Release the 1/2-inch clips holding the spout to the wall with a flat-head screwdriver. Remove the clips from a point approximately 5-feet high to the bottom-most clip. Replace these with longer 1-inch clips so that the spout angles out slightly at the midpoint and down to the bottom. This increased angle allows the water to rush down the side of the tube instead of falling to the elbow joint below.
Purchase a gutter noise guard from a home center or hardware store and install it into the inside of the angle joint piece. This is a foam material in the shape of a ramp with magnets on the back to hold it in place even under the force of water rushing over it. The angle and the soft surface eliminate most of the noise.
Remove the angle joint of the downspout and replace it with flexible corrugated tubing that you can adjust the angle of to eliminate the noise. These allow for reduced noise while allowing you to direct the water out away from the house in any direction desired.
Dig a trench with a shovel 1 and 1/2 feet deep near the base of the downspout and angled downward slightly to a location further away from the house. Run the downspout into the trench and feed the end into corrugated tubing. Crimp the end of the tube to the downspout pipe with a metal strap by tightening the screw in the strap.
Lay the tubing down the trench away from the house. Cover the trench with soil and sod to hide the redirected pipe. The ground insulates the water running through to eliminate noise.