“Thames” is pronounced as “temz” to rhyme with “shemz”, unless you mean the one in Connecticut, which is pronounced “Thames”. “River” in both instances is just “river”.”Thames” is pronounced as “temz” in England, Canada and new Zealand.
It is believed that 1st Havoverian Monarch (King George 1st) had a thick German accent and couldn’t pronounce ‘th’, so he called it the river ‘Temmes”. Nobody wanted to tell the King he was wrong so in deference to him, everyone called it the ‘Temmes” and it has been ever since. (Source: 84, Charing Cross Road by Helene Hanff, Page 160.)
Another source says that the modern spelling of the word Thames illustrates an interesting phenomenon in the history of the English language. The Thames is first mentioned in English around 893 in King Alfred the Great’s Orosius. At the time it was called the Temese, a form believed to come from an earlier, unrecorded English *Tamisa. The spelling Thames, which first appears in 1649, is an example of the kind of “learned” respelling that went on in English from the late Renaissance through the Enlightenment, when the prestige of Latin and Greek prompted scholars to “correct” the form of many English words.
The a in Thames is etymologically correct, since the Latin forms had that vowel, but the h is a “learned” error, added in the mistaken belief that Thames derived from Greek. Such errors were common, and many words that had nothing to do with Greek were respelled to make them look Greek; two other examples are author (ultimately from Latin auctor) and Anthony (from Latin Antonius), with the h added as if these were based on Greek words with a theta (th) in them. In many cases, the pronunciations of these words changed accordingly, yielding what linguists call a spelling pronunciation; author is now pronounced with a (th). The pronunciation of Thames remained unchanged, however, providing an etymologically explicable example of the notorious discrepancy between English spelling and pronunciation.
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