rhotic speech
Definition - The pronouncing of the /r/ sound both before and after vowels. As opposed to non-rhotic speech, which pronounces the /r/ sound only if it is followed by a vowel.
Note: English pronunciation is divided into two main accent groups, the rhotic and non-rhotic, depending on when the /r/ sound is pronounced.

Etymology -
The term derives from the the Greek rho, the letter r (which ultimately derives from the Phoenician roth).

Oxford English Dictionary -
Its first citation is from 1968:
"It was possible to divide respondents into three categories:
A. (non-rhotic) Those who had nonprevocalic r-colouring neither for -er nor for -a;
B. (rhotic) Those who had nonprevocalic r-colouring for -er but not for -a; C. (hyperrhotic)."
(J. C. Wells in Progress Rep. Phonetics Lab. Univ. Coll. London (unpublished) June 56)

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