A question about acoustic wave equation

[huangchao]

Hi, everyone,

I have a question about the acoustic wave equations in two different forms (see the attached). I think the simpler form is more general (in terms of density variation) than the complex one, although the latter looks more general at first sight. But my advisor thinks it's the other way around. So could someone take a look at my derivation and tell me your opinion?

Thanks a lot!

Chao

P.S. The second attachment is the explanation of the core part of the first one.

 

[Studiot]

I am not completely clear whether you regard equation (7) or equation (13) as 'more fundamental - or even quite what you mean by more fundamental ?

 

[huangchao]

I am not completely clear whether you regard equation (7) or equation (13) as 'more fundamental - or even quite what you mean by more fundamental ?

I mean Eq. (7) is more fundamental in terms of that it can deal with strong density variation, while Eq. 13 is just a special case of Eq. 7 and only valid when density variation is small.

 

[Studiot]

I'm still having trouble seeing what you are driving at.

Both your equations are dependent upon equation (1) holding. This requires adiabatic conditions and limits the sound waves to small oscillations. Too great a variation results in sufficient local temperature variation to drain away some of the sound energy by thermal conduction.

So both equations are only applicable to limited density variation.

Incidentally neither analyses are a 'fundamental' way to derive the wave equation for sound.

 

[huangchao]

I'm still having trouble seeing what you are driving at.

Both your equations are dependent upon equation (1) holding. This requires adiabatic conditions and limits the sound waves to small oscillations. Too great a variation results in sufficient local temperature variation to drain away some of the sound energy by thermal conduction.

So both equations are only applicable to limited density variation.

Incidentally neither analyses are a 'fundamental' way to derive the wave equation for sound.

When I say variation, I mean the variation in space. And assume there is no variation in time at every location.

 

[huangchao]

When I say variation, I mean the variation in space. And assume there is no variation in time at every location.

or the variation in time at every location is sufficient small.

 

[Studiot]

I think I can see what your supervisor is saying.

In you paragraph 4 you restrict equation (7) variables more than you do in equation (13) which would suggest equation (13) is the more general, since you are suggesting you can apply it to a wider range of circumstances.

 

[huangchao]

I think I can see what your supervisor is saying.

In you paragraph 4 you restrict equation (7) variables more than you do in equation (13) which would suggest equation (13) is the more general, since you are suggesting you can apply it to a wider range of circumstances.

Could you say it more specifically? What restriction did I apply to equation (7) other than equation (13)?

 

[Studiot]

From your pdf

equation (7)...density is a weak function of time.. equation (13) density not only a weak function of time but also a weak function of space.

Now if you were to prove that equation (7) holds, regardless of density as a function of spcae, then you would be correct.

 

[huangchao]

From your pdf



Now if you were to prove that equation (7) holds, regardless of density as a function of spcae, then you would be correct.

when I derived equation (7), I didn't make any assumption about the variation of density in space, which means equation (7) holds no matter the density varies in space or not.

 

[Studiot]

I am not familiar with the text cited so haven't seen the derivation fully.

It seems fairly conventional, except that it introduces 's' at an earlier stage than I am used to.

I think the approximations made in simplifying the formula when using the adiabatic modulus and obtaining your equations (1) & (2) contain this assumption.
If density is variable, how can you posit a thermodynamically reversible process?

 

[huangchao]

I am not familiar with the text cited so haven't seen the derivation fully.

It seems fairly conventional, except that it introduces 's' at an earlier stage than I am used to.

I think the approximations made in simplifying the formula when using the adiabatic modulus and obtaining your equations (1) & (2) contain this assumption.
If density is variable, how can you posit a thermodynamically reversible process?

In Kinsler's "Fundamentals of Acoustics", on pp. 119-120, it claims, quote, "Since the above derivation never required a restriction on B or rho0 with respect to space, (5.5.4) (which is the same as equation (7)) is valid for propagation in media with sound speeds that are functions of space, such as found in the atmosphere or the ocean."

BTW, thank you for your patient replies, and if you want to see Kinsler's book, I can use dropbox to share with you, because it's too large to be attached in the forum or email.

 

[Studiot]

Consider

$$\frac{{\partial \rho }}{{\partial t}} + \frac{{\partial \left( {\rho {u_1}} \right)}}{{\partial x}} + \frac{{\partial \left( {\rho {u_2}} \right)}}{{\partial y}} + \frac{{\partial \left( {\rho {u_3}} \right)}}{{\partial z}} = 0$$

the approximation made is that changes in density are small comapred with x, y and z so

$$\frac{{\partial \left( {\rho {u_1}} \right)}}{{\partial x}} \approx \frac{{\rho \partial \left( {{u_1}} \right)}}{{\partial x}} \approx \frac{{{\rho _0}\partial \left( {{u_1}} \right)}}{{\partial x}}$$

It is nearly 1.30 am in London so I am off to sleep for now.

 

[huangchao]

Consider

$$\frac{{\partial \rho }}{{\partial t}} + \frac{{\partial \left( {\rho {u_1}} \right)}}{{\partial x}} + \frac{{\partial \left( {\rho {u_2}} \right)}}{{\partial y}} + \frac{{\partial \left( {\rho {u_3}} \right)}}{{\partial z}} = 0$$

the approximation made is that changes in density are small comapred with x, y and z so

$$\frac{{\partial \left( {\rho {u_1}} \right)}}{{\partial x}} \approx \frac{{\rho \partial \left( {{u_1}} \right)}}{{\partial x}} \approx \frac{{{\rho _0}\partial \left( {{u_1}} \right)}}{{\partial x}}$$

It is nearly 1.30 am in London so I am off to sleep for now.

yes, you are right, but this is the approximation used by equation (13) but not equation (7), so equation (7) is more general, right?

Good night!

 

[Studiot]

Have you considered how you get from the control volume to your equations (1) & (2)?

I think the restrictions are inherent in the derivation of these from the natural variables of the control volume viz pressure and volume along with the statement "a lossless fluid".

I am not keen on this last statement. I presume you mean an irrotational fluid, since you are using vector mechanics. Alternatively you assume a conservative sound field.

 

[huangchao]

Have you considered how you get from the control volume to your equations (1) & (2)?

I think the restrictions are inherent in the derivation of these from the natural variables of the control volume viz pressure and volume along with the statement "a lossless fluid".

I am not keen on this last statement. I presume you mean an irrotational fluid, since you are using vector mechanics. Alternatively you assume a conservative sound field.

I don't think there is any restriction regarding the variation of density in space during the derivation of equation (1) and (2). Do you want to see how equation (1) and (2) are derived in the textbook? I can share it with you via Dropbox or Google Docs.

 

[Studiot]

It would be interesting to compare this modern derivation with older flavours.

Thank you

 

[huangchao]

It would be interesting to compare this modern derivation with older flavours.

Thank you

OK, the link is:
https://docs.google.com/leaf?id=0B3...JiMTQtMDk4NDRhZjliNDEy&hl=en&authkey=COHl_70M

 

[Studiot]

Got it thanks.

Will come back after reading.

 

[Studiot]

I am a little concerned about mixing up a fixed control volume and a fixed mass which is oscillating.

For the thermodynamic equation of state let us consider a fixed mass.
The density, condensation etc are not state variables and an equation connecting them, although convenient is not an equation of state.

First some definitions

Consider a fixed mass of fluid at pressure P₀, volume V₀ and density $$\rho$$₀
subject to variations due sound oscillations.

$$\begin{array}{l} P = {P_0} + p \\ V = {V_0} + v \\ \rho = {\rho _0} + {\rho _d} \\ \end{array}$$

We can define auxiliary variables

$$\begin{array}{l} dilatation = \delta = \frac{v}{{{V_0}}} \\ condensation = s = \frac{{{\rho _d}}}{{{\rho _0}}} \\ \end{array}$$

The elastic bulk modulus is the ratio of the pressure difference causing the strain to the volumetric strain

$$Bulk\bmod ulus = B = - \frac{{dP}}{{\frac{{dV}}{V}}} = - V\frac{{dP}}{{dV}}$$

Doing a bit of work on our fixed mass we realize

$$\begin{array}{l} mass = {\rho _0}{V_0} = {\rho _n}{V_n} = \rho V = {\rho _0}{V_0}\left( {1 + \delta } \right)\left( {1 + s} \right) \\ Thus\left( {1 + \delta } \right)\left( {1 + s} \right) = 1 \Rightarrow s = - \delta \\ \end{array}$$

Now we wish our oscillation to be conservative energywise. Thermodynamically this means that the process of energy is adiabatic so

$$P{V^\gamma } = const$$

Since both P and V are state variables, differentiating the equation yields exact differentials.

$${V^\gamma }dP + \gamma P{V^{\gamma - 1}}dV = 0$$

rearranging and substituting from above

$$- V\frac{{dP}}{{dV}} = \gamma P = B$$

but P=P₀+p so dP = p thus

$$\begin{array}{l} B = - \frac{p}{{\frac{v}{{{V_0}}}}} \\ p = - B\delta = Bs \\ \end{array}$$

This post is long enough, but I have now done your equation 1.

 

[huangchao]

I am a little concerned about mixing up a fixed control volume and a fixed mass which is oscillating.

For the thermodynamic equation of state let us consider a fixed mass.
The density, condensation etc are not state variables and an equation connecting them, although convenient is not an equation of state.

First some definitions

Consider a fixed mass of fluid at pressure P₀, volume V₀ and density $$\rho$$₀
subject to variations due sound oscillations.

$$\begin{array}{l} P = {P_0} + p \\ V = {V_0} + v \\ \rho = {\rho _0} + {\rho _d} \\ \end{array}$$

We can define auxiliary variables

$$\begin{array}{l} dilatation = \delta = \frac{v}{{{V_0}}} \\ condensation = s = \frac{{{\rho _d}}}{{{\rho _0}}} \\ \end{array}$$

The elastic bulk modulus is the ratio of the pressure difference causing the strain to the volumetric strain

$$Bulk\bmod ulus = B = - \frac{{dP}}{{\frac{{dV}}{V}}} = - V\frac{{dP}}{{dV}}$$

Doing a bit of work on our fixed mass we realize

$$\begin{array}{l} mass = {\rho _0}{V_0} = {\rho _n}{V_n} = \rho V = {\rho _0}{V_0}\left( {1 + \delta } \right)\left( {1 + s} \right) \\ Thus\left( {1 + \delta } \right)\left( {1 + s} \right) = 1 \Rightarrow s = - \delta \\ \end{array}$$

Now we wish our oscillation to be conservative energywise. Thermodynamically this means that the process of energy is adiabatic so

$$P{V^\gamma } = const$$

Since both P and V are state variables, differentiating the equation yields exact differentials.

$${V^\gamma }dP + \gamma P{V^{\gamma - 1}}dV = 0$$

rearranging and substituting from above

$$- V\frac{{dP}}{{dV}} = \gamma P = B$$

but P=P₀+p so dP = p thus

$$\begin{array}{l} B = - \frac{p}{{\frac{v}{{{V_0}}}}} \\ p = - B\delta = Bs \\ \end{array}$$

This post is long enough, but I have now done your equation 1.

Hi, Studiot,

I just went through your derivation, and found no problem of it, so the equation (1) is valid and it requires no restrict about the variation of density in space, right?

 

[Studiot]

Depends what you mean by restriction.

Obviously since all three quantities in equation 1 are derived quantities they are subject to any restrictions on the state variables they are derived from.
Your book lists these restrictions towards the beginning of the chapter.
You should also make clear whether you are referring to variation in space in general, (throughout the fluid) or just in the control volume or mass.

 

[huangchao]

Depends what you mean by restriction.

Obviously since all three quantities in equation 1 are derived quantities they are subject to any restrictions on the state variables they are derived from.
Your book lists these restrictions towards the beginning of the chapter.
You should also make clear whether you are referring to variation in space in general, (throughout the fluid) or just in the control volume or mass.

Sure, but what I mean is no restriction of density variation in space, and surely there is a restriction of density variation in time. Just as it's stated in the book (at the bottom of pp. 115), quote, "The essential restriction is that the condensation is small."

 

[Studiot]

Yes but I was mainly thinking of the paragraphs on page 114 , immediately below 5.1.7

"small enough to have acoustic variables uniform throughout"

"changes in density.. ..small compared with the equilibrium value"

These also apply to the underlying variables of pressure and volume.

 

[huangchao]

Yes but I was mainly thinking of the paragraphs on page 114 , immediately below 5.1.7

"small enough to have acoustic variables uniform throughout"

"changes in density.. ..small compared with the equilibrium value"

These also apply to the underlying variables of pressure and volume.

OK, I didn't find any problem of it, it's just a definition of fluid element.

 

[Studiot]

It's more than just a definition of a fluid element.

The elemtal variation in density etc could be large or small so long as the element remained instantaneously homogenous.

However the variation is deliberately restricted to small values.

 

[Studiot]

I prefer to stick to 'first principles' wherever possible and work from there.
This philosophy can be continued to derive the wave equation but that is for another night as it's bed time here.
cheers for now.

 

[huangchao]

I prefer to stick to 'first principles' wherever possible and work from there.
This philosophy can be continued to derive the wave equation but that is for another night as it's bed time here.
cheers for now.

In my point of view, just the variation of density in time (or the condensation 's') is deliberately restricted to small values, not the density variation in space.

Anyway, thank you so much and good night!