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Chapter 7: Problem 25

The speed of sound in a gas, \(c,\) is a function of the gas pressure, \(p,\) and density, \(\rho .\) Determine, with the aid of dimensional analysis, how the velocity is related to the pressure and density. Be careful when you decide on how many reference dimensions are required.

Short Answer

Expert verified
The speed of sound is proportional to the square root of the pressure divided by density: \( c \propto \sqrt{\frac{p}{\rho}} \).

Step by step solution

01

Identify Key Variables

The problem involves the speed of sound, denoted as \( c \), the gas pressure \( p \), and the gas density \( \rho \). These are the primary variables required for the analysis.
02

Determine Dimensional Formulas

Assign the basic dimensional formulas to each variable. For velocity \( c \), the dimensional formula is \( [LT^{-1}] \). The pressure \( p \) has the formula \( [ML^{-1}T^{-2}] \), and the density \( \rho \) has the formula \( [ML^{-3}] \).
03

Formulate the Dimensional Equation

Assume that the velocity \( c \) is proportional to some powers of \( p \) and \( \rho \), expressed as \( c = k \cdot p^a \cdot \rho^b \), where \( k \) is a dimensionless constant, and \( a \) and \( b \) are the powers to be determined.
04

Express in Dimensional Terms

Using the dimensional formulas, express the equation as: \([LT^{-1}] = [ML^{-1}T^{-2}]^a \cdot [ML^{-3}]^b\). This implies \([LT^{-1}] = [M^aL^{-a}T^{-2a}][M^bL^{-3b}]\).
05

Match Dimensions

Combine the terms and equate the powers of each dimension on both sides:1. For mass: \( a + b = 0 \)2. For length: \( -a - 3b = 1 \)3. For time: \( -2a = -1 \)
06

Solve for Exponents

Solve the system of equations:From equation 3, find \( a = \frac{1}{2} \).Substitute \( a \) into equation 1: \( \frac{1}{2} + b = 0 \) leads to \( b = -\frac{1}{2} \).Verify using equation 2: \(-\frac{1}{2} - 3(-\frac{1}{2}) = 1\) holds true.
07

Determine the Functional Relationship

Substitute the values of \( a \) and \( b \) back into the original equation to get the relationship: \( c = k \sqrt{\frac{p}{\rho}} \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Speed of Sound
The speed of sound is crucial in understanding how sound waves travel through different mediums. In gases, it depends largely on two factors: the pressure and the density of the gas. Physically, the speed of sound is how fast the sound waves travel through the air or any other medium.
  • It is typically measured in meters per second \( m/s \).
  • Faster in solids compared to gases due to molecular closeness.
In air, for instance, it is approximately 343 \( m/s \) at room temperature. However, this can change based on different conditions like temperature and humidity.
Understanding how the speed of sound is affected by pressure and density is significant for numerous applications, from aviation to music.
Gas Pressure
Gas pressure is a force exerted by the gas particles per unit area on the walls of its container. It is a critical factor in determining the speed of sound. You can think of it as how hard the gas molecules are pushing against each other and their container.
Pressure directly influences how quickly sound can travel through a gas.
  • Measured in pascals (\[ Pa \]) within the SI units.
  • Derives from both the concentration of particles and their temperature.
The higher the pressure in a gas, the faster sound can travel through it. This relationship stems from the fact that higher pressure typically means more tightly packed particles, aiding quicker vibration transfer.
Density
Density refers to how much mass of a gas is present in a given volume. It's mathematically represented as mass per unit volume, often measured in kilograms per cubic meter (\[ kg/m^3 \]). Like pressure, density significantly affects the speed of sound.
When the density increases, you would generally expect the medium to be more rigid and thus potentially increase the speed of sound. However, in a gas, the reality is more nuanced.
  • An increase in density from increased mass tends to slow sound down due to higher inertia.
  • An increase caused by pressure tends to speed sound up due to closer particle interactions.
Therefore, understanding density's role requires considering its interaction with other factors like pressure.
Velocity-Pressure-Density Relationship
The relationship between velocity (or speed of sound), pressure, and density of a gas is elegantly expressed through dimensional analysis. The analysis shows that the speed of sound \( c \) is directly linked to both gas pressure \( p \) and density \( \rho \).
This relationship can be expressed by the formula:
  • \( c = k \sqrt{\frac{p}{\rho}} \)
where \( k \) is a dimensionless constant. This equation reveals that:
  • Higher pressure \( p \) tends to increase the speed of sound.
  • Higher density \( \rho \) tends to decrease it.
This balance is the basis for understanding sound propagation in different environments and conditions.

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Most popular questions from this chapter

Flow patterns that develop as winds blow past a vehicle, such as a train, are often studied in low-speed environmental (meteorological wind tunnels. (See Video \(\mathrm{V} 7.16 .\) ) Typically, the air velocities in these tunnels are in the range of \(0.1 \mathrm{m} / \mathrm{s}\) to \(30 \mathrm{m} / \mathrm{s}\). Consider a cross wind blowing past a train locomotive. Assume that the local wind velocity, \(V,\) is a function of the approaching wind velocity (at some distance from the locomotive), \(U,\) the locomotive length, \(\ell\) height, \(h,\) and width, \(b,\) the air density, \(\rho,\) and the air viscosity, \(\mu\). (a) Establish the similarity requirements and prediction equation for a model to be used in the wind tunnel to study the air velocity, \(V\) around the locomotive. (b) If the model is to be used for cross winds gusting to \(U=25 \mathrm{m} / \mathrm{s}\), explain why it is not practical to maintain Reynolds number similarity for a typical length scale 1: 50 .

Shown in the following table are several flow situations and the associated characteristic velocity, size, and fluid kinematic viscosity. Determine the Reynolds number for each of the flows and indicate for which ones the inertial effects are small relative to viscous effects.

The pressure drop per unit length, \(\Delta p_{\ell},\) for the flow of blood through a horizontal small-diameter tube is a function of the volume rate of flow, \(Q,\) the diameter, \(D,\) and the blood viscosity, \(\mu\) For a series of tests in which \(D=2 \mathrm{mm}\) and \(\mu=0.004 \mathrm{N} \cdot \mathrm{s} / \mathrm{m}^{2}\) the following data were obtained, where the \(\Delta p\) listed was measured over the length, \(\ell=300 \mathrm{mm}\) $$\begin{array}{cc} Q\left(\mathbf{m}^{3} / \mathbf{s}\right) & \mathbf{\Delta} p\left(\mathbf{N} / \mathbf{m}^{2}\right) \\ \hline 3.6 \times 10^{-6} & 1.1 \times 10^{4} \\ 4.9 \times 10^{-6} & 1.5 \times 10^{4} \\ 6.3 \times 10^{-6} & 1.9 \times 10^{4} \\ 7.9 \times 10^{-6} & 2.4 \times 10^{4} \\ 9.8 \times 10^{-6} & 3.0 \times 10^{4} \end{array}$$ Perform a dimensional analysis for this problem, and make use of the data given to determine a general relationship between \(\Delta p_{\ell}\) and \(Q(\text { one that is valid for other values of } D, \ell, \text { and } \mu)\).

At a sudden contraction in a pipe the diameter changes from \(D_{1}\) to \(D_{2} .\) The pressure drop, \(\Delta p,\) which develops across the contraction is a function of \(D_{1}\) and \(D_{2},\) as well as the velocity, \(V\) in the larger pipe, and the fluid density, \(\rho,\) and viscosity, \(\mu .\) Use \(D_{1}, V,\) and \(\mu\) as repeating variables to determine a suitable set of dimensionless parameters. Why would it be incorrect to include the velocity in the smaller pipe as an additional variable?

The drag characteristics of an airplane are to be determined by model tests in a wind tunnel operated at an absolute pressure of 1300 kPa. If the prototype is to cruise in standard air at 385 \(\mathrm{km} / \mathrm{hr},\) and the corresponding speed of the model is not to differ by more than \(20 \%\) from this (so that compressibility effects may be ignored), what range of length scales may be used if Reynolds number similarity is to be maintained? Assume the viscosity of air is unaffected by pressure, and the temperature of air in the tunnel is equal to the temperature of the air in which the airplane will fly.
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