Why do magnets silence anvils? Today I've been to a forge. The blacksmith demonstrated that if he beats the anvil without a magnet attached, it make horrible noises. As soon as he reattached it, the sound of the anvil was way more pleasant.  
The anvil had a weight of 250 kg,  the magnet had the size of a mozzarella.  
What is causing this? 
 A: Ok, reading up about it, here's what I think is happening. Just a guess.
I think when an anvil rings, there are parts of the anvil that vibrate vigorously, and parts that are self-damping. Or perhaps there's a surface standing wave, some areas with nodes and some areas with antinodes. In any case, the vibrations happen in specific areas of the anvil. By putting a magnet in a certain place, you can strategically damp the vibrations.
There's an easy way to test this theory: move the magnet around and see what happens! And if it can be done safely, place your hand on different areas of the anvil, and feel if some areas have stronger vibrations.
A: Blacksmith Forum
This answer is just to illustrate how one blacksmith reduced the noise level without resorting to magnets or weights. Over time, and literally being hammered every day, even a well tied down anvil must loosen it's bolts and  restraining materials and start to vibrate and produce the noise mentioned.

The more mass involved, the less vibration and noise. Materials that absorb or change the frequency of vibration help also. I am a firm believer in bolting the anvil down tight and putting something in-between the anvil and stand. Many on the forum have had good luck using industrial adhesive or silicone caulk, you know, stuff that comes in a tube and you apply with a caulking gun. A good layer of that on the bottom of the anvil, set it in place, bolt it down good, and voila! I lucked into a roll of some kind of rubber underlayment for cork flooring tiles that some workers were throwing away after a job. It is about 5mm thick. I just cut a piece to the shape of my anvils base and put it in-between before bolting the anvil down. The type of stand that is a big box full of sand with the anvil bolted to a plate setting on top of the sand works pretty well to deaden anvil noise. Just make sure to put a layer of something between the anvil and the plate.

A: I did an anvil ring noise suppression test using a couple small magnets, and the results were surprising.  I made a YouTube video of it:  https://www.youtube.com/watch?v=mAWk3SONnw8
A: Researchers at the Ohio State University have shown that heat can be controlled with a magnetic field, and that it should also be possible to affect sound waves with a magnetic field.  Apparently phonons, the quasi-particles associated with compressional waves and vibrations in a lattice, have magnetic properties.  Here is an account of the study: https://news.osu.edu/news/2015/03/23/heatmag/.
Essentially, the vibrations of the thinner parts of an anvil are like heat, the vibration of atoms and molecules.  Acoustic waves are also like heat.  They cause materials to vibrate, and they are produced by vibrating material.
The Ohio State researchers were able to control heat flowing through diamagnetic conventionally non-magnetic material.  The study "said that the magnetic field caused some of the phonons passing through the material to vibrate out of sync so that they bumped into one another…" 
I suspect that ferromagnetic material such as an anvil would be particularly susceptible to magnetic damping of internal vibrations, particularly if the magnet is large and placed on the thinner parts of the anvil.
