Since the speed of sound vibrations increases with the density of the medium it travels through how fast would they travel through some ultradense substance like white dwarf star material?
What about black hole material? If one could tap on a black hole without being sucked in or shredded by tidal forces could the vibrations travel through it at the speed of light?
What is the specific formula for determining sound velocity according to density?
The speed of sound in a medium is simply:
c = SQRT(B/[symbol]r[/symbol])
where [symbol]r[/symbol] is the density and B is the bulk modulus (which is basically a measure of the elasticity of the material). So, technically, the speed of sound decreases with increasing density. However, denser materials also tend to be stiffer (and apparently stiffness increases more quickly than density), so the heavier and stiffer materials also tend to have a higher speed of sound. Air, for example, has a speed of sound about 350 m/s, water about 1500 m/s, and steel about 5000 m/s. This progression isn’t always the case, however. As an example, lead is both more dense and less stiff than steel, and has a speed of sound of about 1300 m/s (which is why lead goes thunk and steel goes ting).
As for your actual question about white dwarf star material, I really don’t know. I woud hazard a guess, however, that the speed of sound would be very low, because matter is crushed together, and is both very dense and of low stiffness (because it’s not in a crystalline structure with rigid bonds between adjacent atoms). Unless maybe the atomic forces come into play at that level, forming a rigid substance?
Ahh, I see. So diamond would then have the highest speed out of all ordinary earthly available materials?
A black hole has infinite density, and therefore the (theoretical) speed of sound is zero. But then, the dimensions of this infinite mass are zero, so all bets are off 
It’s for sure right up there, although I don’t know for sure if its the highest. Diamond has a speed of sound of about 12,000 m/s. Beryllium, however, appears to be higher: 12,800 m/s. Also check out speeds of sound for fiber materials, particularly carbon fiber (up to 20,000 m/s!).
In addition, diamond has a Young’s modulus of over 1,000,000 MPa and a density of about 3.5 g/cc (3500 kg/m[sup]3[/sup]); speed of sound should be related to the square root of the ratio of these properties. Take a look at this chart comparing density and stiffness. If you scroll down to the “ceramics” section, you’ll see diamond right at the top of the chart. However, the chart also implies that there are “wood products” with lower stiffness but higher density, and a higher overall ratio. I found some info on the speed of sound in various woods (given in ft/s, not m/s), and these numbers don’t approach that of diamond or beryllium, so I suspect that the high stiffness-to-density ratio doesn’t give the full story for non-isotropic (i.e., fiberous) materials like wood. An observation which, of course, points back to those carbon fiber numbers.
Diamonds are not really all that special in most properties. Yes, they are near the end of the scale in a few measures, but they are not at all the extremum in any of them. A great deal of marketing hype has been made to convince consumers about diamond’s special properties. I applaud zut’s post for the points raised as well as the links given.
Not quite. Most cosmologists no longer believe that the density of a black hole is infinite, not that it has zero spatial extent. See this page:
In other words, for all intents and purposes, whatever lies beyond the Swartzchild radius is the black hole.
Sheesh. That should read “…nor that it has…”
The question of what’s the speed of sound inside a star isn’t merely theoretical these days. There are sound waves bouncing back and forth inside the Sun and these have the consequence that it’s vibrating. Rather in the way geophysicists deduce what’s inside the Earth by following the shock waves from earthquakes, these vibrations can be used to figure out the internals of the Sun. How the waves speed up and slow down as the density varies with depth is an important part of this.
(To avoid getting a kick from angua, I should note that some of her Birmingham colleagues are part of BiSON, another helioseismology project.)
I’m pretty sure that there are no such observations for white dwarfs, but theorists have considered the equivalent phenomenon for the even more extreme case of neutron stars.
The speed of sound in the very early, dense, radiation-dominated Universe was a third the speed of light, and if I recall correctly, it can be even higher than that in a neutron star (but still less than light, of course). Even defining the volume of a black hole as the volume of its event horizon, small black holes still have higher density than neutron stars (larger holes have much lower density; a supermassive black hole like those found in the centers of galaxies can be more like the density of water or air). But the “speed of sound” isn’t well-defined with regards to a black hole, since sound refers to the vibration of particles, and black holes aren’t made up of particles. If you drop a large object into a black hole, or otherwise gravitationally disturb it, you can make it “ring” like a spherical bell, but the characteristic speed of that ringing is the speed of light (which is also the speed of gravity), not a sound speed.