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In my book it's written that speed of sound will in increase with increase in density of the medium as molecules with get closer to each other, but after some browsing on internet I found out about Laplace's formula which states that speed of sound in a medium is inversely proportional to density of the medium?Which of these is correct and why?

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    $\begingroup$ What medium? Gases or solids? $\endgroup$ May 29, 2020 at 15:23
  • $\begingroup$ Have a look at these values: aplustopper.com/speed-of-sound-in-various-substances $\endgroup$ May 29, 2020 at 15:26
  • $\begingroup$ @JohnRennie can you pls explain this for all the mediums $\endgroup$ May 29, 2020 at 15:51
  • $\begingroup$ None of your statements are valid. The speed of sound goes like the inverse of the square root of density of the medium. $\endgroup$
    – nasu
    Jan 20, 2021 at 3:26

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The speed of a sound wave (a longitudinal wave) in a gas can be shown, using Newton's second law, to be given by $$v=\sqrt{\frac{\gamma p}{\rho}}$$ in which $\gamma =\frac{\text {molar heat capacity at constant pressure}}{\text{molar heat capacity at constant volume}}$

$\gamma =1.4$ for diatomic molecules such as oxygen or nitrogen.

$p$ is the pressure and $\rho$ is the density.

At first sight you might say that the equation shows the speed of sound in (let's say) a diatomic gas to be inversely proportional to the square root of the density. This would be true if we had a cylinder full of nitrogen (relative molecular mass 28), and then replaced the nitrogen with oxygen (relative molecular mass 32) at the same pressure. The speed of sound in the cylinder would drop by a factor of $\sqrt \frac {28}{32}$.

But suppose that we didn't change the gas, but squashed it into a smaller volume by pushing in a piston very slowly so that the temperature change was negligible. It's easy to see that $$\frac{\text{new density}}{\text{old density}} =\frac{\text{old volume}}{\text{new volume}}$$ and from Boyle's law, that $$\frac{\text{new pressure}}{\text{old pressure}} =\frac{\text{old volume}}{\text{new volume}}$$ So the speed of sound would be unaffected by such a compression!

But we could increase the pressure without changing the density by increasing the temperature of the gas in the cylinder without changing its volume. We can then see from our equation that the speed of sound in the gas will increase. [This makes sense because molecules at a higher temperature have a greater random speed and will on average move faster between collisions, so passing on faster the superimposed velocities that represent the sound.]

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In that answer for elastic field in a solid, you can see that the density is multiplying the time derivative of the displacement: $$\rho\frac{\partial^2 u_x}{\partial t^2}$$ The second derivatives of the displacements with respect to position are in the left side of the equations.

So, the differential equations show that the wave velocity is proportional to the inverse of density.

Intuitively, it is because the acceleration is smaller for the same force, as mass increases, and density is related to the atomic weight per volume. The materials are not compressible.

It is different for gases because they are compressible. Increasing the density, the number of molecules per volume also increases. The average speed of the molecules are function of temperature, and doesn't change with the bigger density. So, the probability of interaction between molecules increases, what is necessary to propagate the sound.

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The speed of sound goes like the square root of stiffness/density. Thus, if the density increases and the stiffness doesn’t change much, the speed of sound will decrease as you divide by a smaller number and thus take the square root of a larger number. In air the stiffness doesn’t change as quickly as density (with temperature) so the increase in density “wins” over the change in stiffness.

Note that keeping stiffness in mind is important: solids have a higher density than air but also much higher stiffness, resulting in greater speed of sound.

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