/sci/ - Science & Math

[Anonymous No.5688903]

Who knows why is the propagation velocity of sound is defined like this?
I do not understand what does constant entropy mean...
I'm not sure what should I think when a problem talks about entropy, neither entalpy. How do you deal with these?

[Anonymous No.5688911]

>>5688903
Entropy can be a very difficult quantity to get your head around. Sometimes you can think of "constant entropy" as the same thing as "no flow/exchange of heat" (no heat exchange is also called an "adiabatic process"). So the equation basically says:

"The speed of sound (squared) equals the change in pressure when the density is changed with no exchange of heat".

I'm not entirely familiar with the equation, but to my knowledge it can be understood as, as you change the density, in air for example, how much does the air molecules push on their nearby neighbors? If the answer is a lot, then the molecules will quickly exert a high pressure on the next molecules and so forth so the sound will travel fast. You can imagine the sound travelling through molecules in the air (or another medium) as a longitudinal wave travelling through a spring. Pic related.

[Anonymous No.5688925]

>>5688911
>>"The speed of sound (squared) equals the change in pressure when the density is changed with no exchange of heat".
I want to reach your understanding, really, thank you

[Anonymous No.5688926]

>>5688911
Do you have any idea about why is squared?

[Anonymous No.5688930]

>>5688911
>"The speed of sound (squared) equals the change in pressure when the density is changed with no exchange of heat".
Which is important because if instead you assume no change in temperature, you get the wrong answer.

[Anonymous No.5688934]

>>5688926
Because when you solve the
http://en.wikipedia.org/wiki/Wave_equation
the velocity of the waves you get is the square root of the coefficient.

[Anonymous No.5688935]

>why is squared?
(N/m^2)/(kg/m^3)=Nm/kg=m^2/s^2

[Psistar !!bRma1Ao62vX No.5688941]

The speed of a wave can be derived from the wave-equation, right?

Which, in a unidimensional problem with no losses, is in the form of a d'Alembert equation:
<span class="math"> c^2 \frac{\partial p}{\partial x^2} - \frac{\partial p}{\partial t^2} = 0 [/spoiler], where <span class="math">c[/spoiler] has the dimension of a velocity.

Assuming small changes in pressure, volumic mass, and a small speed (compared to the speed we're looking for), we can write the developement:
<span class="math"> \rho(x,t) = \rho_0 + \mu (x,t), \mu << \rho_0[/spoiler]

Now, we also have: <span class="math">\rho(x,t)= \rho_0 + \frac{\partial \rho}{\partial p} p(x,t) [/spoiler] (it's the differential)
Physicists have hesitated between two interpretations of <span class="math">\frac{\partial \rho}{\partial p}[/spoiler]: is the process adiabatic or is it isothermal? (I think it was Laplace and Newton who didn't agree on that)
Turns out the adiabatic hypothesis describes the phonomenon more accurately.

Finally, when trying to find the wave equation, you have the term c^2 written as <span class="math">c^2 = \frac{1}{\rho_0 \chi_s [/spoiler], with <span class="math">\chi_s[/spoiler] the isentropic compressibility.

[Anonymous No.5688943]

>>5688926
Like I said, I'm not quite familiar with the equation. Like someone already said, it might come from solving the wave equation for the speed of light. By natural means though, you can kind of work it out by dimensional analysis and it goes like this. The dimensions of pressure is force per area, or Newton per square meter:

<span class="math">[P]=Nm^{-2}[/spoiler]

The dimensions of density is mass per volume, or kg per cubic meter.

<span class="math">[ \rho ]=kgm^{-3}[/spoiler]

Pressure per density is then

<span class="math">\frac{Nm^{-2}}{kgm^{-3}=Nmkg^{-1}[/spoiler]

But Newton is acceleration times mass or meter per second squared times kg, so

<span class="math">Nmkg^{-1}=ms^{-2}mkgkg^{-1}=m^{2}s^{-2}[/spoiler]

which is just the dimension of velocity squared. The technique of dimensional analysis though doesn't give you any information about constant factors like 2 or pi.

[Psistar !!bRma1Ao62vX No.5688944]

>>5688941
sorry for the end:
<span class="math"> c^2 = \frac{1}{\rho_0 \chi_s}[/spoiler]

[Psistar !!bRma1Ao62vX No.5688945]

>>5688941
also <span class="math"> p(x,t)[/spoiler] is the variation in pressure.

[Anonymous No.5688950]

When did you learn all this things, I feel stupid

[Anonymous No.5688962]

>>5688943
Does dimensional analysis always work, why?
We will reach that subject in a few weeks, it looks so powerfull...

[Anonymous No.5688963]

>>5688950

Any fluid mechanics course. The wave equation can be derived from navier-stokes for a compressible fluid assuming small perturbations from equilibrium, and trashing squared terms (which by hypothesis are very small), like >>5688941 introduced

[Anonymous No.5688966]

>>5688963
Not any, in my course they just threw the equation to our faces

[Anonymous No.5688979]

>>5688962
It doesn't help you get the constants correct, but it will often be a powerful tool to tell you how some quantities scale with each other, just like the example I gave you. By analysing the dimensions, I can tell that <span class="math">\frac{dp}{d \rho}[/spoiler] must scale as <span class="math">v^{2}[/spoiler]. I didn't take any courses in dimensional analysis though, it's a technique I learned to use after I got more comfortable with physics in general.

[Anonymous No.5689058]

the derivation is online here

http://en.wikipedia.org/wiki/Acoustic_wave_equation#Derivation

[Psistar !!bRma1Ao62vX No.5689218]

>>5688950
During my second year of classe préparatoire (in France)