The circuit is divided in two stages the first stage is built with two 2N4401 transistors and second is built with 555 timer IC. The IC is working in monostable mode here. When the electret microphone detects a sound it will convert it in to electrical signal. This electrical signal is amplified by the transistors and then goes at the trigger pin2 of the 555 IC, hence the output of the IC goes high at pin 3 and activates the LED and relay for a certain time period. After that time period the output of the IC at pin3 goes low hence making the relay and LED deactivated.

The ON time can be increase or decrease by increasing or decreasing the value of 47uF capacitor. The operating voltage of the circuit is 9 to 12 volt DC but you can also operate it with 5 or 6 volts, but make sure to use the same voltage relay too.
 
 

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The figure below shows a project / schematic of a clap switch circuit. The circuit will switch ON the relay and an LED when a loud sound is detected by the microphone for example clap, whistle or any other loud sound. You can connect any DC or AC appliance with the output of the relay to make it switch ON by sound.
Clap Switch Circuit Schematic Diagram