How can em waves be continuous?

Question

Suppose we have a charge that is undergoing simple harmonic motion (an oscillating motion).

Since it is an accelerating motion too, the charge radiates em waves. But here's the interesting part :- when the charge reaches its mean position, it's acceleration is zero and thus it should not radiate any energy. This means that the charge is not radiating energy continuously but in the form of small aggregates ("pockets") which is not what is predicted in em theory (or at least what is written in my physics textbook about the classical em theory).

The em theory predicts that an accelerating charge would produce em waves continuously (and not in the form of photons) but the above scenario is completely opposite of it .

Doesn't it indicate a kind of contradiction to the classical theory and is in favour of what Einstein predicted ?

Edit :- While reading about the photoelectric effect, I read that the effect couldn't be explained by the em theory but when Einstein predicted that the energy comes in the form of so called "packets" , the theory and observation matches.

Now the case I considered ensures that there is no energy radiated when the charge is at its mean position. So doesn't it imply that energy indeed come in the form of "packets" ?

Answer 1

A continuous function can be zero at some point without becoming discontinuous. So the fact that the field generated by the particle becomes zero just for an instant doesn't contradict the statement.

Any harmonic field oscillates and becomes zero once every half-period.

Following Larmor's formula to compute the mean power radiated by the particle, what matters is the average of the squared acceleration, so the fact that this acceleration becomes zero from time to time is largely irrelevant.

Answer 2

$$\sin(\omega t) = 0$$

When $$t=0$$

Which would correspond to the zero of acceleration.

The poynting vector has similar properties, there are points in space where the power radiated is zero