Speed of sound and density relation?

[amk_dbz]

Speed of sound at a specific temperature is independent of Pressure as Pressure varies directly with density.
So wouldn't any change in density vary the pressure such that it has no net effect on speed of sound? but my book says the speed increases with decrease in density...

 

[mfb]

It depends how you change the density. If you decrease the density by heating the gas, you increase the temperature, which increases the speed of sound.
If you decrease it by lowering the pressure, you do not change the speed (in a significant way).

If you try to change the density of a gas, heating/cooling gas is usually easier than changing the pressure.

 

[nasu]

Speed of sound at a specific temperature is independent of Pressure as Pressure varies directly with density.
So wouldn't any change in density vary the pressure such that it has no net effect on speed of sound? but my book says the speed increases with decrease in density...

The speed of sound depends on both density and stiffness of the medium. It decreases with increased density IF the elasticity of the medium does not change.
For a gas (ideal gas model) the bulk modulus is proportional to the pressure.
The density is inverse proportional to the pressure. So their ratio (and in consequence, the speed of sound) is a constant (at a given temperature).

 

[amk_dbz]

It depends how you change the density. If you decrease the density by heating the gas, you increase the temperature, which increases the speed of sound.
If you decrease it by lowering the pressure, you do not change the speed (in a significant way).

If you try to change the density of a gas, heating/cooling gas is usually easier than changing the pressure.

Suppose density is changed by adding water vapours?

 

[Philip Wood]

Good question.

Speed of sound will go up. Imagine some of the air molecules in a given container are replaced one-for-one by water molecules. If the temperature stays the same, so, too, will the pressure. [The partial pressure of the water vapour will be the same as that of the air it replaces, assuming ideal gas behaviour.] But the mass of each water molecule is 18/28 of the mass of a nitrogen molecule (and 18/32 of the mass of an oxygen molecule, so the density will be less. So, too will be the speed of sound, given by
$$v=\sqrt{\frac{\gamma p}{\rho}}.$$

 

[amk_dbz]

Thank you sir for helping. :-) :)

 

[Philip Wood]

It would have been more helpful if I'd pointed out that when you have a relationship between 3 or more variables, x, y and z (say), it's ambiguous to say that x depends on y in such-and-such a way, unless you say whether z is constant or allowed to vary in some way.

Thus the speed of sound is indeed given by $v=\sqrt{\frac{\gamma p}{\rho}}$, but it's misleading to say that v is proportional to $\sqrt{p}$. This is the case if the temperature of a fixed number of gas molecules in a container of fixed volume is raised. But the speed of sound will stay the same if the gas pressure is raised by pumping more gas of the same sort, at the same temperature, into the container, as the density will rise proportionately to the pressure.

Another example:Will more total power be dissipated in two identical resistors connected in parallel, or the same two connected in series? The answer depends on whether the same p.d. is applied across the combination, or some other condition applies.

 

[truesearch]

I agree completely with Philip Wood. The essential property of atoms that is related to the speed of sound is the speed of molecules. The speed of molecules is most obviously related to temperature. The equation γP/ρ 'hides' the temperature significance.
It is easy to see how students can (mistakenly) go away with the idea that the speed of sound depends on Pressure and density of a gas.

 

[Philip Wood]

Indeed, substituting $p=\tfrac{1}{3} \rho c_{rms}^2$ into $v =\sqrt{\frac{\gamma p}{\rho}}$ we get $v=\sqrt{\gamma \tfrac{1}{3} c_{rms}^2}$.
For a diatomic gases like oxygen and nitrogen, $\gamma=1.4$, so $v=0.68c_{rms}$.

So the speed of sound is about 2/3 the rms speed of the molecules. This is not surprising, because on a molecular level, a sound wave arises from periodically varying extra velocities being superimposed (by, for example, a loudspeaker cone) on the much larger random velocities of the molecules. How quickly the disturbance travels through the gas depends on the rms speed of the molecules, as they pass the superimposed small velocities on to their neighbours in collisions.