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Is the English "Woe" and the German "Wo" related?

I just heard a colleague say, "Wo ist mein ..." and I thought of the band Woe is me.

Are these words just false cognates... or is there some common root?

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    There are innumerable false cognates like that. Especially when you have short words, the chances of two words whose meanings have superficially nothing to do with each other actually having nothing to do with each other are rather high. – LjL Dec 11 '18 at 16:15
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In addition to Wilson's comment:

The English /oː/[oʊ̯] in indigenous words is derived from a Proto-Germanic diphthong that is reconstructed as pgm. *ai (pgm. *ai > Old English ā > modern /oː/). The same diphthong is the root of German /aɪ̯/(usually spelled ⟨ei⟩) and /eː/ (can be spelled ⟨eh⟩). That's why German Stein and English stone are cognates as well as weh and woe.

Where and wo are not as easy to explain. By also considering Dutch waar you can reconstruct a Proto Germanic *hwē¹r > West-Germanic *hwār.
hwār must have been palatalized in English. In German a long /aː/ became /o:/ in some dialects. This seems to be a problematic case overall, since there is a short vowel in the Norse cognate. See also https://www.dwds.de/wb/wo

  • This explains a lot of cases here English has o and German has e, as in the OP's question. – Wilson Dec 11 '18 at 10:40
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They have nothing to do with eachother, except that they kind of sound similar. But consider similar words with similar meanings!

English Woe => German Weh

German Wo => English Where

These two are pairs of words which are historically related.

As to the phrase "Woe is me", one German translation could be "Weh mir". And in the closely related Yiddish, "vey iz mir" is sometimes heard.

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