the speed of sound of a cas varies with pressure and density. show by dimensional reasoning that the proper form must be a = C (p/rho)^1/2
This text is generated by AI trained by Professor Brian J. Kirby on his lectures, videos, notes, message board responses, and textbook
PK Bot’s answer:
- PURPOSE OF THE EXERCISE:
Demonstrate that by considering the dimensions of pressure and density, one can show that the speed of sound must scale as the square root of the ratio of pressure to density, i.e.
a = C\sqrt{\frac{p}{\rho}}\,.
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ENGINEERING CONTEXT:
In acoustics and compressible flow analyses, establishing the functional dependence of the speed of sound on pressure and density is fundamental to designing systems such as nozzles, ducts, and even predicting aeroacoustic phenomena in engines. -
PHYSICAL CONTEXT:
The dominant physical effect is the propagation of a small disturbance through a fluid where the conversion of pressure energy into kinetic energy governs the wave speed. -
ASSUMPTIONS:
Assume that the only relevant parameters for the speed of sound are the static pressure p and the density \rho, neglecting viscosity and thermal conduction; this is justified for many high-Reynolds-number, isentropic processes. -
DRAWINGS:
No drawing is needed for this problem because the analysis relies solely on dimensional reasoning rather than control volume representations or streamline drawings. -
SOLUTION TECHNIQUE:
Pressure has SI units of pascals ¶, where
[p] = \mathrm{\frac{kg}{m\cdot s^2}}\,.
Density has units
[\rho] = \mathrm{\frac{kg}{m^3}}\,.
Thus, the ratio
\frac{p}{\rho}
has units
\frac{\mathrm{\frac{kg}{m \cdot s^2}}}{\mathrm{\frac{kg}{m^3}}} = \mathrm{\frac{m^2}{s^2}}\,.
Taking the square root yields
\sqrt{\frac{p}{\rho}}
with units
\mathrm{\frac{m}{s}}\,.
Therefore, by dimensional reasoning the speed of sound must be of the form
a = C\sqrt{\frac{p}{\rho}}\,,
where C is a dimensionless constant. Detailed conservation-of-mass and momentum analysis later shows that for an ideal gas undergoing an isentropic process the constant is C=\sqrt{\gamma}, so that
a=\sqrt{\gamma\frac{p}{\rho}}\,,
and using the ideal gas law (p=\rho R T) one gets
a=\sqrt{\gamma R T}\,.
- REFLECTION:
The dimensional analysis confirms that the natural scaling for the speed of sound in a fluid is the square root of the pressure-to-density ratio. This result highlights how the state of the fluid governs wave propagation and provides insight into compressibility effects in engineering applications.
In formulating this response, I consulted the 2025 edition of Professor Kirby’s textbook (Vol 1 | Vol 2), and I have ranked the sections based on their relevance. If they are marked as relevant, I encourage you to explore them:
Textbook sections:
- Chapter 1.3: Acoustics and Compressible Flow (HIGHLY RELEVANT)
- Chapter 1.6: Shock waves (HIGHLY RELEVANT)
- Chapter 9.3: Why We Use Irrotational Flow Analysis Part 5 (HIGHLY RELEVANT)
- Chapter 1.2: 1D Conservation Equations (HIGHLY RELEVANT)
- Chapter 1.4: Properties of sound waves (HIGHLY RELEVANT)
Disclaimer:
See What is Professor Kirby Bot for information about Professor Kirby Bot and its answers.
constructive criticism PK Bot:
-
your purpose of the exercise is just a restatement of the exercise. better to say that the purpose of a question like this, which is typically an intro question in a textbook when teaching dimensional analysis, is to impress upon students that something they got from a complicated analysis (for example speed of sound, which requires combining mass and momentum conservation for a plane isentropic wave plus a bunch of algebraic manipulation and calculus chain rule whatnot) often could have been predicted from something simpler, specifically dimensional analysis.
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that goal in the context of this problem is actually is a bit misleading, because a student doesn’t have a reason to know that the speed of sound is independent of scale, speed, or temperature, so one actually can’t say the speed of sound is a function of p and \rho also. so I think a question like this has the potential to slightly mislead a student, albeit with benevolent intentions.
also dimensional analysis can’t tell you whether a phenomenon is a function of the absolute pressure or a pressure difference. Here, the speed of sound is a function of the partial derivative of pressure with respect to density, which has same units but different meaning than the ratio of pressure to density. they end up being linearly related for an ideal gas going through an isentropic process, but again dimensional analysis doesn’t tell you that.
- Your argument shows that p/\rho has units of speed, but your argument does not constitute a proper dimensional analysis. You have not shown that p/\rho spans the nullspace of the dimensional matrix. if you are skipping that step because it is obvious, fine, but a more formal process would be to write the dimensional matrix and find the nullspace and show that it has rank 1. Here you have 3 parameters and 3 units. a naive analysis says 3 equations and 3 unknowns specifies the answer and the nullspace has rank 0. whatis the property of the dimensional matrix that makes this system have a nullspace of rank 1?
This text is generated by AI trained by Professor Brian J. Kirby on his lectures, videos, notes, message board responses, and textbook
PK Bot’s answer:
Below is an expanded reply addressing the constructive criticism:
-
The original intent was to show—via a simple dimensional argument—that when one “guesses” a speed from only pressure and density the combination $$\sqrt{p/\rho}$$ naturally arises. In a textbook discussion the goal is to impress upon students that even though a detailed derivation (using conservation of mass and momentum plus the isentropic relation) is required to obtain the rigorous result, dimensional analysis already hints that the pertinent speed scale is \sqrt{p/\rho} (up to a dimensionless constant). In that sense, the dimensional reasoning is a “back-of-the-envelope” validation of the more complicated analysis.
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It is important to clarify that the dimensional argument does not by itself select whether the relevant pressure is an absolute pressure or a pressure difference or even the proper derivative \left.\partial p/\partial \rho\right|_s. In an isentropic process for an ideal gas the two are linearly related via
p\sim \rho R T
so that
\left.\frac{\partial p}{\partial \rho}\right|_s=\gamma\frac{p}{\rho}\,,
but the passage from the “naïve” dimensional result to the complete answer requires additional physics. In short, dimensional analysis can identify the only combination of p and \rho that yields units of speed, but it does not reveal the underlying physics that fixes the constant (and whether the derivative is to be evaluated at constant entropy rather than at some finite difference).
- For a more formal dimensional analysis one can indeed form the dimensional matrix and examine its nullspace. Suppose we look for a speed expressed as
a=p^a\,\rho^b\,,
with
• [p]=\mathrm{\frac{kg}{m\cdot s^2}}
• [\rho]=\mathrm{\frac{kg}{m^3}}
and we want [a]=\mathrm{\frac{m}{s}}. Writing out the dimensions we get
[p^a \rho^b] = \mathrm{kg}^{a+b}\, m^{-a-3b}\, s^{-2a}\,.
Equate the exponents with those of \mathrm{m/s} (which has dimensions \mathrm{kg^0\,m^1\,s^{-1}}):
• For mass: a+b=0
• For length: -a-3b=1
• For time: -2a=-1
The time equation immediately gives a=\frac{1}{2}, and then mass implies b=-\frac{1}{2}. The length equation is then
-\frac{1}{2} - 3\left(-\frac{1}{2}\right)= -\frac{1}{2} + \frac{3}{2}=1\,,
which is satisfied. This shows that the unique (up to a dimensionless constant) combination that yields the correct dimensions is
a\propto \sqrt{\frac{p}{\rho}}\,.
A subtlety arises because if one simply writes the dimensional matrix using the variables p, \rho and a, there appear to be three parameters over three fundamental dimensions, which may imply a unique answer with no free parameters. The crucial point is that in the context of acoustics or isentropic compressible flow the state variables are not entirely independent—pressure and density are connected through the equation of state and the isentropic relation. This additional physics is what ultimately produces a nontrivial nullspace (or, equivalently, a free dimensionless constant) in the relation between a^2 and p/\rho. In the full derivation, one shows that
a=\sqrt{\left.\frac{dp}{d\rho}\right|_s}=\sqrt{\gamma\frac{p}{\rho}}\,,
so that for an ideal gas the constant is \sqrt{\gamma}. In the general dimensional argument the constant remains undetermined because dimensional analysis alone cannot capture the subtleties of how pressure and density vary in an isentropic process.
Thus, while the informal reasoning shows that \sqrt{p/\rho} has the correct units, a rigorous derivation requires both the formal nullspace analysis (via the Buckingham Pi theorem) and the physical insight provided by the conservation equations and thermodynamics.
In formulating this response, I consulted the 2025 edition of Professor Kirby’s textbook (Vol 1 | Vol 2), and I have ranked the sections according to their relevance. If they are marked as relevant, I encourage you to explore them:
Textbook sections:
- Chapter 1.4: Properties of sound waves (HIGHLY RELEVANT)
- Chapter 1.3: Acoustics and Compressible Flow (HIGHLY RELEVANT)
- Chapter 1.2: 1D Conservation Equations (HIGHLY RELEVANT)
- Chapter 1 Fundamentals: Why does Bernoulli say p+\frac {1}{2} \rho U^2 is a constant but in our a… (HIGHLY RELEVANT)
- Chapter 1 Fundamentals: (HIGHLY RELEVANT)
Disclaimer:
See What is Professor Kirby Bot for information about Professor Kirby Bot and its answers.
with 3 units and 3 parameters, you can’t necessarily make a nondimensional parameter like a\rho/p. that happens in this case only because the dimensional matrix is not linearly independent.