Only an infinitesimal piston does no acoustic work at all

Evidence

Earlier, we said that it would become apparent why even a small piston has a component of its velocity in phase with the force. This can be understood by dividing each small piston into two smaller ones, which shows that the time delay from one to the other, however small, at least leads to a small in-phase component. In this way, only an infinitesimal piston does no work at all, which is in fact the case. — Eric J. Heller, Why You Hear What You Hear- An…, 659

Categorization

• QUESTION: Worth chasing the full derivation of why the in-phase component (the resistive or real part of impedance) appears only when the piston is subdivided into parts with time delays between them.