The verb wax, meaning to grow, has only a few surviving uses in English. The moon waxes and wanes. And people wax . . . some adjective, usually describing their gestures or their speech. Note: adjective, not adverb. Its often misused. If John is waxing eloquently, maybe he is reciting the Gettysburg Address while polishing his Camaro. If Mary is waxing poetically, maybe she is reciting Hamlets soliloquys while polishing the coffee table. That is, shes waxing the table, and shes being poetic about it. But wax, meaning to grow, always takes an adjective, just as grow does. You can grow angry. You cant grow angrily. That makes no sense. Therefore you can wax poetic, or wax lugubrious, or wax nostalgic, or wax suspicious.
The word is very old indeed. It was our original general word for growth, of plants, children, animals, nations, whatever. It is cognate with German wachsen, to grow. It used to be a strong verb, toothe German linguists term for verbs that dont fool around with piddly dental past tenses, but change their own darned vowels, take that! The Anglo Saxon weaxan had the past weox: he weox eald, he grew old; cf. German er wuchs, he grew. Does it have any relations in Italy and Greece? Quite a few. The old Romans used the letter v for the sound we denote by a w: so that Caesars boast, Veni, vidi, vici, might still at that time have been pronounced wainy, weedy, weeky, which does not actually impress the ear with power; and that may explain why he had to cross the Rubicon. In any case, we look for Latin cousins of our w words among their v words. And sure enough, we have vigere, to thrive; and the kinfolk in that clan, from which we derive Modern English vegetable and vigor. So if a man waxes vigorous, we can say he waxes waxy: a heck of a lot of growing going on there.