My gutter runs along the bottom of the roof and not around the bay window. Because of this there is very little sound insulation between the gutter and inside the bedroom and so we can hear water every time it rains! So annoying. Only options recommended to me at this point really have been to insulate the ceiling which sounds like a big job, but I am thinking of trying to get some thick string to run along the base of the roof and inside the gutter to slow the flow of water. Does anyone think this sounds like a crazy idea? Of even know how I would attach string to my roof and gutter please?
The sound should be the water entering the downspout's top, but could also transmit from the water hitting the very bottom elbow at the ground. One solution is to remove a few gutter nails or screws back from the window & install rubber bushings or pads to insulate or isolate the gutter's connection to the house. Then, of course put your gutter screws or nails back in...a little loose will assist the padding.
A Second measure, is to back the gutter & downspout, give them more density. Like a stainless steel kitchen sink's noise reduction pad(s). As an example is Grace's Vycor Plus Ice & Water Shield self-adhesive flashing. The 4" wide by 75' long for $20 at home improvement stores should accomplish this quite well. You can build it up to dramatically deaden or eliminate that sound & then paint it to match.
I think you might find the majority of the noise comes from the downspout not the gutter. Also, once the heavy string or rope becomes soaked it will no longer slow anything as the water coming in will force water out at the same rate. The other problems would be how to fasten it in place without penetrating the roof, and how fast will it rot being outside.
A cheaper / easier solution might be to remove the gutter and downspout entirely and put stone under the eave to prevent splash-up.
If the sound you're complaining about is simply the sound of water dripping the 3 inches between the roof edge flashing and the bottom of the gutter, the solution may be to install a sloped flashing to allow the water to flow into the gutter rather than drop.
A flexible plastic or slightly angled metal flashing slid under the shingles and left in contact with the side or bottom of the gutter would do this.
You should confirm whether it's the downspout you are hearing, or the water entering the gutter. One of you can, carefully, use a ladder and a hose to see if the noise happens just when water enters the downspout, or as it flows along the gutter.
If the noise is from water entering the gutter, there are several foam inserts you can get. These are designed to prevent leaves clogging the gutter, but they may also slow the flow and prevent any "pinging" noises.