There are lots of place-names ending in -mouth. If the part before this ending has more than one syllable, then you can pretty much guarantee that the pronunciation of the ending is maʊθ. Here are some examples:
Avonmouth = ˈeɪvənmaʊθ
Cockermouth = ˈkɒkəmaʊθ
Oystermouth = ˈɔɪstəmaʊθ
If, however, there is only one syllable before the ending, then most of the time (but not always!) the pronunciation is məθ. Examples:
Bournemouth Dartmouth Falmouth Portsmouth Sidmouth
For names in Scotland the maʊθ pronunciation is used for all names, no matter how many syllables they have.