# What is the speed of sound? | everyday einstein

We live in a universe of space and time. Events occur at a particular time, and in some location in space. In this way, space and time can be seen as a background against which things happen. Throughout most of human history, this background was seen as absolute.

Each event occurs at a unique point in space and time, and in principle everyone can agree what that point is. Intuitively, it makes a lot of sense. In our everyday lives the Earth seems to be an unmoving rock, and acts as a point of reference for everything we do. Sure, we now know the Earth moves around the Sun, but it doesn’t feel that way.

When Galileo and others began proposing a more sophisticated view of physics, the intuitive view of spacetime as an immutable background remained. Galileo argued that the motion of objects was relative to each other, but that motion could always be measured relative to the “fixed” space. For Galileo, speed was relative, but spacetime was not.

When Newton developed his theories of physics and gravity, he also assumed that spacetime was fixed and absolute.

The success of Newton’s physics seemed to confirm the absolute nature of spacetime, and the assumption remained largely unquestioned for two centuries. But as we came to understand light, the idea became less intuitive.

According to Maxwell’s equations, the speed of light is the same for all light. That’s because electromagnetic waves propagate at the same rate.

But water waves propagate through water, and sound waves through air, so what do light waves propagate through?

The most popular idea was that light moved through a luminiferous aether. This aether couldn’t be observed directly, but it was thought to be stationary relative to the background of space.

Some proposed that this aether could in fact be the absolute frame of reference for the universe.

But if that’s the case, then your measurement of the speed of light should depend upon your motion relative to the aether.

Suppose you were on the platform of a train moving at 10 m/s (20 mph). If you measured the speed of sound in the direction you are moving, you would get a speed of 330 m/s.

That’s because the sound is traveling through the air at 340 m/s, but you are traveling through the air in the same direction at 10 m/s, so the sound is moving 330 m/s relative to you. In the same way, if you measured the speed of sound in the opposite direction, you would get 350 m/s because of your motion.

This is a key feature of waves traveling through a medium: they can be different in different directions because of your motion through the medium.

Then in 1887, Albert Michelson and Edward Morley performed an experiment to measure this difference in the speed of light. But what they found was the speed of light was always the same.

No matter what direction light traveled, no matter how they oriented their experiment, the speed of light never changed. This was not only surprising, it violated the fundamental assumption of an absolute reference frame.

It seemed the speed of light (and only the speed of light) is absolute, and this made no sense at all. The relative nature of “now.”

This is the puzzle Einstein sought to resolve in his paper “On the Electrodynamics of Moving Bodies.” In this paper Einstein noted that in order for the speed of light to be an absolute constant, either Maxwell’s equations or Newton’s concept of space and time had to be wrong. Somewhat surprisingly, Einstein argued for the latter.

Specifically, he argued that the “grid” of space and time was relative to the observer. He demonstrated this by looking at a property known as simultaneity. In Newton’s view, two events seen to occur at the same time will be seen to be simultaneous for all observers. But Einstein showed that the constancy of light required this concept of “now” to be relative.

Different observers moving at different speeds will disagree on the order of events.

Rather than a fixed background, space and time is a relation between events that depends upon where and when the observer is. This relativity of space and time led to strange predictions, such as time dilation, which were later found to be true. It’s a concept that’s still difficult to fully understand, but is absolutely necessary for modern devices such as GPS.

Tomorrow: Einstein looks at the connection between matter and energy, and finds that relativity could explain the light of the stars.

Paper:  Einstein, Albert. Zur Elektrodynamik bewegter Körper. Annalen der Physik 17 (10): 891–921 (1905)

## Sound Pulses Exceed Speed of Light Photo taken by Davide Guglielmo (brokenarts).
There are no usage restrictions for this photo

A group of high school and college teachers and students has transmitted sound pulses faster than light travels—at least according to one understanding of the speed of light.

The results conform to Einstein's theory of relativity, so don't expect this research to lead to sound-propelled spaceships that fly faster than light. Still, the work could help spur research that boosts the speed of electrical and other signals higher than before.

The standard metric for the speed of light is that of light traveling in vacuum. This constant, known as c, is roughly 186,000 miles per second, or roughly one million times the speed of sound in air. According to Einstein's work, matter and signals cannot travel faster than c.

• PVC science
• However, physicist William Robertson at Middle Tennessee State University in Murfreesboro, along with a high school teacher, two college students and two high school students, managed to, depending on how you look at it, transmit sound pulses faster than c using little more than a plastic plumbing pipe and a computer's sound card.
• “This experiment is truly basement science,” Robertson told LiveScience.

The key to understanding their results, reported online Jan. 2 in the journal Applied Physics Letters, is envisioning every pulse of sound or light as a group of intermingled waves. This pulse rises and falls with energy over space, with a peak of strength in the middle.

Robertson and his colleagues transmitted sound pulses from the sound card through a loop made from PVC plumbing pipe and connectors from a hardware store. This loop split up and then recombined the tiny waves making up each pulse.

This led to a curious result. When looking at a pulse that entered and then exited the pipe, before the peak of the entering pulse even got into the pipe, the peak of the exiting pulse had already left the pipe.

If the velocities of each of the waves making up a sound pulse in this setup are taken together, the “group velocity” of the pulse exceeded c.

“I believe that this is the first experimental demonstration of sound going faster than light,” Robertson said. Past research has proven it possible to transmit electrical and even light pulses with group velocities exceeding c.

Common thing?

## What Is "Supersonic"?

Scientific American presents Everyday Einstein by Quick & Dirty Tips. Scientific American and Quick & Dirty Tips are both Macmillan companies.

Every so often, you might hear somebody mention that something or other has reached supersonic speeds. Obviously that must be pretty fast, but exactly how fast is it?

Speed Vocabulary

Let’s start with same basic terminology. Most people use the word “speed” to describe how fast something is moving. However, in physics, speed is just part of the story. The direction of movement is also crucial in most physics calculations. Scientists use the term “velocity” to describe both the magnitude (or speed) of your movement combined with the direction.

The final term we need to understand to talk about supersonic speed is something called the Mach number. The Mach number is the ratio of the velocity of an object relative to some medium and the speed of sound in that medium.

Let’s look at an example. According to the IAAF, world record holder Usain Bolt can run at a maximum velocity of 12.27 meters per second (m/s). The speed of sound is around 340.

3 m/s (technically the speed of sound varies somewhat with temperature, but we’ll just stick with this number to make things easier). If you take the ratio of these two numbers, 12.27 m/s divided by 340.3 m/s, you get 0.036.

In other words, Usain Bolt can run at Mach 0.036.

Those of you who have been listening to the Math Dude podcast and are good with fractions will notice that as Usain’s velocity increases, the ratio of his velocity to the speed of sound approaches 1. If he were able to run at 340.3 m/s, he would reach Mach 1.

## 17.2 Speed of Sound – University Physics Volume 1

By the end of this section, you will be able to:

• Explain the relationship between wavelength and frequency of sound
• Determine the speed of sound in different media
• Derive the equation for the speed of sound in air
• Determine the speed of sound in air for a given temperature

Sound, like all waves, travels at a certain speed and has the properties of frequency and wavelength. You can observe direct evidence of the speed of sound while watching a fireworks display (Figure 17.4). You see the flash of an explosion well before you hear its sound and possibly feel the pressure wave, implying both that sound travels at a finite speed and that it is much slower than light.

The difference between the speed of light and the speed of sound can also be experienced during an electrical storm. The flash of lighting is often seen before the clap of thunder.

You may have heard that if you count the number of seconds between the flash and the sound, you can estimate the distance to the source. Every five seconds converts to about one mile.

The velocity of any wave is related to its frequency and wavelength by

where v is the speed of the wave, f is its frequency, and λλ is its wavelength. Recall from Waves that the wavelength is the length of the wave as measured between sequential identical points.

For example, for a surface water wave or sinusoidal wave on a string, the wavelength can be measured between any two convenient sequential points with the same height and slope, such as between two sequential crests or two sequential troughs.

Similarly, the wavelength of a sound wave is the distance between sequential identical parts of a wave—for example, between sequential compressions (Figure 17.5). The frequency is the same as that of the source and is the number of waves that pass a point per unit time.

Table 17.1 shows that the speed of sound varies greatly in different media. The speed of sound in a medium depends on how quickly vibrational energy can be transferred through the medium.

For this reason, the derivation of the speed of sound in a medium depends on the medium and on the state of the medium.

In general, the equation for the speed of a mechanical wave in a medium depends on the square root of the restoring force, or the elastic property, divided by the inertial property,

v=elasticpropertyinertialproperty.v=elasticpropertyinertialproperty.

Also, sound waves satisfy the wave equation derived in Waves,

∂2y(x,t)∂x2=1v2∂2y(x,t)∂t2.∂2y(x,t)∂x2=1v2∂2y(x,t)∂t2.

• Recall from Waves that the speed of a wave on a string is equal to v=FTμ,v=FTμ, where the restoring force is the tension in the string FTFT and the linear density μμ is the inertial property. In a fluid, the speed of sound depends on the bulk modulus and the density,
• The speed of sound in a solid the depends on the Young’s modulus of the medium and the density,
• In an ideal gas (see The Kinetic Theory of Gases), the equation for the speed of sound is

where γγ is the adiabatic index, R=8.31J/mol·KR=8.31J/mol·K is the gas constant, TKTK is the absolute temperature in kelvins, and M is the molecular mass. In general, the more rigid (or less compressible) the medium, the faster the speed of sound.

This observation is analogous to the fact that the frequency of simple harmonic motion is directly proportional to the stiffness of the oscillating object as measured by k, the spring constant. The greater the density of a medium, the slower the speed of sound.

This observation is analogous to the fact that the frequency of a simple harmonic motion is inversely proportional to m, the mass of the oscillating object. The speed of sound in air is low, because air is easily compressible.

Because liquids and solids are relatively rigid and very difficult to compress, the speed of sound in such media is generally greater than in gases.

Medium
v (m/s)
Gases at 0°C0°C
Air 331
Carbon dioxide 259
Oxygen 316
Helium 965
Hydrogen 1290
Liquids at 20°C20°C
Ethanol 1160
Mercury 1450
Water, fresh 1480
Sea Water 1540
Human tissue 1540
Solids (longitudinal or bulk)
Vulcanized rubber 54
Polyethylene 920
Marble 3810
Glass, Pyrex 5640
Aluminum 5120
Steel 5960

Table 17.1 Speed of Sound in Various Media

Because the speed of sound depends on the density of the material, and the density depends on the temperature, there is a relationship between the temperature in a given medium and the speed of sound in the medium. For air at sea level, the speed of sound is given by

v=331ms1+TC273°C=331msTK273Kv=331ms1+TC273°C=331msTK273K

where the temperature in the first equation (denoted as TCTC) is in degrees Celsius and the temperature in the second equation (denoted as TKTK) is in kelvins.

The speed of sound in gases is related to the average speed of particles in the gas, vrms=3kBTm,vrms=3kBTm, where kBkB is the Boltzmann constant (1.38×10−23J/K)(1.38×10−23J/K) and m is the mass of each (identical) particle in the gas.

Note that v refers to the speed of the coherent propagation of a disturbance (the wave), whereas vrmsvrms describes the speeds of particles in random directions. Thus, it is reasonable that the speed of sound in air and other gases should depend on the square root of temperature.

While not negligible, this is not a strong dependence. At 0°C0°C, the speed of sound is 331 m/s, whereas at 20.0°C20.0°C, it is 343 m/s, less than a 4%4% increase. Figure 17.6 shows how a bat uses the speed of sound to sense distances.

Figure 17.6 A bat uses sound echoes to find its way about and to catch prey. The time for the echo to return is directly proportional to the distance.

As stated earlier, the speed of sound in a medium depends on the medium and the state of the medium. The derivation of the equation for the speed of sound in air starts with the mass flow rate and continuity equation discussed in Fluid Mechanics.

Consider fluid flow through a pipe with cross-sectional area A (Figure 17.7). The mass in a small volume of length x of the pipe is equal to the density times the volume, or m=ρV=ρAx.m=ρV=ρAx. The mass flow rate is

## (PDF) Speed of sound in a Bose-Einstein condensate

• 4
• of this system, as a function of the temperature, in the
• usual manner 
• E(T) = 3~2
• 2mV 2/3Nh1−T
• Tc3+8
• 3ra3N
• V π i+
• 5~2
• 2mV 2/3NhT
• Tc3−8
• 3ra3N
• V π i+
• 2πa~2
• mV N2h3−4
• 3ra3N
• V π i.(16)
• P(T) = ~2
• mV 5/3N+2~2
• 3mV 5/3NhT
• Tc3
• −17
• 3ra3N
• V π +2πa~2
• mV 2N2h3−2ra3N
• V π i.(17)
• The speed of sound (cs(T)) is given by
• c2
• s(T) = 5~2
• 3m2V2/3+10~2
• 3m2V2/3hT
• Tc3
• −68
• 15ra3N
• V π +4πa~2
• m2VNh3−5
• 2ra3N
• V π i.(18)
• IV. DISCUSSION AND CONCLUSIONS
• Our last expression allows us to predict the speed of
• sound, as a function of the temperature, of course, this
• happens only in the regime T≤Tc. Notice that our
• prediction does not include a dependence upon the am-
• plitude of the disturbance, a fact that matches with the
• experimental output [7, 15].
• It has already been recognized that many theoretical
• studies, in the realm of BEC, ignore the role that the
• thermally excited atoms play in the deﬁnition of the char-
• acteristics of the gas . The assumption of vanishing
• temperature is not correct, as a matter of fact thermody-
• namics tells us that in a practical sense the achievement
• of this temperature is impossible . Therefore, the
• deduction of the speed of sound under the assumption
• of T= 0 is an approximation, the one should be im-
• proved. A point that has to be underlined in the present
• manuscript concerns this issue. Indeed, a ﬂeeting glimpse
• at (12) tells us that the second term on the right hand–
• side takes into account the contribution to the kinetic
• energy of the thermal cloud of the system as a function

of the temperature, i.e., we do not assume T= 0.

1. In the context of the assumptions here accepted, of
2. course, we have, as mentioned before, discarded the pos-
3. sibility of having, in this kinetic term, transitions be-
4. tween excited states and the ground state. In order to
5. have this case we must consider that not only the ﬁrst
6. excited state is populated, but also higher states. From
7. this last comment we expect a very small contribution to
8. the speed of sound stemming from this neglected possibil-
9. ity. An additional simpliﬁcation has been introduced, but
10. now in relation with (3). If we consider (14) we, imme-
11. diately, notice that we have only considered interactions
12. between particles in the ground state with particles in
13. the same state. Of course, more possibilities are present,
14. for example, a particle in the ground state might interact
15. with a particle in the ﬁrst excited state. Since the num-
16. ber of particles in the ground state is much larger than
17. those in excited states we expect that the probability of
18. having a ground state–ground state interaction is larger
19. than having a ground state–ﬁrst excited state interac-
20. tion, and this last one is larger than the ﬁrst state–ﬁrst
21. state interaction. In this sense has to be understood the
22. assumption.
23. Let us now confront our theoretical prediction against
24. the extant experimental results. We will consider the
25. case of a BEC comprised by sodium atoms. This choice
26. is done since we have already experimental results in this
27. direction; N= 5 ×106,n= 1021 m−3,Tc= 2 ×10−6K,

m= 35.2×10−27Kg, l∼10−2m. Finally, the scatter-

ing length, for sodium, has already been measured ,

namely, a= 2.75 ×10−9m. We need also an assumption

for our temperature, here we assume T= 0.9Tc. These

values imply

cs= 2.2×10−3m/s. (19)

A careful look at the present measurement outputs en-

tails that our result is not a bad one , i.e., it provides

• the correct order of magnitude. The shortcomings of the
• present manuscript, at least in the realm of its compat-
• ibility with the experimental results, are also shared by
• other approaches [15, 18]. The most intriguing interro-
• gant in this sense is related to the fact that in the region
• of the thermodynamical space in which the assumption of
• MFT is strongly satisﬁed the theoretical prediction has
• its worst behavior . Several conjectures could be put
• forward, in order to solve this puzzle. The inclusion of
• additional terms, for instance, in connection with the dis-
• cussion about the number of signiﬁcant terms related to
• (3) would, surely, modify the result. In this direction an
• alternative way is related to the deduction of the speed of
• sound resorting to the generalized Gross–Pitaevski equa-
• tion (usually known as Zaremba–Nikuni–Griﬃn equation
• (ZNG) ) the one takes into account a coupling be-
• tween the condensate and non–condensate components of
• the corresponding system. A physical motivation behind
• this statement can be found in the fact that sound can
• be understood as density waves. Clearly, changes in the
• density, which involve changes in the separation among
• particles, are determined by the interactions among the
• constituents of the gas. This explains in a very simple
• way why the idea of interaction plays a relevant role in
• the determination of the speed of sound. The ZNG equa-
• tion  includes in the dynamics of a BEC a coupling
• between the condensate and the noncondensate part of