There are "arrows" which travel at supersonic speeds, but they are not fired by bows, but by large calibre tank cannons. They are technically knowns as APDSFS rounds (Armour Piercing Discarding Sabot Fin Stabilized)
The "petals" of the sabot serve to keep the round centred in the barrel during firing, and also fill the barrel so the propellant gasses don't flow past the round, providing the greatest energy transfer to the projectile. Once it clear the barrel, the sabot pets peel away, leaving the "dart" to carry on to the target. Soviet and Russian APDSFS rounds have similar properties, but use a different form of sabot:
Russian Bm15 round for a 125mm tank cannon
The specific shape of the APDSFS penetrator is related to factors such as cross sectional density, the need to maintain velocity downrange and the ability to penetrate armour, especially composite armours and various forms of spaced armours. The long, thin "dart" shape provides low drag (preventing the velocity from dropping too much with range), and concentrates the energy in a very tiny area on impact, improving the ability to penetrate armour. Extra fun can be had from using high density depleted uranium, as the material becomes "pyrophoric", spontaneously igniting in the presence of oxygen once it passes through the armour (the frictional heat provides the energy for ignition).
Now you have not mentioned what, exactly a supersonic arrow is needed for, but it would have several of the properties of an APDSFS penetrator. The dart shape for low drag and armour penetration is a must. If you are actually using a bowstring and improbably large bow to provide the launching energy, a stiff metal dart will have less tendency to flex like a wooden arrow or quarrel, allowing for greater energy transfer to the projectile.
Flexing of an arrow when launched