

Speed of Sound, Density, Elasticity
Name: Mavis
Status: educator
Grade: 68
Location: Canada
Country: N/A
Date: 11/4/2005
Question:
I came across a statement in a book that I home school out of, (The Weaver
Curriculum Volume 1 Grade 712 supplement page 63). This is the whole
statement:
The speed of sound depends on two factors: elasticity and density. The
more elastic a medium, the greater the speed. The more dense the medium
the slower the speed of sound. For example steel is 6000 times more dense
than air but 2,000,000 times more elastic than air, so sound travels 16
times faster in steel than in air.
This text does not say where they got their information from and it seems
to contradict other statements that sounds travels faster through solids
and liquids because of their density. Also can you explain to me how
solids and liquids are more elastic than gases? What does that mean?
Replies:
Hi Mavis,
Yes, it is true that the speed of propagation of a wave motion depends on
the elastic and inertial properties of the medium, like:
v = sqrt (elastic property/inertial property)
For example, the speed of wave motion along a string is given by:
v = sqrt (T/u) where T is the tension in the string (in Newtons in the
metric system and pounds in the obsolete English system) and u is the mass
density of the string (in kg/m or slug/foot).
For sound propagating through a gas, the speed is given by
v = sqrt (B/u) where B is the bulk modulus B =  dp/(dV/V).
The bulk modulus tells how "stiff" the material is. A large bulk modulus
means it takes a large change in pressure (dp) to make an appreciable
percentage change in volume (dV/V).
We can now plug in some numbers to get some results. For steel,
B = 6E10 N/m^2 and u = 8E3 kg/m^3 giving v = 7.5E6 m/s. (In my notation,
8E3 = 8000, etc.)
For air, B = 1E5 N/m^2 (for a gas, B = the pressure) and
u = 1.3 kg./m^3 giving v=277 m/s.
You may notice that this is the incorrect answer since the speed of sound in
air is close to 343 m/s. The book I am using to get numbers and ideas
(Serway and Beichner, 5th edition) neatly bypasses this worrisome point (it
does not calculate the speed of sound in air). I luckily remembered the
professor in my very first physics course at the U of Chicago (57 years
ago!) told the story of how Isaac Newton realized he got the wrong answer
and could never figure out why.
We now know why! The sound wave compresses and expands the air very rapidly
so heat does not have time to be transferred. This increases the bulk
modulus since it is harder to compress hot gas (and expand cold gas. When
this effect is taken into account the answer agrees with experiment!
To answer your question directly, by large elasticity they must mean a large
bulk modulus, tension, pressure, or some other physical quantity that makes
it more difficult to deform the medium. Density is simpler; it is just the
inertial property that makes it difficult to accelerate the medium.
I do hope this is useful.
Best, Dick Plano, Professor of Physics emeritus, Rutgers University
The speed of sound varies as the square root of the ratio of the "elastic
properties" / "bulk properties".
The more elastic a substance and the least dense, the faster the speed of
sound in general. So the speed in increasing order is: gas < liquid < solid.
There are some crossovers of course. At one extreme are heavy gases (slow)
and hard light solids, like diamond (fast). There is also a difference in the
case of solids and liquids whether the vibrations are longitudinal (medium
vibrates in the direction of the sound propagation) and transverse (medium
vibrates perpendicular to the direction of the sound propagation). Solids
and liquids are more elastic than gases because when molecules in those
phases are moved from their equilibrium position there is a restoring force.
There is no (or very small) restoring force in a gas. Think of sound
propagating like a sling shot  the 'stronger' the rubber band (restoring
force) and the lighter the stone (bulk property) the faster the stone will
travel when the stone is released  it is a weak analogy but the direction
of the effects are parallel. The following sites can lead you to more
details if you wish to pursue them.
http://hyperphysics.phyastr.gsu.edu/hbase/sound/sprop.html#c1
http://hyperphysics.phyastr.gsu.edu/hbase/sound/souspe2.html#c1
http://www.newton.dep.anl.gov/askasci/phy00/phy00625.htm
Vince Calder
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