Amersham rhymes with:
wickersham, wickersham, abraham, allingham, amsterdam, anagram, barthram, bartram, beckham, bellingham, beltram, billingham, birmingham, blakenham, buckingham, burlingham, burningham, buttram, cablegram, callaham, centigram, centram, cockerham, coram, cottingham, cunningham, dagenham, dataram, diagram, diaphragm, dillingham, dithyramb, electrocardiogram, engram, epigram, everingham, fillingham, flimflam, framingham, friedsam, frothingham, gillingham, goddam, goddamn, gooderham, histogram, hologram, ingraham, kilogram, kingham, landingham, laningham, leadingham, logjam, logogram, mailgram, mammogram, microgram, milgram, milligram, monogram, newingham, nottingham, oxfam, pegram, polygram, potsdam, priam, program, programme, reprogram, rockingham, rotterdam, sonogram, stringham, telegram, tungsram, vallandingham, vanlandingham, vanlaningham, westerdam, whittingham, wickerham, wickersham, willingham, winningham, wolfgram, wolfram
Amersham sounds like:
american, americana, americano, amerson, aneurism