batteries only have to provide current through the resistor which will be about 3 volts divided by 10 kilohms or 0.3 ma.  Two AA batteries can provide this much current for hundreds of hours.  If necessary you can use C or D batteries for even greater lifetime.  This circuit is simply and can be easily modified to handle other sound sources if needed.