Indeed this is not possible for a classical wave. However, in a quantum description there exists a so-called "vacuum state" for which the expectation value of the electric field operator vanishes at all times, but the expectation value of the square of this operator does not. In the classical approximation of quantum optics, the expectation value of the electric field operator corresponds to the amplitude of the classical wave, whereas the expectation value of the square of the electric field operator corresponds to so-called shot (or quantum) noise on the classical wave.
Of course in nature there is no such thing as a "classical wave", so this description has to break down at some point. This is the case for vacuum fluctuations which simply do not have a classical explanation.