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The speed of sound depends on the density of the medium in which it is travelling and increases when the density increases. For example, in solids sound travels faster than in liquid and even faster than in gas, and the density is highest in solids, lower in liquids and lowest in gas.

So iron has a density of about $7\,800\ \mathrm{kg/m^3}$, while mercury has $13\,600\ \mathrm{kg/m^3}$, but the speed of sound is $1\,450\ \mathrm{kg/m^3}$ in mercury and $5\,130\ \mathrm{kg/m^3}$ in iron, so mercury has a higher density, but sound travels slower in it. Why is this?

Abhishek Mhatre
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5 Answers5

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The speed of sound in a liquid is given by:

$$ v = \sqrt{\frac{K}{\rho}} $$

where $K$ is the bulk modulus and $\rho$ is the density. The bulk modulus of mercury is $2.85 \times 10^{10}\ \mathrm{Pa}$ and the density is $13534\ \mathrm{kg/m^3}$, so the equation gives $v = 1451\ \mathrm{m/s}$.

The speed of sound in solids is given by:

$$ v = \sqrt{\frac{K + \tfrac{4}{3}G}{\rho}} $$

where $K$ and $G$ are the bulk modulus and shear modulus respectively. The bulk modulus of iron is $1.7 \times 10^{11}\ \mathrm{Pa}$, the shear modulus is $8.2 \times 10^{10}\ \mathrm{Pa}$ and the density is $7874\ \mathrm{kg/m^3}$, so the equation gives $v = 5956\ \mathrm{m/s}$.

You give a slightly different figure for the speed of sound in iron, but the speed does depend on the shape and the figure you give, $5130\ \mathrm{m/s}$, is the speed in a long thin rod. There are more details in the Wikipedia article I've linked.

John Rennie
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    This answer would be improved by giving easy to understand definitions of bulk modulus and shear modulus, and how they relate to/differ in different states of matter. – Patrick M Mar 04 '14 at 19:00
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    I agree with PatrickM. This doesn't tell us why, it just tells us how to compute it. The answer to "why?" is that the speed depends not only on density, but also a mysterious thing call bulk modulus, which represents ... what? – GreenAsJade Mar 05 '14 at 03:13
  • Is it correct that K and G are medium specific constants, which basically have been measured originally? If so, are you basically not saying that the speeds are different because we have measured them to be different? – Thorbjørn Ravn Andersen Mar 05 '14 at 09:14
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    @ThorbjørnRavnAndersen: $K$ is the bulk modulus, which is a measure of how much force is required to compress the material. $G$ is the shear modulus, which is a measure of how much force is required to bend the material. These are characteristics of the solid. – John Rennie Mar 05 '14 at 09:17
  • @JohnRennie so you agree? – Thorbjørn Ravn Andersen Mar 05 '14 at 09:18
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    Re comments on expanding the answer: I didn't expect the question to attract anything like this amount of attention. I'm not sure expanding this answer to a full description of the theory of sound propagation is appropriate, but if anyone wants to ask a related question I'd be happy to answer. Please read the Wikipedia link first though, and try to make any new question specific rather than a generic tell me how it works. – John Rennie Mar 05 '14 at 09:19
  • @ThorbjørnRavnAndersen: we regard the moduli as fundamental because they are the same for all lumps of iron (larger than a few thousand atoms) and they are directly related to interatomic forces. By contrast the velocity depends on the size and shape of the piece of iron. In my answer I referred to the fact that a long thin rod gives a different speed of sound to a more symmetric piece. – John Rennie Mar 05 '14 at 09:32
  • @JohnRennie I suggested an edit after attempting to understand it myself. Feel free to correct any inaccuracies or revert it if you still don't think this is the correct place to answer 'why'? – Patrick M Mar 05 '14 at 17:26
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    @PatrickM: thanks :-) However I feel that if we were going to extend the answer (and I'm not sure we should) then a more mathematical treatment would be better. – John Rennie Mar 05 '14 at 17:47
  • @JohnRennie I would have to leave any deeper math to you, as I am but a layman. I did post my thoughts as a separate answer; hopefully they provide some understanding to others such as myself. – Patrick M Mar 05 '14 at 18:51
  • @JohnRennie _ is the bulk modulus, which is a measure of how much force is required to compress the material_ Not exactly. Bulk modulus is defined as $$ K=-V{\frac {dP}{dV}} $$, so it measures materials resistance to compression - how much it's pressure increases when volume is decreased. What you are talking about is inverse of bulk modulus, aka "compressibility" - change in volume under applied pressure. And no, $K$ applies not only to solids, but to fluids and gasses as well, because almost everything is compressible – Agnius Vasiliauskas Sep 19 '19 at 06:29
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John Rennie has provided an exact mathematical treatment of the equations behind the calculation of the speed of sound. I don't want to detract from that treatment, and of course the Wikipedia articles we both draw from provide a broader treatment; but an intuitive understanding of the 'why' has been equally helpful for me, in the past. The following is my attempt to comprehend and explain the 'why' to the question.

There are more factors affecting the speed of sound in a substance other than the density of the medium. This is reflected in the equations for determining the speed of sound, most notably the presence of the bulk modulus and shear modulus in different places in the equations for sound in a solid and a liquid.

The bulk modulus is a measure of a substance's resistance to uniform compression. It is measured in pascals, which is the same unit for pressure. Uniform compression means the substance is experiencing equal pressure in all directions (as in atmospheric or underwater pressure). Thus, the bulk modulus tells you how much the substance will shrink — that is, decrease in volume and increase in density — when subject to a given pressure.

Now the shear modulus is a measure of stiffness. Specifically, it measures how a material responds to forces acting in opposite directions, as in friction holding a block in place or moving your hands away from each other to tear a piece of paper in half. Imagine trying to subject a liquid or a gas to a shearing force and it becomes clear that the shear modulus is meaningless for forms of matter other than solids. Simply put, gasses and liquids don't resist shearing forces.

For that reason, the shear modulus factors into the speed of sound in a solid, but not into the speed of sound in a liquid. Wikipedia summarizes this on the speed of sound section linked to above as:

In a fluid the only non-zero stiffness is to volumetric deformation (a fluid does not sustain shear forces).

In a more general sense, different mediums have different responses to different forces. Wave propagation is essentially a transfer of energy through a medium — that energy transfer is accomplished by a compression force at the molecular level.

For a macroscopic metaphor, imagine propagating a wave through a slinky. That slinky looks to be made of steel, but it is still very flexible, which is to say it is not stiff. Imagine repeating the experiment in that video with a solid steel bar of the same width and length. Assuming the students could still move the mass with the same vigor, you would not be able to observe either a transverse or a longitudinal wave motion. The solid steel, despite being the same material with the same density as the slinky, is much stiffer, and therefore propagates waves very differently.

Similarly, the difference in the stiffness between liquid mercury and solid iron are enough to overcome the greater density of the mercury to make sound propagate faster in the iron.

Patrick M
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"The speed of sound is variable and depends on the properties of the substance through which the wave is traveling. In solids, the speed of transverse (or shear) waves depend on the shear deformation under shear stress (called the shear modulus), and the density of the medium. Longitudinal (or compression) waves in solids depend on the same two factors with the addition of a dependence on compressibility."

Speed of sound is function of more than just density. Shear modulus for iron is 82 GPa, I haven't found any data on the modulus of Mercury, but certainly it differs and that's the most probable reason.

Waffle's Crazy Peanut
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Wojciech
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The square of the sound velocity is proportional to the ratio of an elastic modulus to the mass density of the material.The reason why the sound velocity is usually larger in solids than in liquids and usually larger in liquids than in gases is because of the elastics constants of the material.

What determines the elastic constants of a material is the interatomic bond strength. The stronger the bond, the higher the elastic constants. In liquids, the atoms are weakly bonded together than solids and the elastic constants are low. In solids, the atoms are more tightly bonded together, and the elastic constants are higher.

Vaishakh Rajan K
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Your question should more accurately have been, "Why does sound travel faster in solid iron than in liquid mercury even though mercury has higher density?"

Were the question phrased that way, the answer would be more obvious. At temperatures at which both metals are liquid or both metals are solid, sound travels faster in the denser metal.

Bram
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    "sound travels faster in the denser metal" The point is that mercury is denser than iron at STP. There are other factors which play a part, in this case the elastic modulus or compressibility. – dmckee --- ex-moderator kitten Mar 04 '14 at 15:49