From what I understand (and I could be wrong,) the "frog mouth" helm was originally a German style, known as the "Stechhelm" and used for a type of tournament called the "Stechzeug" which was a joust in an open field with sharp lances. It was the most protective and heaviest type of helmet and was bolted directly to the cuirass with a long metal bracket. I'm not sure when it spread to the rest of Europe, and how common it was, but I have definitely seen drawings of Henry VIII at the tournament with knights wearing frog-mouth helms. I'm under the impression that the frog-mouth was discarded in favor of the close-helm sometime in the late 1500s.

However it lives on - in heraldry. It seems extremely common on top of coats of arms. Here for instance is the Barbados coat of arms:

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Here is the Rhodesian coat of arms:

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The Knapp coat of arms, which features the helms not only in the crest but on the shield itself:

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And so on and so on. I've seen many coats of arms topped with close-helms and armets as well, but the frogmouth helm seems the most common. How did such a relatively specialized kind of helmet become so widely-used in heraldry?