# Why does sound move faster in solids?

I know that the molecules are closer together in solids, and I know thicker springs also respond carry waves faster than thinner springs, but for some reasons I can't understand why. The molecules will have a larger distance to move before colliding with another molecule, but in a thicker medium wouldn't that time just be spent relaying the message between multiple atoms? Why is the relaying between a lot of tight knit atoms faster than one molecule moving a farther distance and colliding with another?

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Sound moving through solids is kind of like the falling domino effect. Would the energy pass quicker through dominoes that are more closely spaced or dominoes that are spaced farther apart? It seems to me there is that little bit of time that it takes as one domino is falling to hit the next one and cause that one to fall - which is shortened when the dominoes are closer together. Thus I predict that the energy would travel faster as the spacing between dominoes (AKA molecules) is decreased. – Greg May 23 '13 at 1:53
But aren't there more dominoes to be hit? – user24082 May 23 '13 at 2:09
Here's a way to think of it. If I shove my open palm towards you, will you feel it more quickly if: a) there is water between us, and a wave ripples towards you and eventually splashes you, or b) if there is a solid rod between us, and it smacks into you? – mbeckish May 23 '13 at 2:58

I assume "faster in solids" means faster than in gases.

The speed of a mechanical wave is in general proportional to $\sqrt{k/m}$, where $k$ is some measure of the restoring force (e.g., the tension in a string, or a Young's modulus), and $m$ is some measure of inertia (e.g., the mass per unit length of a string, or the density of the medium).

Compared to a gas, a solid has a density that is greater by about a factor of $10^3$. However, gases are very compressible, while solids and liquids are extremely incompressible. The incompressibility factor overwhelms the inertial factor.

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The difference between solids and gases appears in the momentum conservation equation: $\rho\frac{d\vec v}{dt}=\vec S$ where $\vec S$ is a source term that expresses the rate at which momentum is exchanged between neighboring volumes, a "restoring force".

In gases, $\vec S=-\vec \nabla p$, where the pressure $p$ relates to density and temperature through equations of state, which describe the interaction between gas molecules. In statistical physics, this term is written as a collision operator. It means that the transmission of the disturbance, or the restoring force, is driven by collisions between molecules. These collisions are rare in average, so each molecule travels a long way before transferring its momentum.

In solids, $\vec S$ is the electrostatic force on an atom that is displaced from its equilibrium position. A very small displacement is enough to imply a large force because of the proximity of other atoms repelling each other. Also, the distance at which atoms "feel" each other is generally higher in solids than in gases (charge interaction instead of dipole interaction). Overall, atoms in solids don't need to travel long before they transmit their momentum.

That's why the disturbance propagation is faster in solids than gases.

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Density also matters, not just the restoring force. – Ben Crowell May 24 '13 at 4:11
@BenCrowell Well the density is accounted for in the restoring force as it is a sum of all individual pushes on a given fluid volume. What effect of density are you referring to? – fffred May 24 '13 at 6:53
I mean the common definition of density, mass density. The effect you've ignored is inertia. – Ben Crowell May 25 '13 at 19:07

Think of it this way. Elasticity is a property of material that allows it to store energy and release it without dissipating. Solids have high elasticity, therefore, they can store and release energy quite efficiently. Liquids and gases have low elasticity. They are also viscous and dissipate energy instead of transmitting it. Please note that I am not connecting speed of waves to viscosity but merely pointing out its dissipative nature.

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Viscosity and dissipation don't related to the speed. – Ben Crowell May 23 '13 at 14:58
@Ben, I agree. In fact, I said that in last sentence of my answer. – Amey Joshi May 23 '13 at 15:21