Why should the time constant of an RC circuit for demodulation purpose be higher than the time period of the carrier wave?

Modulation is defined as the process by which some characteristics of a carrier signal is varied in accordance with a modulating signal. The base band signal is referred to as the modulating signal and the output of the modulation process is called as the modulation signal.
Amplitude modulation is defined as the process in which is the amplitude of the carrier waveis varied about a means values linearly with the base band signal. The envelope of the modulating wave has the same shape as the base band signal.

After receiving an AM signal, it can be demodulated to recover the low frequency signal. One of the simplest types of AM demodulating circuits is the envelope detector. In order to accurately recover the low frequency signal the envelope detector must satisfy an important condition; the time constant of the envelope detector network must be much longer than the period of the high frequency signal but much shorter than the period of the low frequency signal. Once the signal is demodulated, the high frequency signal iseliminated and what remains is the low frequency component. The general idea of the envelope detection method to regain the low frequency signal.

It is composed of a resistor, capacitor, and diode. The time constant set by the values of the resistor and capacitor needs to be much less than the period of the audio signal but much greater than the period of the RF (high frequency) signal. From the observation, we conclude that, the time constant of an RC circuit for demodulation purpose be higher than the time period of the carrier wave.