Steam engines are quite quiet...the loudest part of them
is actually the blowers that feed air to the boilers. If
you're amidships you can sometimes hear the air rushing
into the vents or you might hear some sound as it pushes
back up out of the stack; I wouldn't expect to hear
same at the taffrail.

There, you'd hear the wake. If the vessel was lightly
loaded, you might hear the screw splashing about in
the water. Or, if it were stormy you'd hear the screw
come up out of the water (and feel a shudder) then
silence as it went back in.

In those days, vessels often streamed a 'patent log'
which consisted of a torpedo-shaped impeller (think
giant bass lure) attached to a cable. That runs into
a gear-box sort of affair which clicks off nautical
miles and tenths. It will whir and click...the log
line will hum and thrum in the wind.

Don't also, forget the flapping of the ensign flying
on the stern, and the higher frequency sound of
signal flags.

If there's an old-fashined wheel-house astern (as there
would be on vessels before the advent of the midships
bridge and the steering engine) you'll also hear any
chatter from there; this is very limited, as such
places were to be kept 'sterile' and free from
distractions. But orders and responses might come
via speaking tube between the officer of the watch
and the engine room.

Bell time would have been rung each half-hour, both
on a ship-strike clock in the wheelhouse and on the
ship's bell (usually forward).

If it's the first three days out, you might enjoy the
sound of the first mate returning his supper of codfish
to the sea.

Note that one difference between ships with steam and
those without is that when ships steam into the wind
(and steam-sail ships would frequently do so) they
increase the apparent wind by their forward speed.
If you steam at 10 knots into a 25-knot wind (probably
optimistic on an underpowered inefficient old ship with
a full ship rig and lots of windage) you get 35 knots.

And that will set up quite a howl in the rig and bare
shrouds.