I'm surprised the dripping is that loud but there is a simple solution. Drop a some rope, such as heavy sash cord, down the drainpipe and attach top end in such a way that at least a couple of inches of the rope is sitting inside the gutter itself. The way this works is that the water will tend to run along the rope, which should ensure the water runs along the wall of the drainpipe instead of falling through the air.
Bleeding radiators is quite simple, and can usually be done by homeowners. All that's required is a radiator key, a towel and/or bowl, and a bit of time.
Why Bleed Radiators
Even with closed systems, air can still find its way into the system. When it does, it will collect at the highest points available (the top of radiators). Trapped air can cause radiators to heat less effectively, so removing it will help reduce your heating costs.
How to Bleed Radiators
To bleed the radiator, you'll need a special tool called a "radiator key". A radiator key is a small key like object that will fit on the bleed valve of the radiators, and is used to open/closed the valve.
![enter image description here](https://i.stack.imgur.com/pTyiIs.jpg)
The first step in bleeding the radiators, is to turn off the system. Next, you'll have to locate the bleed valve on each radiator. This valve should be near the top of the radiator, and will look something like this.
![enter image description here](https://i.stack.imgur.com/cNvfim.png)
Insert the key into the valve, and hold the towel and/or bowl below the valve. Slowly turn the key anticlockwise, until you hear a hissing noise or water starts to drip from the valve. Once water starts to drip, close the valve by turning the key clockwise.
![enter image description here](https://i.stack.imgur.com/C9NM8.jpg)
WARNING: Never open the bleed valve too fast, too far, or completely remove the valve stem.
CAUTION: Water dripping from bleed valve may be hot. Contact with aforementioned water could lead to injury.
Repeat this procedure for each radiator in the system.
Repressurize the System
After bleeding the radiators (or even before bleeding the radiators), you may notice the pressure in the system is too low. The typical rule of thumb is that for a 2 story home, the pressure should be between 12-15 lbs. or 25 - 30 ft. If you notice the pressure is a bit low, you'll want to top the system off.
There should be a supply pipe connected to the boiler, with a valve that is fully closed. This pipe should come from the cold water supply system of the house. Once you've located the supply and valve. Open the valve slowly, keeping an eye on the gauges, close the valve again when the gauge reaches the optimal level.
Best Answer
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.