hi, I used to live in london as well as in bournemouth so here you go:
you can take a train and when you book in a week advance it cost around 20 pounds return, but you need to provide a return date as well,
you can take a bus, from victoria coach station,www.nationalexpress.com one way its 16 pounds, no need to book but when you do so online you can find "funny fare" and it might be even for 3 pounds.
last way is to go on www.megabus.com and that is really very chea, can be even 1 pound. Try! aha, train takes 1,5 hour to waterloo, National express coach goes in 3 hours and magabus more than 3 hours.
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