# Speed of propagation of waves

## Main Question or Discussion Point

the equation for wave of speed(v) relating tension(F) of string is v= sqrt(F/μ) says that as mass density μ increases, velocity of propagation decreases.

but why does sound wave's propagation speed increases in a denser medium like water compared to air?

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yes. thats why the sound wave propagation is much faster in water than air...

so similarly, for the equation v = sqrt(T/μ), where μ is the mass density of the string, T is tension,

why is it in this case, when μ increases, the velocity decreases?

Doc Al
Mentor
yes. thats why the sound wave propagation is much faster in water than air...

so similarly, for the equation v = sqrt(T/μ), where μ is the mass density of the string, T is tension,

why is it in this case, when μ increases, the velocity decreases?
Increased mass density by itself acts to decrease the speed of propagation. (Think of it as making the material harder to wiggle.) You also must consider the elastic properties of the medium, such as the bulk modulus. If the denser medium has a correspondingly greater bulk modulus, then the wave speed could increase.

(In the case of the wave on a string, the elastic property is represented by the tension. Increased tension leads to faster wave speed.)

oh. so it's like increased mass density sort of increases the inertia of the string and hence, when the propagation is transverse, it slows down.

so if we transfer to the case of the sound in water, because sound travels longitudinally in propagation, a increase in density would actually make it travel faster as it takes less time to hit its neighbouring buddy.

so the crux here is the style of propagation ?

Doc Al
Mentor
oh. so it's like increased mass density sort of increases the inertia of the string and hence, when the propagation is transverse, it slows down.
Exactly. More mass density = more inertia.

so if we transfer to the case of the sound in water, because sound travels longitudinally in propagation, a increase in density would actually make it travel faster as it takes less time to hit its neighbouring buddy.
No. The increase in density (mass/volume) acts to slow down the wave, just like it does for the wave on a string. To find the speed of the wave, you need to know the bulk modulus versus the density.

so the crux here is the style of propagation ?
No. The crux is elastic property versus inertial property.

the equation for wave of speed(v) relating tension(F) of string is v= sqrt(F/μ) says that as mass density μ increases, velocity of propagation decreases.

but why does sound wave's propagation speed increases in a denser medium like water compared to air?
You have given the equation for the speed of a transverse wave along a string.The vibrating string sets the surroundings into vibrations with the same frequency.Sound, however is not a transverse wave ,it is a longitudinal wave and the speed of sound is given by root elastic modulus /density.You can set the string into longitudinal vibration by displacing it longitudinally as a violinist might do,the frequencies are much higher than those produced by the transverse vibrations.

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oh. so the increased in density by itself in the case of water would actually also slow down the speed of propagation of sound waves?

but because the bulk modulus "outweighs" this density, the result is a faster propagation of sound waves in water?

To get a better idea of how sound travels think of a longitudinal wave travelling along a line of identical toy cars joined by identical springs.To make the wave travel faster there are two main things you can do:
1.Use stronger springs(analogous to increasing elastic modulus)
2.Use smaller mass cars(analogous to reducing the density)

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Doc Al
Mentor
oh. so the increased in density by itself in the case of water would actually also slow down the speed of propagation of sound waves?

but because the bulk modulus "outweighs" this density, the result is a faster propagation of sound waves in water?
Correct. But note, as Dadface explained above, you can't directly compare the m/L for a transverse wave on a string to mass density for water. But the idea is the same: elastic property versus inertial property.

ah isee.. thanks everyone

There has been a lot of discussion about how the speed of sound depends on density, but nobody has explained why a b-flat carinet (a wind instrument) has the same pitch at Aspen, Colorado (elevation 2450 m) as it does at sea level. The air density at Aspen is only 75% of sea level-pressure. Why doesn't the sound velocity change at Aspen?
Bob S

Doc Al
Mentor
There has been a lot of discussion about how the speed of sound depends on density, but nobody has explained why a b-flat carinet (a wind instrument) has the same pitch at Aspen, Colorado (elevation 2450 m) as it does at sea level. The air density at Aspen is only 75% of sea level-pressure. Why doesn't the sound velocity change at Aspen?
What matters is the air temperature, not the pressure. The ratio of bulk modulus and density depends only on the temperature (given the usual idealized assumptions). See: http://hyperphysics.phy-astr.gsu.edu/hbase/sound/souspe3.html#c2"

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What matters is the air temperature, not the pressure. The ratio of bulk modulus and density depends only on the temperature (given the usual idealized assumptions). See: http://hyperphysics.phy-astr.gsu.edu/hbase/sound/souspe3.html#c2"
OK. But what happens when a clarinetist plays a long note, and the exhaled CO2 (GMW = 44) gets into the instrument? Does the pitch go flat?
Bob S

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