# Why is the speed of sound lower at higher altitudes?

At sea level the speed of sound is 760mph, but at altitudes like the Concorde would fly at (55,000ft) the sound barrier is at 660mph, so 1000th slower. Does it have to do with lower pressure?

From a non-technical viewpoint, I would say that the simplest way I understand this is the following one.

Yes, it has to do with pressure. Actually, it might be easier to think that it has to do with density. Consider the dominoes effect. If the dominoes are more apart from each other, that is, if the density of dominoes is lower, it will take longer for a certain domino to "communicate" to the next one that it has suffered a mechanical push from the previous one. Analogously, if the density of the air is lower, the propagation of the mechanical pulse will take longer, that is, its speed will be lower.

• The speed of sound is $v=\sqrt{\gamma P/\rho}$ so you can't just say it's to do with pressure or with density. You need to know how $P$ and $\rho$ are related to make any definitive statement. See How can the speed of sound increase with an increase in temperature? for more. – John Rennie Jan 12 '16 at 18:01
• Thanks for your comment, @JohnRennie. If you consider the ideal gas, you will see that the pressure is proportional to the density, giving no dependence to the speed on the density. As a better approximation, consider the Virial Expansion (en.wikipedia.org/wiki/Virial_expansion). One can see that the speed must decrease if the density decreases, from that perspective. – pscm Jan 12 '16 at 18:11
• No, $P \propto T\rho$. That's why as costrom says the temperature is the main factor. The speed is $v = \sqrt{\gamma RT}$. – John Rennie Jan 12 '16 at 18:29
• This is certainly true for ideal gases. Please have a look at the Virial expansion. When you consider one extra term only, that is, the second term in the expansion, the Newton-Laplace formula gives a different dependence for the sound speed on the density, as far as I understand. Although the question was made in the context of ideal gases, the general dependence of the speed on the thermodynamic parameters is such that density can directly be a main factor as good as the temperature, except if you're restricted to the ideal gas case. @JohnRennie – pscm Jan 12 '16 at 19:12

it has to do with the temperature lapse with altitude. since the speeed of sound is related to temperature by:

$a = \sqrt{\gamma RT}$, where $\gamma$ and R are gas properties and T is temperature

and the temperature profile follows (generally) like the left of these three plots:

The area of interest for airliners is in the lowermost region where the temperature is steadily decreasing with altitude (at a rate of ~6.5K/km)

• And, for an ideal(ish) gas specifically does not have a pressure dependence... – Jon Custer Jan 12 '16 at 0:30
• @JonCuster: Could you elaborate on what you mean? For an ideal gas, $c_s^2=\gamma P/\rho$, so one can say that $c_s:=f(P)$. Unless you're talking about something else. – Kyle Kanos Jan 12 '16 at 3:28

A sound wave moving through a gas requires a small scale bulk movement of gas molecules back and forth as pressure at any locations builds or falls. Therefore, the sound wave can not possibly move through the gas at a speed greater than that of the individual molecules themselves, and in fact must move at a lower speed than that due to the random nature of molecular movement. Since molecular speed in a gas is a direct function of average molecular kinetic energy and that is a direct function of temperature, the speed of sound in a gas will also be a function of temperature.

At the cruise altitude of the Concorde the atmospheric temperature is below that of the standard atmospheric temperature at sea level, so the speed of sound is also lower. That is the simple answer.

One might also note that the temperature is lower as one ascends primarily because the pressure is lower as one ascends and then conclude that the speed difference is equally attributable to lower pressure (or lower density for that matter). However, it is primarily and more directly related to molecular speed and thus temperature than it is to anything else.