page_content stringlengths 12 2.63M | metadata unknown |
|---|---|
Certain crystals may be exploited as transducers because they have the following properties. (a) The crystal's physical dimensions change when an electric field is applied to it. (b) Deformations caused by mechanic forces cause electric polarization on the surfaces of the crystal. These effects are subsumed under the t... | {
"Header 1": "6.1 The Piezoelectric Transduction Principle",
"token_count": 2036,
"source_pdf": "datasets/websources/Physics_v1/Physics/acoustics-for-engineers-2nd_troy-lecture.pdf"
} |
A relatively high capacitance of about 1–3 µF is valuable from a practical standpoint because it allows for a few meters of cable before amplification becomes necessary.
Fig. 6.6. (a) Two-mass longitudinal vibrator, (b) crystal microphone
Figure 6.6 (b) shows a traditional microphone ... | {
"Header 1": "6.1 The Piezoelectric Transduction Principle",
"token_count": 704,
"source_pdf": "datasets/websources/Physics_v1/Physics/acoustics-for-engineers-2nd_troy-lecture.pdf"
} |
All dielectric materials experience mechanic deformations when exposed to an electric field. There is a basic quadratic relationship between the strength of the field and the stress of the material, which can, however, be linearized with the bias of a permanent electric field. It is in this way that the electrostrictiv... | {
"Header 1": "6.3 The Electrostrictive Transduction Principle",
"token_count": 801,
"source_pdf": "datasets/websources/Physics_v1/Physics/acoustics-for-engineers-2nd_troy-lecture.pdf"
} |
Among the electrostrictive sound emitters and receivers there are many shapes in addition to those used in piezoelectric transducers. It is indeed the great advantage of influenced piezoelectricity that many relevant materials are freely formable. Piezopolymeres can even be manufactured into foils (films) as thin as a ... | {
"Header 1": "6.4 Electrostrictive Sound Emitters and Receivers",
"token_count": 257,
"source_pdf": "datasets/websources/Physics_v1/Physics/acoustics-for-engineers-2nd_troy-lecture.pdf"
} |
Historically this principle was called the electrostatic principle. The result of the arrangement shown in Fig. 6.13 is a quadratic force law. Two conducting plates, one fixed and one movable, are exposed to an electric voltage, u. The plates become electrically charged, resulting in an electrostatic pulling force.
!... | {
"Header 1": "6.5 The Dielectric Transduction Principle",
"token_count": 2030,
"source_pdf": "datasets/websources/Physics_v1/Physics/acoustics-for-engineers-2nd_troy-lecture.pdf"
} |
If such a material, for example teflon or flourcarbon, is positioned behind the membrane, an external polarization voltage becomes superfluous. Actually, these electrets easily mimic polarization voltages of 100 V.
Figure 6.20 illustrates the principle. Electret microphones, including an integrated impedance converte... | {
"Header 1": "6.5 The Dielectric Transduction Principle",
"token_count": 321,
"source_pdf": "datasets/websources/Physics_v1/Physics/acoustics-for-engineers-2nd_troy-lecture.pdf"
} |
In addition to the transducer principles that we already dealt with in this and the prior sections, there are some further techniques, even without magnetic and electric fields that we would like to briefly mention.
The first is based on the fact that a wire with a varying current passing through it heats up, causes ... | {
"Header 1": "6.7 Further Transducer and Coupler Principles",
"token_count": 341,
"source_pdf": "datasets/websources/Physics_v1/Physics/acoustics-for-engineers-2nd_troy-lecture.pdf"
} |
So far in this book we have dealt with vibrations. These are processes that vary as functions of time. We were able to describe relevant types of vibrations with common differential equations. This chapter now focuses on waves, which are processes that vary with both time and space. Their mathematical description requi... | {
"Header 1": "The Wave Equation in Fluids",
"token_count": 1096,
"source_pdf": "datasets/websources/Physics_v1/Physics/acoustics-for-engineers-2nd_troy-lecture.pdf"
} |
We assume an idealized medium as a model of the real physical medium. Only compression and expansion but no shear stress are allowed in this medium, which limits us to a category of media called *fluids*. Many gases and liquids can be treated as fluids, which only experience longitudinal waves. The following features o... | {
"Header 1": "7.1 Derivation of the One-Dimensional Wave Equation",
"token_count": 2009,
"source_pdf": "datasets/websources/Physics_v1/Physics/acoustics-for-engineers-2nd_troy-lecture.pdf"
} |
The mass flowing in during a time interval, dt, is
$$dm_{\rm in} = A \ \varrho_{-} \ \overrightarrow{v} \ \overrightarrow{dt} \,, \tag{7.20}$$
and the mass flowing out during the same time interval is
$<sup>^3</sup>$ The characteristic field impedance, $Z_{\rm w},$ also known as wave impedance, will be intro... | {
"Header 1": "7.1 Derivation of the One-Dimensional Wave Equation",
"token_count": 680,
"source_pdf": "datasets/websources/Physics_v1/Physics/acoustics-for-engineers-2nd_troy-lecture.pdf"
} |
The two wave equations that we have just derived are valid for a onedimensional axial wave in a tube. The walls of the tube do not play any role since we have a purely longitudinal wave that actually propagates in parallel to the walls.
Our equations are also valid for a bundle of tubes – shown in Fig. 7.5 – and noth... | {
"Header 1": "7.2 Three-Dimensional Wave Equation in Cartesian Coordinates",
"token_count": 730,
"source_pdf": "datasets/websources/Physics_v1/Physics/acoustics-for-engineers-2nd_troy-lecture.pdf"
} |
Now we will present solutions of the wave equation. To keep the discussion simple, we restrict ourselves to one dimensional cases, specifically the wave equation in the following form, as predicted at the beginning of this chapter<sup>7</sup>.
$$\frac{\partial^2 p}{\partial x^2} = \frac{1}{c^2} \frac{\partial^2 p}{\p... | {
"Header 1": "7.2 Three-Dimensional Wave Equation in Cartesian Coordinates",
"Header 2": "7.3 Solutions of the Wave Equation",
"token_count": 811,
"source_pdf": "datasets/websources/Physics_v1/Physics/acoustics-for-engineers-2nd_troy-lecture.pdf"
} |
First we look at the forward progressing wave, expressed as
$$\underline{p}(x) = \underline{p}_{\rightarrow} e^{-j\beta x}, \text{ and } \underline{v}(x) = \underline{v}_{\rightarrow} e^{-j\beta x}.$$
(7.40)
Following Euler, we obtain
$$\varrho \frac{\partial v}{\partial t} = -\frac{\partial p}{\partial x} \quad ... | {
"Header 1": "7.4 Field Impedance and Power Transport in Plane Waves",
"token_count": 708,
"source_pdf": "datasets/websources/Physics_v1/Physics/acoustics-for-engineers-2nd_troy-lecture.pdf"
} |
As previously mentioned, there is only axial wave propagation in tubes with much smaller diameters than wavelengths, $d \ll \lambda$ .
Since the ratio of $\underline{p}$ and $\underline{v}$ inside the tube is only dependent on the terminating impedance of the tube, $\underline{Z}_0 = \underline{p}_0/\underline{... | {
"Header 1": "7.4 Field Impedance and Power Transport in Plane Waves",
"Header 2": "7.5 Transmission-Line Equations and Reflectance",
"token_count": 1845,
"source_pdf": "datasets/websources/Physics_v1/Physics/acoustics-for-engineers-2nd_troy-lecture.pdf"
} |
The acoustic measuring tube, also known as Kundt's tube, can be used to measure field impedances and reflectances in the terminal plane of a tube. Such a tube is schematically shown in Fig. 7.7. The tube can be excited by a sound source from one end, a sinusoidal sound in our case. The other end is terminated by the im... | {
"Header 1": "7.6 The Acoustic Measuring Tube",
"token_count": 1606,
"source_pdf": "datasets/websources/Physics_v1/Physics/acoustics-for-engineers-2nd_troy-lecture.pdf"
} |
The wave equations derived in the preceding chapter allow for calculation of arbitrary sound fields with any possible, physically meaningful boundary conditions. We had restricted ourselves to one-dimensional waves so far. These can, for instance, be observed in tubes with a diameter being small compared to the wavelen... | {
"Header 1": "Horns and Stepped Ducts",
"token_count": 405,
"source_pdf": "datasets/websources/Physics_v1/Physics/acoustics-for-engineers-2nd_troy-lecture.pdf"
} |
This chapter deals with the condition where the area function varies only gradually, and perpendicular areas are areas of approximately constant phase. This case is captured by the so-called Webster equation or Horn equation. Figure 8.2 illustrates the derivation of this differential equation. Please consider the eleme... | {
"Header 1": "8.1 Webster's Differential Equation – the Horn Equation",
"token_count": 1954,
"source_pdf": "datasets/websources/Physics_v1/Physics/acoustics-for-engineers-2nd_troy-lecture.pdf"
} |
The field impedance $\underline{Z}_{\rm f}$ of the conical sound field results from dividing (8.13) by (8.14) and is
$$\underline{Z}_{f}(x) = \frac{\underline{p}_{\rightarrow}(x)}{\underline{v}_{\rightarrow}(x)} = \frac{1}{\frac{1}{\rho c} + \frac{1}{j\omega\rho x}} = \varrho c \frac{j\frac{2\pi x}{\lambda}}{1 + ... | {
"Header 1": "8.1 Webster's Differential Equation – the Horn Equation",
"token_count": 370,
"source_pdf": "datasets/websources/Physics_v1/Physics/acoustics-for-engineers-2nd_troy-lecture.pdf"
} |
The area function of the exponential horn – see Fig. 8.5 – is given by
$$A(x) = A_0 e^{2\epsilon x}, (8.19)$$
with ² > 0 being the so-called flare coefficient.
Fig. 8.5. Exponential horn
By differentiation we get
$$\frac{1}{A(x)}\frac{\mathrm{d}A}{\mathrm{d}x} = \frac{\mathrm{d... | {
"Header 1": "8.3 Exponential Horns",
"token_count": 1405,
"source_pdf": "datasets/websources/Physics_v1/Physics/acoustics-for-engineers-2nd_troy-lecture.pdf"
} |
The acoustic power that is sent out by an electro-acoustic transducer or any other sound source, is proportional to the real part of the impedance, r rad = Re{Z rad} that terminates the source at its acoustic output port. Since this impedance is formed by the sound field coupled to the source, we call it radiation impe... | {
"Header 1": "8.4 Radiation Impedances and Sound Radiation",
"token_count": 830,
"source_pdf": "datasets/websources/Physics_v1/Physics/acoustics-for-engineers-2nd_troy-lecture.pdf"
} |
We shall now discuss the situation at the position of the step in a tube – shown in Fig. 8.7. Left and right of the step, we have tubes with constant, though different diameters. As already mentioned at the beginning of this chapter, perpendicular modes at this position may be neglected because they cannot propagate as... | {
"Header 1": "8.5 Steps in the Area Function",
"token_count": 945,
"source_pdf": "datasets/websources/Physics_v1/Physics/acoustics-for-engineers-2nd_troy-lecture.pdf"
} |
With $\underline{q}$ as the second wave quantity, we can easily deal with stepped ducts by means of electric analogies. Furthermore, we can include acoustic concentrated elements into our consideration within the same analogue circuits – refer to Section 2.6. This means, in fact, that the theories of analysis and syn... | {
"Header 1": "8.5 Steps in the Area Function",
"Header 2": "8.6 Stepped Ducts",
"token_count": 1590,
"source_pdf": "datasets/websources/Physics_v1/Physics/acoustics-for-engineers-2nd_troy-lecture.pdf"
} |
The wave equation, ∇<sup>2</sup>p = ¨p/c<sup>2</sup> as derived in Sections 7.1 and 7.2 theoretically determines all possible sound fields in idealized fluids, that is, gases and liquids. The special task of computing sound fields for particular cases requires solutions of the wave equation for particular boundary cond... | {
"Header 1": "Spherical Sound Sources and Line Arrays",
"token_count": 459,
"source_pdf": "datasets/websources/Physics_v1/Physics/acoustics-for-engineers-2nd_troy-lecture.pdf"
} |
The wave equation allows for a one-dimensional, point-symmetric solution. This is a sound wave where all parameters only depend on the distance from the origin, r. The solution does not depend on the direction of propagation, which is always radial and directed either outward or toward the center. This type of wave is ... | {
"Header 1": "9.1 Spherical Sound Sources of 0<sup>th</sup> Order",
"token_count": 2014,
"source_pdf": "datasets/websources/Physics_v1/Physics/acoustics-for-engineers-2nd_troy-lecture.pdf"
} |
\tag{9.13}$$
The term $4\pi$ in the denominator of equation (9.11) denotes the full spherical angle, $\Omega_{\Sigma} = 4\pi$ . If a 0<sup>th</sup>-order spherical source with the source strength $\underline{q}_0$ radiates into a smaller spherical angle, $\Omega_1$ , that is only a section of the available volu... | {
"Header 1": "9.1 Spherical Sound Sources of 0<sup>th</sup> Order",
"token_count": 322,
"source_pdf": "datasets/websources/Physics_v1/Physics/acoustics-for-engineers-2nd_troy-lecture.pdf"
} |
In the $0^{\rm th}$ -order spherical sound field we have
$$\frac{\underline{\underline{p}}(r)}{\underline{\underline{v}}(r)} = \frac{1}{\frac{1}{\varrho c} + \frac{1}{\mathrm{j}\omega \varrho r}}, \quad \text{or} \quad \underline{\underline{g}}_{\rightarrow} \frac{\mathrm{e}^{-\mathrm{j}\beta r}}{r} = \frac{\underli... | {
"Header 1": "Point Sources of $0^{th}$ Order (Monopoles)",
"token_count": 596,
"source_pdf": "datasets/websources/Physics_v1/Physics/acoustics-for-engineers-2nd_troy-lecture.pdf"
} |
A rigid sphere may oscillate according to the sketch in Fig. 9.2 (b), and a sound field created in this way is called a 1<sup>st</sup>-order spherical sound field. Such a sound field is no longer point-symmetric, which means that the shells around the sphere do not represent areas of equal phase. This may also be expre... | {
"Header 1": "9.2 Spherical Sound Sources of 1st Order",
"token_count": 260,
"source_pdf": "datasets/websources/Physics_v1/Physics/acoustics-for-engineers-2nd_troy-lecture.pdf"
} |
Two point sources with equal strength but of opposite phase, that is $\underline{q}_1 = -\underline{q}_0$ and $\underline{q}_2 = +\underline{q}_0$ , are positioned a distance 2d apart, forming a so-called *dipole*. Due to the linearity of the wave equation, the sound field of this arrangement is given by superpositi... | {
"Header 1": "Point Sources of 1st Order (Dipoles)",
"token_count": 1470,
"source_pdf": "datasets/websources/Physics_v1/Physics/acoustics-for-engineers-2nd_troy-lecture.pdf"
} |
Any sound-field can be considered to be composed of a series of orthogonal spherical harmonics of different orders. These spherical harmonic waves are eigen-functions of the wave equation in spherical coordinates. The first two, the spherical waves of 0<sup>th</sup> and 1<sup>st</sup> order, have been introduced in the... | {
"Header 1": "9.3 Higher-Order Spherical Sound Sources",
"token_count": 402,
"source_pdf": "datasets/websources/Physics_v1/Physics/acoustics-for-engineers-2nd_troy-lecture.pdf"
} |
In an arrangement like the one depicted in Fig. 9.6, we look at a reference point at a distance of r<sup>0</sup> À 2h. The sum of the contributions of all monopoles of the array is
$$\underline{p}_{\rightarrow}(r=r_0,\delta) = \frac{\mathrm{j}\omega \,\varrho \,\underline{q}_0}{4\pi} \sum_{i=1}^n \,\frac{\mathrm{e}^{... | {
"Header 1": "9.4 Line Arrays of Monopoles",
"Header 2": "Line Array of Identical and Equidistant Monopoles",
"token_count": 619,
"source_pdf": "datasets/websources/Physics_v1/Physics/acoustics-for-engineers-2nd_troy-lecture.pdf"
} |
First we perform a limit operation by letting the distance between the individual monopoles, 2d, and, consequently, b, go to zero. With the length of line array, 2h, kept constant, we then get n → ∞. To also keep the total source strength, n q <sup>0</sup> = q 0 (x) 2h, constant, we normalize by n, with q 0 (x) being a... | {
"Header 1": "9.4 Line Arrays of Monopoles",
"Header 2": "Continuously Loaded Line Array",
"token_count": 377,
"source_pdf": "datasets/websources/Physics_v1/Physics/acoustics-for-engineers-2nd_troy-lecture.pdf"
} |
In the preceding section we assumed a continuous source-strength load, q 0 (x), having the dimension [volume velocity/length]. We continued to presume that each point on the line acts as a monopole – shown in Fig. 9.8. With the following integration we can now calculate the sound pressure at the observation point, $r_... | {
"Header 1": "9.5 Analogy to Fourier Transforms as Used in Signal Theory",
"token_count": 1209,
"source_pdf": "datasets/websources/Physics_v1/Physics/acoustics-for-engineers-2nd_troy-lecture.pdf"
} |
When a reversible transducer or transducer array is operated as sound emitter, its directional characteristic is equivalent to its directional sensitivity characteristic when operated as a receiver.
This can be shown by using the following two elements,
M ... a transducer with arbitrary directional characteristics ... | {
"Header 1": "9.6 Directional Equivalence of Sound Emitters and Receivers",
"token_count": 874,
"source_pdf": "datasets/websources/Physics_v1/Physics/acoustics-for-engineers-2nd_troy-lecture.pdf"
} |
In the last chapter the sound field produced by point-source arrangements was calculated by linear superposition of individual spherical sound fields. For continuously loaded line arrays we further substituted the source strength of the individual sources with a length-specific source-strength load, q 0 0 (x). The sour... | {
"Header 1": "Piston Membranes, Diffraction and Scattering",
"token_count": 409,
"source_pdf": "datasets/websources/Physics_v1/Physics/acoustics-for-engineers-2nd_troy-lecture.pdf"
} |
A small vibrating piston results in a 1st-order spherical sound field– see Fig. 9.5 (b). Yet, building this piston into an infinitely extended, plane and rigid baffle prohibits hydrodynamic shorting between the front and back of the baffle. This results in a hemispherical sound field of 0th order radiating into half of... | {
"Header 1": "10.1 The Rayleigh Integral",
"token_count": 581,
"source_pdf": "datasets/websources/Physics_v1/Physics/acoustics-for-engineers-2nd_troy-lecture.pdf"
} |
Fraunhofer's approximation applies when the distance from the reference point to the membrane is very large in comparison to the linear dimensions of the radiating membrane. Figure 10.3 depicts the situation to be discussed.
Fig. 10.3. Fraunhofer's approximation
The quantity r<sup>0<... | {
"Header 1": "10.2 Fraunhofer's Approximation",
"token_count": 552,
"source_pdf": "datasets/websources/Physics_v1/Physics/acoustics-for-engineers-2nd_troy-lecture.pdf"
} |
This section deals with the sound field produced by rigid membranes, so-called piston membranes in infinitely extended, rigid, flat baffles. In other words, we speak of baffled pistons with identic v (x, y) everywhere on the piston. Such piston membranes can serve as models for loudspeakers in large baffles as long as ... | {
"Header 1": "10.3 The Far Field of Piston Membranes",
"token_count": 926,
"source_pdf": "datasets/websources/Physics_v1/Physics/acoustics-for-engineers-2nd_troy-lecture.pdf"
} |
Because the sound field close to a membrane can not be computed by Fraunhofer's approximation, the Rayleigh integral itself must be solved. Discrete numerical methods are commonly used to accomplish this.
#### Zone Construction after Huygens and Fresnel
In this discussion, we introduce a traditional method of solvi... | {
"Header 1": "10.4 The Near Field of Piston Membranes",
"token_count": 1231,
"source_pdf": "datasets/websources/Physics_v1/Physics/acoustics-for-engineers-2nd_troy-lecture.pdf"
} |
We will now discuss the sound pressure of a diverging wave, |p <sup>→</sup> (r0)|, on the middle axis of a circular piston membrane. This is an example of the near field of a circular piston membrane in an infinitely extended, rigid, plane baffle with radius R.
For a given r<sup>0</sup> and wavelength, λ, the number ... | {
"Header 1": "10.4 The Near Field of Piston Membranes",
"Header 2": "On-Axis Sound Pressure and Radiation Impedance of Circular Piston Membranes",
"token_count": 1197,
"source_pdf": "datasets/websources/Physics_v1/Physics/acoustics-for-engineers-2nd_troy-lecture.pdf"
} |
The calculations performed above for pistons can be expanded to the phenomenon of *diffraction* caused by circular holes in baffles – see Fig. 10.11. We assume that a plane wave hits such a baffle perpendicularly, and that there is
a constant velocity, v (x, y), normal to the plane of the area of the hole. The situat... | {
"Header 1": "10.5 General Remarks on Diffraction and Scattering",
"token_count": 859,
"source_pdf": "datasets/websources/Physics_v1/Physics/acoustics-for-engineers-2nd_troy-lecture.pdf"
} |
The wave equation as derived in Section 7.1 and used so far in this book, is valid for sound propagation in lossless media. The Helmholtz form of this equation is
$$\frac{\partial^2 \underline{p}}{\partial x^2} - (j\beta)^2 \underline{p} = 0.$$
(11.1)
Its solution for the forward-progressing plane wave can be expre... | {
"Header 1": "Dissipation, Reflection, Refraction, and Absorption",
"token_count": 905,
"source_pdf": "datasets/websources/Physics_v1/Physics/acoustics-for-engineers-2nd_troy-lecture.pdf"
} |
The assumptions for lossless sound propagation, as put forward in Section 7.1 are as follows,
- No inner friction, which means no viscosity
- Negligible thermal conduction
- Behavior as a perfect gas
It goes without saying that these assumptions are not strictly valid in real fluids. Therefore we have to consider t... | {
"Header 1": "11.1 Dissipation During Sound Propagation in Air",
"token_count": 1002,
"source_pdf": "datasets/websources/Physics_v1/Physics/acoustics-for-engineers-2nd_troy-lecture.pdf"
} |
From the previous discussion it is easy to comprehend that sound propagation in media that are perforated by many narrow tubes and connected cavities, must be profoundly damped. Media of this consistency are called porous<sup>4</sup>. With the set as shown in Fig. 11.2, the characteristic flow resistance, $\Xi$ , of p... | {
"Header 1": "11.2 Sound Propagation in Porous Media",
"token_count": 1389,
"source_pdf": "datasets/websources/Physics_v1/Physics/acoustics-for-engineers-2nd_troy-lecture.pdf"
} |
Reflection and refraction are important phenomena in wave propagation. They can be treated together by assuming a situation where a wave hits the boundary between two media, medium 1 and medium 2, with different speeds of sound, c<sup>1</sup> and c2. If we take both the boundary and the wave as infinite in space, the s... | {
"Header 1": "11.3 Reflection and Refraction",
"token_count": 792,
"source_pdf": "datasets/websources/Physics_v1/Physics/acoustics-for-engineers-2nd_troy-lecture.pdf"
} |
This section deals with the wave effects at the boundary between two fluid media in more detail and, by appropriate extrapolation, will also approximately cover the effect of waves encountering a wall.
#### Boundary Between Two Fluids
As a starting point we take a case as depicted in Fig. 11.5. The characteristic i... | {
"Header 1": "11.4 Wall Impedance and Degree of Absorption",
"token_count": 1686,
"source_pdf": "datasets/websources/Physics_v1/Physics/acoustics-for-engineers-2nd_troy-lecture.pdf"
} |
Porous absorbers are very important for practical applications. They are, for example, built from fibrous materials such as fabric, mineral wool or cocos fibre that is compressed into mats or plates as well as from porously extruded artificial foams. To arrive at an estimate of the absorptive behavior, the Rayleigh mod... | {
"Header 1": "11.5 Porous Absorbers",
"token_count": 2030,
"source_pdf": "datasets/websources/Physics_v1/Physics/acoustics-for-engineers-2nd_troy-lecture.pdf"
} |
So far in this book we dealt with sound propagation in terms of the wave equation. This procedure becomes very complicated, however, when treating sound fields inside rooms with complicated shapes like concert halls or churches. An approximate method called *geometrical acoustics* is often useful in these cases.
This... | {
"Header 1": "11.5 Porous Absorbers",
"Header 2": "Geometric Acoustics and Diffuse Sound Fields",
"token_count": 657,
"source_pdf": "datasets/websources/Physics_v1/Physics/acoustics-for-engineers-2nd_troy-lecture.pdf"
} |
The behavior of rays at plane reflecting surfaces is particularly relevant for geometrical acoustics. Plane means here, that any unevenness of the surface is small compared to the wavelengths of the sound considered. The reflection law, Θ<sup>1</sup> = Θ<sup>0</sup> 1 , holds, and may even be applied to slightly curved... | {
"Header 1": "12.1 Mirror Sound Sources and Ray Tracing",
"token_count": 1872,
"source_pdf": "datasets/websources/Physics_v1/Physics/acoustics-for-engineers-2nd_troy-lecture.pdf"
} |
We will now move beyond the case of two parallel walls and consider a rectangular (cuboid) room with six reflecting boundaries, namely, four walls, one floor and one ceiling. This room will illustrate an important rule in room acoustics that which states that the reflection density, n 0 , increases with t 2 in many roo... | {
"Header 1": "12.3 Impulse Responses of Rectangular Rooms",
"token_count": 2005,
"source_pdf": "datasets/websources/Physics_v1/Physics/acoustics-for-engineers-2nd_troy-lecture.pdf"
} |
\tag{12.20}$$
The rays transport this energy to the wall on an average of n 0 times per second, that is
$$\overline{n}' W_{\mathrm{d}}'' V = \frac{W_{\mathrm{d}}'' c}{4} A \cdot 1 s$$
, with $\overline{n}' = \frac{A c}{4V}$ . (12.21)
This leads to the expression for the mean free-path length as
$$\bar{l} = \fr... | {
"Header 1": "12.3 Impulse Responses of Rectangular Rooms",
"token_count": 1216,
"source_pdf": "datasets/websources/Physics_v1/Physics/acoustics-for-engineers-2nd_troy-lecture.pdf"
} |
We will now introduce three well known and frequently used applications of the diffuse-sound-field model.
#### Reverberation-time Acoustics
The reverberation time, T, estimated with either Eyring's or Sabine's formula, is considered to be a relevant parameter of the acoustic quality of spaces, which includes, among... | {
"Header 1": "12.6 Application of Diffuse Sound Fields",
"token_count": 347,
"source_pdf": "datasets/websources/Physics_v1/Physics/acoustics-for-engineers-2nd_troy-lecture.pdf"
} |
Absorption measurements are taken in reverberation chambers, which are rooms where a diffuse sound field has been realized using highly reflective, obliquely oriented walls and planes<sup>4</sup> .
The equivalent absorptive area, A<sup>α</sup><sup>1</sup> , of the empty chamber must be determined beforehand, usually ... | {
"Header 1": "12.6 Application of Diffuse Sound Fields",
"Header 2": "Measurement of Spatially Averaged Absorption",
"token_count": 1044,
"source_pdf": "datasets/websources/Physics_v1/Physics/acoustics-for-engineers-2nd_troy-lecture.pdf"
} |
Sound isolation is the confinement of sound to a space in such a way that transmission to neighboring spaces is totally or partially prevented. Sound isolation is predominantly based on reflection caused by impedance discontinuities in possible transmission paths. Dissipation and absorption may also play a role in soun... | {
"Header 1": "Isolation of Air- and Structure-Borne Sound",
"token_count": 244,
"source_pdf": "datasets/websources/Physics_v1/Physics/acoustics-for-engineers-2nd_troy-lecture.pdf"
} |
An important difference between solids and fluids is that solids experience shear forces. This means that solids can store energy through both volume changes and changes of form. Solids consequently experience a number of wave types in addition to longitudinal waves since transverse movement with respect to the directi... | {
"Header 1": "13.1 Sound in Solids – Structure-Borne Sound",
"token_count": 358,
"source_pdf": "datasets/websources/Physics_v1/Physics/acoustics-for-engineers-2nd_troy-lecture.pdf"
} |
The specific combination of transverse and angular motion that characterizes bending waves result in considerable surface velocities and, consequently, the emission of airborne sound. This type of wave is of particular practical relevance because of this effect.
#### The Wave Equation for the Bending Waves
The wave... | {
"Header 1": "13.2 Radiation of Airborne Sound by Bending Waves",
"token_count": 1202,
"source_pdf": "datasets/websources/Physics_v1/Physics/acoustics-for-engineers-2nd_troy-lecture.pdf"
} |
We will restrict ourselves to non-porous walls in this section. In acoustical terms single leaf refers to a panel in which the cross-sectional particle velocity, vx, is identical at all points inside the leaf. This is the case in thin, solid leaves. Elongation waves inside the leaf may thus be neglected there.
Consid... | {
"Header 1": "13.3 Sound-Transmission Loss of Single-Leaf Walls",
"token_count": 2039,
"source_pdf": "datasets/websources/Physics_v1/Physics/acoustics-for-engineers-2nd_troy-lecture.pdf"
} |
As we just discussed, the sound-transmission loss of single-leaf walls is governed by the mass load. This certainly holds below the coincidence frequency, but it is also usually sufficient above it since the bending stiffness, B', of many wall materials is proportional to the mass load, m''.
In cases where the mass o... | {
"Header 1": "13.4 Sound-Transmission Loss of Double-Leaf Walls",
"token_count": 825,
"source_pdf": "datasets/websources/Physics_v1/Physics/acoustics-for-engineers-2nd_troy-lecture.pdf"
} |
In architectural acoustics and related fields the sound-isolation capability of a wall is characterized by an internationally standardized single-number index called the Weighted Sound-Reduction Index, Rw. This index is specific to the wall element considered and independent of its actual installation, for instance, in... | {
"Header 1": "13.5 The Weighted Sound-Reduction Index",
"token_count": 1163,
"source_pdf": "datasets/websources/Physics_v1/Physics/acoustics-for-engineers-2nd_troy-lecture.pdf"
} |
Isolation of structure-borne sound often turns out to be difficult in praxi. On the one hand, materials like steel or concrete have only very low damping coefficients for structure-borne sound waves. On the other hand, it is often not possible to construct adequate isolation measures like soft resilient inlays in optim... | {
"Header 1": "13.6 Isolation of Vibrations",
"token_count": 1510,
"source_pdf": "datasets/websources/Physics_v1/Physics/acoustics-for-engineers-2nd_troy-lecture.pdf"
} |
In architectural acoustics tapping sound, that is, structure-borne sound excited by walking on floors (footfall), is of particular relevance. Such sound can be transmitted to adjacent rooms, especially below the floor, and then be radiated as airborne sound. This also holds for other approximately point-wise sources of... | {
"Header 1": "13.7 Isolation of Floors with Regard to Impact Sounds",
"token_count": 628,
"source_pdf": "datasets/websources/Physics_v1/Physics/acoustics-for-engineers-2nd_troy-lecture.pdf"
} |
So far we have dealt with sound sources that convert electric energy into mechanic/acoustic energy. These sound sources are mainly used to radiate desired sounds because the mechanic/acoustic signals are easily controlled by the electric ones. In addition to desired sound, there is undesired sound that one would reduce... | {
"Header 1": "Noise Control – A Survey",
"token_count": 506,
"source_pdf": "datasets/websources/Physics_v1/Physics/acoustics-for-engineers-2nd_troy-lecture.pdf"
} |
Reasons why noise is generated are manifold, and a few examples of noise sources are listed below – ordered with respect to their sound excitation types. The list is not intended to be complete.
- Airborne sound sources
- excitation by explosion or implosion −→ impulsive sounds
- excitation by turbulent flow −→ non-p... | {
"Header 1": "14.1 Origins of Noise",
"token_count": 210,
"source_pdf": "datasets/websources/Physics_v1/Physics/acoustics-for-engineers-2nd_troy-lecture.pdf"
} |
Radiation of noise follows the general laws of sound radiation that have been discussed throughout this book, especially in Chapters 9, 10 and 11. Radiation has specific directional characteristics that depend on the form of the gas volumes or structures and the type of vibration they experience.
We will now discuss ... | {
"Header 1": "14.2 Radiation of Noise",
"token_count": 1718,
"source_pdf": "datasets/websources/Physics_v1/Physics/acoustics-for-engineers-2nd_troy-lecture.pdf"
} |
Noise that is generated and radiated from one or more sound sources propagates along one or multiple paths before arriving at one or more receivers (sinks). It seems natural to discuss this situation as a system in transmissiontheory terms. Measures for noise control in this system require knowledge
<sup>2</sup> It i... | {
"Header 1": "14.3 Noise Reduction as a System Problem",
"token_count": 503,
"source_pdf": "datasets/websources/Physics_v1/Physics/acoustics-for-engineers-2nd_troy-lecture.pdf"
} |
When making decisions, it is useful to list all possible measures and weight them according to feasibility and effort. An optimized battery of measures can be compiled on this basis. It is important to also consider extreme measures like replacing machines and completely terminating production in a certain setting. Per... | {
"Header 1": "14.3 Noise Reduction as a System Problem",
"Header 2": "Adequate Noise-control Measures",
"token_count": 203,
"source_pdf": "datasets/websources/Physics_v1/Physics/acoustics-for-engineers-2nd_troy-lecture.pdf"
} |
It goes without saying that the best method of noise control is avoiding or reducing the generation of noise in the first place. This type of noise control is called primary (active) noise control, and it starts at the earliest phases of construction. Some relevant aspects are listed and discussed below.
#### Avoidan... | {
"Header 1": "14.4 Noise Reduction at the Source",
"token_count": 391,
"source_pdf": "datasets/websources/Physics_v1/Physics/acoustics-for-engineers-2nd_troy-lecture.pdf"
} |
Noise control along propagation paths is called secondary (passive) noise control. We will now give an overview with respect to air- and structure-borne noise, organized according to the three main noise-reducing effects on the propagation paths, namely, distribution, reflection and absorption.
#### Reduction by Dist... | {
"Header 1": "14.5 Noise Reduction Along the Propagation Paths",
"token_count": 1925,
"source_pdf": "datasets/websources/Physics_v1/Physics/acoustics-for-engineers-2nd_troy-lecture.pdf"
} |
\tag{14.14}$$
Rewriting this yields
$$\frac{\mathrm{d}[p_{\mathrm{rms}}^2(x)]}{\mathrm{d}x} + \frac{\varrho c}{\mathrm{Re}\{\underline{Z}_{\mathrm{wall}}\}} \frac{U}{A} p_{\mathrm{rms}}^2(x) = 0, \qquad (14.15)$$
which, for the forward progressing wave, has the solution
$$p_{\rm rms}^2(x) = p_{\rm rms \to}^2 e^... | {
"Header 1": "14.5 Noise Reduction Along the Propagation Paths",
"token_count": 284,
"source_pdf": "datasets/websources/Physics_v1/Physics/acoustics-for-engineers-2nd_troy-lecture.pdf"
} |
Due to reciprocity, the techniques discussed in this chapter can also be applied to reduce noise on the receiving end. The most important receivers in this context are human beings<sup>7</sup> .
For them personal protection against noise often provides the quickest and cheapest noise solution. Unfortunately, there ca... | {
"Header 1": "14.6 Noise Reduction at the Receiver's End",
"token_count": 494,
"source_pdf": "datasets/websources/Physics_v1/Physics/acoustics-for-engineers-2nd_troy-lecture.pdf"
} |
An arbitrary sinusoidal signal can be written as
$$z(t) = \hat{z}\cos(\omega t + \phi), \qquad (15.1)$$
with the three free parameters amplitude, ˆz, angular frequency, ω, and (zero)phase angle, φ. If the frequency is known and fixed, that is, for a monofrequent signal, only two free parameters are left – amplitude... | {
"Header 1": "15.1 Complex Notation for Sinusoidal Signals",
"token_count": 894,
"source_pdf": "datasets/websources/Physics_v1/Physics/acoustics-for-engineers-2nd_troy-lecture.pdf"
} |
Consider a force, Fz(t), exciting a one-dimensional motion of vz(t) along a path z, for example at the mechanic port of an electro-mechanic transducer. The input energy can be written as follows, whereby from now on the index z is omitted for simplicity,
$$W = \int F(t) dz = \int F(t) \frac{dz}{dt} dt = \int F(t) v(t... | {
"Header 1": "15.2 Complex Notation for Power and Intensity",
"token_count": 1112,
"source_pdf": "datasets/websources/Physics_v1/Physics/acoustics-for-engineers-2nd_troy-lecture.pdf"
} |
The current textbook suffices as the sole teaching material for an introductory course given by experienced academic teachers who are able to provide specific explanations and stress relevant topics according to the prior knowledge and special interests of their students.
Our book is also suitable for self study. In ... | {
"Header 1": "15.3 Supplementary Textbooks for Self Study",
"token_count": 450,
"source_pdf": "datasets/websources/Physics_v1/Physics/acoustics-for-engineers-2nd_troy-lecture.pdf"
} |
Note: Solutions to the problems will be distributed on a peer-to-peer basis via <http://symphony.arch.rpi.edu/∼xiangn/SpringerBook.html> . In case of failure refer to the home page of the book, to be found under http://www.springer.com/engineering/signals/book/978-3-642-03392-6.
#### Problems to Chapter 1 – Recapitul... | {
"Header 1": "15.4 Exercises",
"token_count": 1754,
"source_pdf": "datasets/websources/Physics_v1/Physics/acoustics-for-engineers-2nd_troy-lecture.pdf"
} |
Problem 4.1. Derive the directional characteristics for arbitrary linear combination of two collocated sound sources, one with spherical and one with figure-of-eight characteristic. Sketch the following special cases: sphere, wideangle cardiod, cardiod, super cardiod, figure-of-eight.
... | {
"Header 1": "15.4 Exercises",
"Header 2": "Problems to Chapter 4 – Electromechanic and Electroacoustic Transduction",
"token_count": 1654,
"source_pdf": "datasets/websources/Physics_v1/Physics/acoustics-for-engineers-2nd_troy-lecture.pdf"
} |
Problem 7.1. Given the one-dimensional acoustic wave equation in lossless fluid.
- (a) Show that d <sup>0</sup>Alembert's solution fulfills this equation
- (b) A sawtooth-formed impulse propagates along a tube see Fig. 15.13. Sketch the reflected impulse with the tube being rigidly terminated
![](_page_221_Figure_1... | {
"Header 1": "15.4 Exercises",
"Header 2": "Problems to Chapter 7 – The Wave Equation, Propagation in Tubes",
"token_count": 592,
"source_pdf": "datasets/websources/Physics_v1/Physics/acoustics-for-engineers-2nd_troy-lecture.pdf"
} |
Simplify the equivalent circuit for the frequency range where frequency course remains constant
- (d) Determine the transformation coefficient (area ratio) of the pressure chamber so that the horn radiates the maximum sound power, whereby
Specific impedance of air $\begin{array}{ll} \dots \rho\,c = 406\,\mathrm{Ns/m... | {
"Header 1": "15.4 Exercises",
"Header 2": "Problems to Chapter 7 – The Wave Equation, Propagation in Tubes",
"token_count": 2046,
"source_pdf": "datasets/websources/Physics_v1/Physics/acoustics-for-engineers-2nd_troy-lecture.pdf"
} |
Given a dipole with the dipole momentum µd.
- (a) Sketch the directional characteristic of this dipole
- (b) An additional 0th–order point source, q 0 , is brought to the same position as that of the the dipole. Find the magnitude and phase of q 0 so that, in the far field, the entire set-up possesses the following d... | {
"Header 1": "15.4 Exercises",
"Header 2": "Problems to Chapter 7 – The Wave Equation, Propagation in Tubes",
"token_count": 246,
"source_pdf": "datasets/websources/Physics_v1/Physics/acoustics-for-engineers-2nd_troy-lecture.pdf"
} |
Problem 10.1. Using Fraunhofer 's approximation of the Huygens–Fresnel integral, show that the directional characteristic of an oscillating plane area is expressed by
$$\Gamma(\theta, \phi) = \Gamma(\theta) \ \Gamma(\phi),$$
if v(x, y) can be separated as follows,
$$v(x,y) = v_0 \ \psi_{x(x)} \ \psi_{y(y)},$$
w... | {
"Header 1": "15.4 Exercises",
"Header 2": "Problems to Chapter 10 – Piston Membranes, Diffraction and Scattering",
"token_count": 318,
"source_pdf": "datasets/websources/Physics_v1/Physics/acoustics-for-engineers-2nd_troy-lecture.pdf"
} |
Problem 11.1. A Kundt's tube be considered lossy, that is, with a complex propagation coefficient, γ = α + j β. How would the losses influence the standing waves – and therefore the measurement results?
Problem 11.2. A sound wave is incident with an angle of θ<sup>1</sup> upon a boundary between two different media, ... | {
"Header 1": "15.4 Exercises",
"Header 2": "Problems to Chapter 11 – Dissipation, Reflection, Refraction, and Absorption",
"token_count": 2011,
"source_pdf": "datasets/websources/Physics_v1/Physics/acoustics-for-engineers-2nd_troy-lecture.pdf"
} |
#### Roman-Letter Symbols
| | Roman-Letter Symbols |
|-------------|------------------------------------------------------------|
| a | Acceleration |
| A | Area ... | {
"Header 1": "15.5 Letter Symbols, Notations and Units",
"token_count": 1636,
"source_pdf": "datasets/websources/Physics_v1/Physics/acoustics-for-engineers-2nd_troy-lecture.pdf"
} |
| $\hat{z}$ | Amplitude, peak value |
|-------------------------------------------------|--------------------------------------------------------------... | {
"Header 1": "15.5 Letter Symbols, Notations and Units",
"Header 2": "Specific Mathematical Notations and Terms",
"token_count": 535,
"source_pdf": "datasets/websources/Physics_v1/Physics/acoustics-for-engineers-2nd_troy-lecture.pdf"
} |
Meter ... unit of length
Second ... unit of time
Kilogram ... unit of mass
Ampere ... unit of electric current
**Basic SI-units**
m
kg
Α
| Some often used SI-derived units | | | |
|----------------------------------|----------------------------... | {
"Header 1": "15.5 Letter Symbols, Notations and Units",
"Header 2": "$\\mathbf{Units}^{*)}$",
"token_count": 333,
"source_pdf": "datasets/websources/Physics_v1/Physics/acoustics-for-engineers-2nd_troy-lecture.pdf"
} |
| absorber | statistical, 169 |
|------------------------------------------|-----------------------------------|
| Helmholtz, 158 | technical, 3 |
| membrane, 158, 159 | active, 37 ... | {
"Header 1": "Index",
"token_count": 13172,
"source_pdf": "datasets/websources/Physics_v1/Physics/acoustics-for-engineers-2nd_troy-lecture.pdf"
} |
| 0 | | Preface | 1 |
|---|-----|-----------------------------------------------------------|----|
| 1 | | The Properties of Matter in Bulk | 4 |
| | 1.1 | What is Statistical Mechanics About?<br> | 4 |
| | 1.2 | ... | {
"Header 1": "Statistical Mechanics",
"Header 2": "Contents",
"token_count": 2212,
"source_pdf": "datasets/websources/Physics_v1/Physics/book.pdf"
} |
This is a book about statistical mechanics at the advanced undergraduate level. It assumes a background in classical mechanics through the concept of phase space, in quantum mechanics through the Pauli exclusion principle, and in mathematics through multivariate calculus. (Section 9.2 also assumes that you can can diag... | {
"Header 1": "Statistical Mechanics",
"Header 2": "Preface",
"token_count": 1256,
"source_pdf": "datasets/websources/Physics_v1/Physics/book.pdf"
} |
Statistical mechanics treats matter in bulk. While most branches of physics. . . classical mechanics, atomic physics, quantum mechanics, nuclear physics. . . deal with one or two or a few dozen particles, statistical mechanics deals with, typically, about a mole of particles at one time. A mole is 6.02 × 1023, consider... | {
"Header 1": "Statistical Mechanics",
"Header 2": "1.1 What is Statistical Mechanics About?",
"token_count": 456,
"source_pdf": "datasets/websources/Physics_v1/Physics/book.pdf"
} |
This book begins with a chapter, the properties of matter in bulk , that introduces statistical mechanics and shows why it is so fascinating.
It proceeds to discuss the principles of statistical mechanics. The goal of this chapter is to motivate and then produce a conceptual definition for that quantity of central im... | {
"Header 1": "Statistical Mechanics",
"Header 2": "1.2 Outline of Book",
"token_count": 569,
"source_pdf": "datasets/websources/Physics_v1/Physics/book.pdf"
} |
I mentioned above that statistical mechanics asks questions like "How does the pressure change with volume?". But what is pressure? Most people will answer by saying that pressure is force per area:
$$pressure = \frac{force}{area}.$$
(1.1)
*Fh*
But force is a vector and pressure is a scalar, so how can this formu... | {
"Header 1": "Statistical Mechanics",
"Header 2": "1.3 Fluid Statics",
"token_count": 1206,
"source_pdf": "datasets/websources/Physics_v1/Physics/book.pdf"
} |
Too often, books such as this one degenerate into a study of gases. . . or even into a study of the ideal gas! Statistical mechanics in fact applies to all sorts of materials: fluids, crystals, magnets, metals, polymers, starstuff, even light. I want to show you some of the enormous variety of behaviors exhibited by ma... | {
"Header 1": "Statistical Mechanics",
"Header 2": "1.4 Phase Diagrams",
"token_count": 570,
"source_pdf": "datasets/websources/Physics_v1/Physics/book.pdf"
} |
The problems of fluid flow are neglected in the typical American undergraduate physics curriculum. An introduction to these fascinating problems can be found in the chapters on elasticity and fluids in any introductory physics book, such as
- F.W. Sears, M.W. Zemansky, and H.D. Young, University Physics, fifth editio... | {
"Header 1": "Statistical Mechanics",
"Header 2": "Resources",
"token_count": 443,
"source_pdf": "datasets/websources/Physics_v1/Physics/book.pdf"
} |
#### 1.2 (I\*) Compressibility, expansion coefficient
The "isothermal compressibility" of a substance is defined as
$$\kappa_T(p,T) = -\frac{1}{V(p,T)} \frac{\partial V(p,T)}{\partial p},\tag{1.6}$$
where the volume V (p, T) is treated as a function of pressure and temperature.
- a. Justify the name "compressib... | {
"Header 1": "Statistical Mechanics",
"Header 2": "1.5 Additional Problems",
"token_count": 1493,
"source_pdf": "datasets/websources/Physics_v1/Physics/book.pdf"
} |
In a book on classical or quantum mechanics, the chapter corresponding to this one would be titled "Foundations of Classical (or Quantum) Mechanics". Here, I am careful to use the term "principles" rather than "foundations". The term "foundations" suggests rock solid, logically rigorous, hard and fast rules, such as th... | {
"Header 1": "Statistical Mechanics",
"Header 2": "Principles of Statistical Mechanics",
"token_count": 217,
"source_pdf": "datasets/websources/Physics_v1/Physics/book.pdf"
} |
This section deals with classical mechanics, not statistical mechanics. But any work that attempts to build macroscopic knowledge from microscopic knowledge—as statistical mechanics does—must begin with a clear and precise statement of what that microscopic knowledge is.
The microscopic description of a physical syst... | {
"Header 1": "Statistical Mechanics",
"Header 2": "2.1 Microscopic Description of a Classical System",
"token_count": 1576,
"source_pdf": "datasets/websources/Physics_v1/Physics/book.pdf"
} |
#### 2.1 (Q) Three-body interactions: microscopic
Make up a problem involving two charged point particles and a polarizable atom.
#### 2.2 Mechanical parameters and dynamical variables
Here is a classical mechanics problem: "A pendulum bob of mass m swings at the end of a cord which runs through a small hole in t... | {
"Header 1": "Statistical Mechanics",
"Header 2": "2.1 Microscopic Description of a Classical System",
"Header 3": "Problems",
"token_count": 459,
"source_pdf": "datasets/websources/Physics_v1/Physics/book.pdf"
} |
The title of this section introduces two new terms. . . large and equilibrium. A system is large if it contains many particles, but just how large is "large enough"? A few decades ago, physicists dealt with systems of one or two or a dozen particles, or else they dealt with systems with about 10<sup>23</sup> particles,... | {
"Header 1": "Statistical Mechanics",
"Header 2": "2.2 Macroscopic Description of a Large Equilibrium System",
"token_count": 1072,
"source_pdf": "datasets/websources/Physics_v1/Physics/book.pdf"
} |
Define microstate and macrostate.
There are many microstates corresponding to any given macrostate. The collection of all such microstates is called an "ensemble". (Just as a musical ensemble is a collection of performers.)
Note: An ensemble is a (conceptual) collection of macroscopic systems. It is not the collect... | {
"Header 1": "Statistical Mechanics",
"Header 2": "2.3 Fundamental Assumption",
"token_count": 1234,
"source_pdf": "datasets/websources/Physics_v1/Physics/book.pdf"
} |
Logarithms and dimensions
You cannot take the logarithm of a number with dimensions. Perhaps you have heard this rule phrased as "you can't take the logarithm of 3.5 meters" or "you can't take the logarithm of five oranges". Why not? A simple argument is "Well, what would be the units of $\ln(3.5 \text{ meters})$ ?"... | {
"Header 1": "Statistical Mechanics",
"Header 2": "2.3 Fundamental Assumption",
"Header 3": "2.4 Statistical Definition of Entropy",
"token_count": 1457,
"source_pdf": "datasets/websources/Physics_v1/Physics/book.pdf"
} |
So far in this chapter, we have been dealing very abstractly with a very general class of physical systems. We have made a number of assumptions that are reasonable but that we have not tested in practice. It is time to put some flesh on these formal bones. We do so by using our statistical definition of entropy to cal... | {
"Header 1": "Statistical Mechanics",
"Header 2": "2.5 Entropy of a Monatomic Ideal Gas",
"token_count": 2012,
"source_pdf": "datasets/websources/Physics_v1/Physics/book.pdf"
} |
At last we can find the entropy! It is
$$S = k_B \ln \frac{W}{N! h_0^{3N}} = k_B \ln \left\{ \left( \frac{2\pi m E V^{2/3}}{h_0^2} \right)^{3N/2} \frac{1}{N! (3N/2)!} \left[ \left( 1 + \frac{\Delta E}{E} \right)^{3N/2} - 1 \right] \right\}$$
(2.23)
or
$$\frac{S}{k_B} = \frac{3}{2}N\ln\left(\frac{2\pi mEV^{2/3}}... | {
"Header 1": "Statistical Mechanics",
"Header 2": "2.5 Entropy of a Monatomic Ideal Gas",
"token_count": 1858,
"source_pdf": "datasets/websources/Physics_v1/Physics/book.pdf"
} |
The end result is that the entropy per particle of the pure classical monatomic ideal gas is
$$s(e, v) = k_B \left[ \frac{3}{2} \ln \left( \frac{4\pi m e v^{2/3}}{3h_0^2} \right) + \frac{5}{2} \right].$$
(2.31)
This is called the "Sackur-Tetrode formula".<sup>4</sup> It is often written as
$$S(E, V, N) = k_B N \l... | {
"Header 1": "Statistical Mechanics",
"Header 2": "2.5 Entropy of a Monatomic Ideal Gas",
"token_count": 1140,
"source_pdf": "datasets/websources/Physics_v1/Physics/book.pdf"
} |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.