What is the Speed of Sound in Iron?

What is the speed of sound in iron?
- Mary Fajman (age 12)
Nightingale School, Chicago,Il. U.S.A.

Mary, that’s a great question!
The exact answer depends on the pressure and temperature of the iron as well as it’s exact composition, but I think a pretty good estimate of the speed of sound in iron would be around 5 kilometers per second. (I will look into this further and post any additional information). MS

(published on 10/22/2007)

But, how'd you get the 5 km? What is the formula of getting the speed of sound in water?
- Arlene (age 12)
Philippines

Hi Arlene,

Good question! Knowing why something is true is a lot more important than knowing the answer itself. It turns out, you can understand a lot about waves inside different materials from this simple formula. (Plus, wave effects are really awesome and fun to study!)

For a typical sound wave, the speed of sound is about c = square root of (K/ρ), where K is the stiffness (aka bulk modulus) and ρ is the density of the substance. This powerful formula works pretty well in most solids, liquids, and gases. (If you are wondering how to find the "stiffness" of a gas, you can use K = γ * p, where γ is the adiabatic index, and p is the pressure.)

So, if you want to calculate the wavespeed of some material, you'll probably be pretty close by looking up these two numbers online and using this formula. For example, I found values on wikipedia for water so that c = sqrt(2.2*10^9 Pascals/1 kg/L) = 1483 m/s, which is pretty close to the value of 1482 m/s from wolfram alpha! This is almost 5 times faster than in air, for which I get c = sqrt(10^5 Pascals* 1.4/ 1.28 g/L) = 330 m/s.

However, this speed isn't always exact. The density and stiffness of a substance can depend on various factors, like temperature, impurities, and even the frequency of the sound wave. So, for example, the speed of a higher pitch wave might be faster than that of a lower pitch. Imagine if this effect were stronger in air... then, when you talked you would sound funny, since the different pitches in your voice would get separated out, almost like light in a rainbow!

Let's think some more about this simple equation, √(K/ρ). After all, equations are good for a lot more than plugging in numbers!

First, let's think about what the equation means. As we can see, if you increase the stiffness K of a material, the wave speed will increase. Similarly, if you increase the density of the material, the wave speed will decrease (since ρ is in the denominator). So, to quote from wikipedia, "sound will travel 1.59 times faster in nickel than in bronze, due to the greater stiffness of nickel at about the same density. Similarly, sound travels about 1.41 times faster in light hydrogen (protium) gas than in heavy hydrogen (deuterium) gas, since deuterium has similar properties but twice the density. At the same time, "compression-type" sound will travel faster in solids than in liquids, and faster in liquids than in gases, because the solids are more difficult to compress than liquids, while liquids in turn are more difficult to compress than gases."

Next, let's get some idea of where the equation comes from.  The above facts are nicely explained by the ball-and-spring model of a solid on wikipedia (, where we treat the molecules as balls and the forces between them as springs. Then, if you increase the mass of the balls, they are harder to push (slower wave speed); if you increase the spring stiffness, the springs push the balls more easily (faster wave speed). Note: this model can be made more rigorous, and the equation exactly derived, using Newton's laws with a few reasonable approximations about the medium.

Finally, let's apply the equation to an exciting example that you can perhaps test experimentally. If you take a liquid, say beer, water, or soda, in a kettle, its natural wave speed will be around 1500 m/s. Now, if you add just a tiny bit of tiny bubbles in the water, we can use an effective model of the liquid to predict the new wave speed. If you add much less than a percent by density of air, you wouldn't expect the wave speed to change much. After all, the mass density of the water + air mixture is almost exactly the same. However, the tiny bit of air is vastly more easy to compress (K is much less for air than for water). So, as long as you don't push so hard on the liquid that the air is completely squashed, the entire liquid has an effective stiffness of air, not water!

(This last fact is quite subtle. It's like saying that if you have a stiff spring attached to a weak spring, and you try to squish them both, then the combination will be squished just as easily as the weak spring by itself... as long as you don't completely flatten the weak spring.) So, what's the wavespeed of something with the same density as water but a much tinier stiffness? By our simple equation √(K/ρ) we can see that the wave speed is also much tinier, about 100 times smaller, in fact!

I've heard you can see this effect by tapping on a pot of boiling water... as the bubbles leave, the wave speed increases. Fill a metal pan with each of these liquids, and tap on them with a metal spoon. You will hear a tone which depends on the wave speed, so if the wave speed is different, the tone will be different!

I guess this is more of an answer than you expected, but hopefully it will give you some small idea of how a physicist thinks about equations. :)

David Schmid

(published on 12/07/2013)