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No they are not expected to wear bow ties! A shirt and tie will suffice - perhaps their school blazers?? but jacket is not really required for children.
The boys can wear a smart suit as my son did. Lots of the men do as well.If the boys want to they can hire a dinner jacket etc. on board. They would love the Johnny Rockets Diner for a change.
My son was 14 when we last sailed on navigator and he just wore casual smart trousers and an open necked shirt-however he is looking forward to wearing his dinner suit this year which we bought him for his school pro.