Drought and drouth, nouns derived from the adjective dry plus a suffix, are spellings that represent two phonetic developments of the same Old English word, and are pronounced

[drout] /draʊt/ and

[drouth] /draʊθ/ respectively. The latter pronunciation, therefore, is not a mispronunciation of drought. The now unproductive suffix -th1 and its alternate form -t were formerly used to derive nouns from adjectives or verbs, resulting in such pairs as drouth — drought from dry and highth—height (the former now obsolete) from high.
In American English, drought with the pronunciation

[drout] /draʊt/ is common everywhere in educated speech, and is the usual printed form.

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