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(11.76)$$
Conservation of energy suggests that this is also the rate at which the particle *loses* energy, under the influence of the radiation reaction force $\mathbf{F}_{rad}$ :
$$\mathbf{F}_{\text{rad}} \cdot \mathbf{v} = -\frac{\mu_0 q^2 a^2}{6\pi c}.\tag{11.77}$$
I say "suggests" advisedly, because this equ... | {
"Header 1": "**Example 11.2.**",
"token_count": 1872,
"source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf"
} |
The equation of motion is
$$m\ddot{x} = F_{\text{spring}} + F_{\text{rad}} + F_{\text{driving}} = -m\omega_0^2 x + m\tau \ddot{x} + F_{\text{driving}}.$$
With the system oscillating at frequency ω,
$$x(t) = x_0 \cos(\omega t + \delta),$$
so
$$\ddot{x} = -\omega^2 \dot{x}.$$
Therefore
$$m\ddot{x} + m\gamma... | {
"Header 1": "**Solution**",
"token_count": 1590,
"source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf"
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In the last section, I derived the Abraham-Lorentz formula for the radiation reaction, using conservation of energy. I made no attempt to identify the actual *mechanism* responsible for this force, except to point out that it must be a recoil effect of the particle's own fields acting back on the charge. Unfortunately,... | {
"Header 1": "**11.2.3 The Mechanism Responsible for the Radiation Reaction**",
"token_count": 1941,
"source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf"
} |
Here a and $\dot{a}$ are evaluated at the *retarded* time $(t_r)$ , but it's easy to rewrite the result in terms of the *present* time t:
$$a(t_r) = a(t) + \dot{a}(t)(t_r - t) + \dots = a(t) - \dot{a}(t)T + \dots = a(t) - \dot{a}(t)\frac{d}{c} + \dots$$
and it follows that
$$\mathbf{F}_{\text{self}} = \frac{... | {
"Header 1": "**11.2.3 The Mechanism Responsible for the Radiation Reaction**",
"token_count": 1572,
"source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf"
} |
**Problem 11.22** A particle of mass m and charge q is attached to a spring with force constant k, hanging from the ceiling (Fig. 11.18). Its equilibrium position is a distance h above the floor. It is pulled down a distance d below equilibrium and released, at time t=0.
(a) Under the usual assumptions $(d \ll \lamb... | {
"Header 1": "**11.2.3 The Mechanism Responsible for the Radiation Reaction**",
"Header 2": "**More Problems on Chapter 11**",
"token_count": 2036,
"source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf"
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\end{cases}$$
(b) Show that the retarded vector potential can be written in the form
$$\mathbf{A}(x,t) = \frac{\mu_0 c}{2} \,\hat{\mathbf{z}} \int_0^\infty K\left(t - \frac{x}{c} - u\right) \, du,$$
and from this determine E and B.
(c) Show that the total power radiated per unit area of surface is
$$\frac{\mu... | {
"Header 1": "**11.2.3 The Mechanism Responsible for the Radiation Reaction**",
"Header 2": "**More Problems on Chapter 11**",
"token_count": 523,
"source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf"
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- (a) Repeat Prob. 11.19, but this time let the external force be a Dirac delta function: $F(t) = k\delta(t)$ (for some constant k). Note that the acceleration is now *discontinuous* at t = 0 (though the *velocity* must still be continuous); use the method of Prob. 11.19 (a) to show that $\Delta a = -k/m\tau$ . In t... | {
"Header 1": "Problem 11.31",
"token_count": 1453,
"source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf"
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Classical mechanics obeys the **principle of relativity**: the same laws apply in any **inertial reference frame**. By "inertial" I mean that the system is at rest or moving with constant velocity.1 Imagine, for example, that you have loaded a billiard table onto a railroad car, and the train is going at constant speed... | {
"Header 1": "**12.1.1 Einstein's Postulates**",
"token_count": 2007,
"source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf"
} |
I shall not go into the details here, because I do not want to distract your attention from the two essential points: (1) all Michelson and Morley were trying to do was compare the speed of light in different directions, and (2) what they in fact *discovered* was that this speed is *exactly the same in all directions*.... | {
"Header 1": "**12.1.1 Einstein's Postulates**",
"token_count": 2038,
"source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf"
} |
- (a) Prove that momentum is also conserved in inertial frame $\bar{S}$ , which moves with velocity v relative to S. [Use Galileo's velocity addition rule—this is an entirely classical calculation. What must you assume about mass?]
- (b) Suppose the collision is elastic in S; show that it is also elastic in $\bar{S... | {
"Header 1": "**12.1.1 Einstein's Postulates**",
"token_count": 375,
"source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf"
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In this section I present a series of *gedanken* (thought) experiments that serve to introduce the three most striking geometrical consequences of Einstein's postulates: time dilation, Lorentz contraction, and the relativity of simultaneity. In Sect. 12.1.3 the same results will be derived more systematically, using Lo... | {
"Header 1": "12.1.2 ■ The Geometry of Relativity",
"token_count": 364,
"source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf"
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Naturally, the train has to be going awfully fast before the discrepancy becomes detectable—that's why you don't notice it all the time.
Of course, it's always possible for a naïve witness to be *mistaken* about simultaneity: a person sitting in the back corner of the car would *hear* buzzer *b* before buzzer *a*, si... | {
"Header 1": "**Two events that are simultaneous in one inertial system are not, in general, simultaneous in another.**",
"token_count": 2010,
"source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf"
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So whereas each observer conducted a perfectly sound measurement, from his/her own point of view, the other observer (watching the process) considers that she/he made the most elementary blunder in the book, by using two unsynchronized clocks. That's how, in spite of the fact that his clocks "actually" run slow, he man... | {
"Header 1": "**Two events that are simultaneous in one inertial system are not, in general, simultaneous in another.**",
"token_count": 1622,
"source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf"
} |
We call this **Lorentz contraction**. Notice that the same factor,
$$\gamma \equiv \frac{1}{\sqrt{1 - v^2/c^2}},$$
appears in both the time dilation formula and the Lorentz contraction formula. This makes it all very easy to remember: Moving clocks run slow, moving sticks are shortened, and the factor is always $\... | {
"Header 1": "**Two events that are simultaneous in one inertial system are not, in general, simultaneous in another.**",
"Header 3": "Moving objects are shortened.",
"token_count": 865,
"source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf"
} |
They're *both* right! When you say "the ladder is in the barn," you mean that all parts of it are inside *at one instant of time*, but in view of the relativity of simultaneity, that's a condition that depends on the observer. There are really *two* relevant events here:
- *a*. Back end of ladder makes it in the door... | {
"Header 1": "**Solution**",
"token_count": 489,
"source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf"
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Indeed, in deriving the time dilation formula I took it for granted that the *height* of the train is the same for both observers. I'll now justify this, using a lovely gedanken experiment suggested by Taylor and Wheeler.<sup>8</sup> Imagine that we build
<sup>7</sup>For a related paradox see E. Pierce, *Am. J. Phys.... | {
"Header 1": "**Dimensions perpendicular to the velocity are not contracted.**",
"token_count": 2003,
"source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf"
} |
Because $\mathcal{O}$ and A are at rest in $\mathcal{S}$ , x is the "moving stick," and
$$\bar{d} = \frac{1}{\gamma}x. \tag{12.16}$$
**FIGURE 12.17**
It follows that
$$x = \gamma(\bar{x} + v\bar{t}). \tag{12.17}$$
This last equation comes as no surprise, for the symmetry of... | {
"Header 1": "**Dimensions perpendicular to the velocity are not contracted.**",
"token_count": 2039,
"source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf"
} |
If this boat is then set in motion at speed v, what angle $\theta$ does an individual photon trajectory make with the deck, according to an observer on the dock? What angle does the *beam* (illuminated, say, by a light fog) make? Compare Prob. 12.10.
**FIGURE 12.19**
**FIGURE 12.20... | {
"Header 1": "**Dimensions perpendicular to the velocity are not contracted.**",
"token_count": 874,
"source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf"
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(i) Four-vectors. The Lorentz transformations take on a simpler appearance when expressed in terms of the quantities
$$x^0 \equiv ct, \quad \beta \equiv \frac{v}{c}. \tag{12.21}$$
Using $x^0$ (instead of t) and $\beta$ (instead of v) amounts to changing the unit of time from the *second* to the *meter*—1 meter ... | {
"Header 1": "**12.1.4** ■ The Structure of Spacetime",
"token_count": 2004,
"source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf"
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12.28 $(-a^0b^0 + \mathbf{a} \cdot \mathbf{b})$ or with an overall minus sign $(a^0b^0 - \mathbf{a} \cdot \mathbf{b})$ ; if one is invariant, so is the other. In the literature, both conventions are common, and you just have to be aware of which one is in use. If they write the diagonal components of the Minkowski m... | {
"Header 1": "**12.1.4** ■ The Structure of Spacetime",
"token_count": 1090,
"source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf"
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- (a) Event *A* happens at point (*xA* = 5, *yA* = 3,*z <sup>A</sup>* = 0) and at time *tA* given by *ctA* = 15; event *B* occurs at (10, 8, 0) and *ctB* = 5, both in system S.
- (i) What is the invariant interval between *A* and *B*?
- (ii) Is there an inertial system in which they occur *simultaneously*? If so, find ... | {
"Header 1": "**Problem 12.20**",
"token_count": 2014,
"source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf"
} |
What do you think? Even if she *could* travel faster than light, could she return before she set out? Could she arrive at some intermediate destination before she set out? Draw a space-time diagram representing this trip.
**Problem 12.23** Inertial system $\bar{S}$ moves in the x direction at speed $\frac{3}{5}c... | {
"Header 1": "**Problem 12.20**",
"token_count": 360,
"source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf"
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As you progress along your world line, your watch runs slow; while the clock on the wall ticks off an interval dt, your watch only advances $d\tau$ :
$$d\tau = \sqrt{1 - u^2/c^2} \, dt. \tag{12.37}$$
(I'll use u for the velocity of a particular object—you, in this instance—and reserve v for the relative velocity o... | {
"Header 1": "**12.2.1** ■ Proper Time and Proper Velocity",
"token_count": 2054,
"source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf"
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Thus
$$\mathbf{p} \equiv m\eta = \frac{m\mathbf{u}}{\sqrt{1 - u^2/c^2}};$$
(12.46)
this is the **relativistic momentum** of an object of mass m traveling at (ordinary) velocity $\mathbf{u}$ . <sup>13</sup>
Relativistic momentum is the spatial part of a 4-vector,
$$p^{\mu} \equiv m\eta^{\mu},\tag{12.47}$$
<su... | {
"Header 1": "**12.2.1** ■ Proper Time and Proper Velocity",
"token_count": 618,
"source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf"
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Mass is *not* conserved—a fact that has been painfully familiar to everyone since 1945 (though the so-called "conversion of mass into energy" is really a conversion of *rest* energy into *kinetic* energy).
Note the distinction between an **invariant** quantity (same value in all inertial systems) and a **conserved** ... | {
"Header 1": "In every closed<sup>14</sup> system, the total relativistic energy and momentum are conserved.",
"token_count": 1978,
"source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf"
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**FIGURE 12.27**
#### Solution
In this case,
$$E_{ ext{before}} = m_{\pi}c^2, \qquad \mathbf{p}_{ ext{before}} = 0,$$
$E_{ ext{after}} = E_{\mu} + E_{\nu}, \qquad \mathbf{p}_{ ext{after}} = \mathbf{p}_{\mu} + \mathbf{p}_{\nu}.$
Conservation of momentum requires that $\mathbf{p... | {
"Header 1": "In every closed<sup>14</sup> system, the total relativistic energy and momentum are conserved.",
"token_count": 2026,
"source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf"
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(12.60)$$
The numerator, of course, is the classical answer—it's approximately right, if $(F/m)t \ll c$ . But the relativistic denominator ensures that u never exceeds c; in fact, as $t \to \infty$ , $u \to c$ .
To complete the problem we must integrate again:
$$x(t) = \frac{F}{m} \int_0^t \frac{t'}{\sqrt{1 + ... | {
"Header 1": "In every closed<sup>14</sup> system, the total relativistic energy and momentum are conserved.",
"token_count": 2034,
"source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf"
} |
**Example 12.11.** The typical trajectory of a charged particle in a uniform magnetic field is **cyclotron motion** (Fig. 12.31). The magnetic force pointing toward the center,
$$F = QuB$$
,
provides the centripetal acceleration necessary to sustain circular motion. Beware, however... | {
"Header 1": "In every closed<sup>14</sup> system, the total relativistic energy and momentum are conserved.",
"token_count": 2049,
"source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf"
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8.45).<sup>22</sup>
**Problem 12.37** In classical mechanics, Newton's law can be written in the more familiar form $\mathbf{F} = m\mathbf{a}$ . The relativistic equation, $\mathbf{F} = d\mathbf{p}/dt$ , cannot be so simply expressed. Show, rather, that
$$\mathbf{F} = \frac{m}{\sqrt{1 - u^2/c^2}} \left[ \mathbf{a... | {
"Header 1": "In every closed<sup>14</sup> system, the total relativistic energy and momentum are conserved.",
"token_count": 1991,
"source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf"
} |
Now, a line charge $\lambda_{tot}$ sets up an *electric* field
$$E = \frac{\lambda_{\text{tot}}}{2\pi \epsilon_0 s},$$
so there is an electrical force on q in $\bar{S}$ , to wit:
$$\bar{F} = qE = -\frac{\lambda v}{\pi \epsilon_0 c^2 s} \frac{qu}{\sqrt{1 - u^2/c^2}}.$$
(12.84)
But if there's a force on q in... | {
"Header 1": "In every closed<sup>14</sup> system, the total relativistic energy and momentum are conserved.",
"token_count": 473,
"source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf"
} |
We have learned, in various special cases, that one observer's electric field is another's magnetic field. It would be nice to know the *general* transformation rules for electromagnetic fields: Given the fields in S, what are the fields in S¯? Your first guess might be that **E** is the spatial part of one 4-vector an... | {
"Header 1": "**12.3.2 How the Fields Transform**",
"token_count": 2005,
"source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf"
} |
12.91 and 12.92 are not the most general transformation laws, for we began with a system S<sup>0</sup> in which the charges were at rest and where, consequently, there was no magnetic field. To derive the *general* rule, we must start out in a system with both electric and magnetic fields. For this purpose S itself wil... | {
"Header 1": "**12.3.2 How the Fields Transform**",
"token_count": 2018,
"source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf"
} |
If $\mathbf{E} = \mathbf{0}$ in $\mathcal{S}$ , then
$$\bar{\mathbf{E}} = -\gamma v(B_z \,\hat{\mathbf{y}} - B_y \,\hat{\mathbf{z}}) = -v(\bar{B}_z \,\hat{\mathbf{y}} - \bar{B}_y \,\hat{\mathbf{z}}),$$
or
$$\bar{\mathbf{E}} = \mathbf{v} \times \bar{\mathbf{B}}.\tag{12.111}$$
In other words, if either $\math... | {
"Header 1": "**12.3.2 How the Fields Transform**",
"token_count": 2054,
"source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf"
} |
As Eq. 12.109 indicates, **E** and **B** certainly do *not* transform like the spatial parts of the two 4-vectors—in fact, the components of **E** and **B** are stirred together when you go from one inertial system to another. What sort of an object is this, which has six components and transforms according to Eq. 12.1... | {
"Header 1": "**12.3.2 How the Fields Transform**",
"Header 3": "12.3.3 ■ The Field Tensor",
"token_count": 1887,
"source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf"
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This leads to **dual tensor**, $G^{\mu\nu}$ :
$$G^{\mu\nu} = \left\{ \begin{array}{cccc} 0 & B_x & B_y & B_z \\ -B_x & 0 & -E_z/c & E_y/c \\ -B_y & E_z/c & 0 & -E_x/c \\ -B_z & -E_y/c & E_x/c & 0 \end{array} \right\}.$$
(12.120)
$G^{\mu\nu}$ can be obtained directly from $F^{\mu\nu}$ by the substitution $\math... | {
"Header 1": "**12.3.2 How the Fields Transform**",
"Header 3": "12.3.3 ■ The Field Tensor",
"token_count": 610,
"source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf"
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Now that we know how to represent the fields in relativistic notation, it is time to reformulate the laws of electrodynamics (Maxwell's equations and the Lorentz force law) in that language. To begin with, we must determine how the *sources* of the fields, ρ and **J**, transform. Imagine a cloud of charge drifting by; ... | {
"Header 1": "**12.3.4 Electrodynamics in Tensor Notation**",
"token_count": 2045,
"source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf"
} |
In terms of $F^{\mu\nu}$ and the proper velocity $\eta^{\mu}$ , the Minkowski force on a charge q is given by
$$K^{\mu} = q\eta_{\nu}F^{\mu\nu}. \tag{12.128}$$
For if $\mu = 1$ , we have
$$\begin{split} K^{1} &= q \eta_{v} F^{1v} = q (-\eta^{0} F^{10} + \eta^{1} F^{11} + \eta^{2} F^{12} + \eta^{3} F^{13}) \... | {
"Header 1": "**12.3.4 Electrodynamics in Tensor Notation**",
"token_count": 2043,
"source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf"
} |
The corresponding *contravariant* gradient would be $\partial^{\mu}\equiv\partial/\partial x_{\mu}$ . *Prove* that $\partial^{\mu}\phi$ is a (contravariant) 4-vector, if $\phi$ is a scalar function, by working out its transformation law, using the chain rule.
**Problem 12.57** Show that the potential representat... | {
"Header 1": "**12.3.4 Electrodynamics in Tensor Notation**",
"token_count": 394,
"source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf"
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**Problem 12.59** Inertial system $\bar{S}$ moves at constant velocity $\mathbf{v} = \beta c(\cos\phi \,\hat{\mathbf{x}} + \sin\phi \,\hat{\mathbf{y}})$ with respect to S. Their axes are parallel to one another, and their origins coincide at $t = \bar{t} = 0$ , as usual. Find the Lorentz transformation matrix $\L... | {
"Header 1": "**12.3.4 Electrodynamics in Tensor Notation**",
"Header 3": "**More Problems on Chapter 12**",
"token_count": 2020,
"source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf"
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12.67 and 12.109) to rewrite $\bar{\mathbf{F}}$ in terms of $\mathbf{F}$ , and $\bar{\mathbf{E}}$ in terms of $\mathbf{E}$ and $\mathbf{B}$ . From these, deduce the formula for $\mathbf{F}$ in terms of $\mathbf{E}$ and $\mathbf{B}$ .
**Problem 12.69** A charge q is released from rest at the origin, in th... | {
"Header 1": "**12.3.4 Electrodynamics in Tensor Notation**",
"Header 3": "**More Problems on Chapter 12**",
"token_count": 1138,
"source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf"
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#### A.1 ■ INTRODUCTION
In this Appendix I sketch proofs of the three fundamental theorems of vector calculus. My aim is to convey the *essence* of the argument, not to track down every epsilon and delta. A much more elegant, modern, and unified—but necessarily also much longer—treatment will be found in M. Spivak's ... | {
"Header 1": "**Vector Calculus in Curvilinear Coordinates**",
"token_count": 1392,
"source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf"
} |
Suppose that we have a *vector* function,
$$\mathbf{A}(u, v, w) = A_u \,\hat{\mathbf{u}} + A_v \,\hat{\mathbf{v}} + A_w \,\hat{\mathbf{w}},$$
and we wish to evaluate the integral **A** · *d***a** over the surface of the infinitesimal volume generated by starting at the point (*u*, v, w) and increasing each of the c... | {
"Header 1": "**A.4 DIVERGENCE**",
"token_count": 1212,
"source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf"
} |
To obtain the curl in curvilinear coordinates, we calculate the line integral,
$$\oint \mathbf{A} \cdot d\mathbf{l},$$
around the infinitesimal loop generated by starting at (*u*, v, w) and successively increasing *u* and v by infinitesimal amounts, holding w constant (Fig. A.4). The surface is a rectangle (at leas... | {
"Header 1": "**A.5 CURL**",
"token_count": 1550,
"source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf"
} |
Suppose we are told that the divergence of a vector function $\mathbf{F}(\mathbf{r})$ is a specified scalar function $D(\mathbf{r})$ :
$$\nabla \cdot \mathbf{F} = D,\tag{B.1}$$
and the curl of $\mathbf{F}(\mathbf{r})$ is a specified vector function $\mathbf{C}(\mathbf{r})$ :
$$\nabla \times \mathbf{F} = \ma... | {
"Header 1": "The Helmholtz Theorem",
"token_count": 2029,
"source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf"
} |
\quad (B.10)$$
For example, in electrostatics $\nabla \cdot \mathbf{E} = \rho/\epsilon_0$ and $\nabla \times \mathbf{E} = \mathbf{0}$ , so
$$\mathbf{E}(\mathbf{r}) = -\nabla \left( \frac{1}{4\pi\epsilon_0} \int \frac{\rho(\mathbf{r}')}{n} d\tau' \right) = -\nabla V$$
(B.11)
(where V is the scalar potential), w... | {
"Header 1": "The Helmholtz Theorem",
"token_count": 375,
"source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf"
} |
In our units (the Système International), Coulomb's law reads
$$\mathbf{F} = \frac{1}{4\pi\epsilon_0} \frac{q_1 q_2}{v^2} \hat{\boldsymbol{\imath}} \quad (SI). \tag{C.1}$$
Mechanical quantities are measured in meters, kilograms, seconds, and charge is in **coulombs** (Table C.1). In the **Gaussian system**, the con... | {
"Header 1": "**Units**",
"token_count": 1706,
"source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf"
} |
| | SI ... | {
"Header 1": "**Units**",
"token_count": 545,
"source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf"
} |
Definitions:
$$\begin{cases} \mathbf{D} = \epsilon_0 \mathbf{E} + \mathbf{P} \\ \mathbf{H} = \frac{1}{\mu_0} \mathbf{B} - \mathbf{M} \end{cases} \qquad \mathbf{D} = \mathbf{E} + 4\pi \mathbf{P} \\ \mathbf{H} = \mathbf{B} - 4\pi \mathbf{M} \end{cases}$$
Linear media:
$$\begin{cases} \mathbf{P} = \epsilon_0 \chi_e \mathb... | {
"Header 1": "**Units**",
"Header 2": "D and H",
"token_count": 338,
"source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf"
} |
Energy:
$$U = \frac{1}{2} \int \left( \epsilon_0 E^2 + \frac{1}{\mu_0} B^2 \right) d\tau \quad U = \frac{1}{8\pi} \int \left( E^2 + B^2 \right) d\tau$$
Poynting vector:
$$\mathbf{S} = \frac{1}{\mu_0} (\mathbf{E} \times \mathbf{B})$$
$$\mathbf{S} = \frac{c}{4\pi} (\mathbf{E} \times \mathbf{B})$$
Larmor formula:
$$P = ... | {
"Header 1": "**Units**",
"Header 2": "**Energy and power**",
"token_count": 720,
"source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf"
} |
M., xvi | lifetime, 487 | 130–131, 575 |
| Ampère dipole, 269, 294 | polarizability, 169–170 | Cauchy's formula, 424 |
| Ampère's law, 233, 243, | Bohr magneton, 263 | Causality, 441, 446–447, 489, |
| 332–337, 567 ... | {
"Header 1": "**Units**",
"Header 2": "**Energy and power**",
"token_count": 2941,
"source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf"
} |
A. M., 380 | Einstein velocity addition rule, | external field, 196 |
| Dirac delta function, 45–52, 164 | 507–508, 523–524 | dielectric sphere in external |
| Dirichlet's theorem, 134 | Einstein's postulates, 501–507 | field, 192–194 |
| Discharge of ca... | {
"Header 1": "**Units**",
"Header 2": "**Energy and power**",
"token_count": 1107,
"source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf"
} |
I., 300 | induced, 313–314, 316–321 | Electromagnetic induction, |
| Dual tensor, 564, 573 | macroscopic, 179–181, 199 | 312–332 |
| Duality transformation, | microscopic, 179–181 | Electromagnetic mass, 495 |
| 353–354, 477 ... | {
"Header 1": "**Units**",
"Header 2": "**Energy and power**",
"token_count": 4240,
"source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf"
} |
A., xvi, 492, 506 | in cavity, 282–283 | Maxwell stress tensor, 362–366 |
| Lorentz contraction, 506, | circular loop, 227 | Maxwell's equations, 241, |
| 514–518, 522 | dipole, 255, 263–265 | 332–339, 567, 570 ... | {
"Header 1": "**Units**",
"Header 2": "**Energy and power**",
"token_count": 7734,
"source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf"
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*See also* Energy against back emf, 328 in charging a capacitor, 106–107
by magnetic forces, 215, 218–220, 373–378 in moving a charge, 91–92 in moving a dielectric, 202–204 in moving a wire loop, 305–308
in polarizing a dielectric, 197–202 in setting up a charge configuration, 91–94 Work energy theorem, 543–544 Wor... | {
"Header 1": "**Units**",
"Header 2": "**Energy and power**",
"token_count": 1599,
"source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf"
} |
(3)
$$\nabla(fg) = f(\nabla g) + g(\nabla f)$$
(4)
$$\nabla (\mathbf{A} \cdot \mathbf{B}) = \mathbf{A} \times (\nabla \times \mathbf{B}) + \mathbf{B} \times (\nabla \times \mathbf{A}) + (\mathbf{A} \cdot \nabla)\mathbf{B} + (\mathbf{B} \cdot \nabla)\mathbf{A}$$
(5)
$$\nabla \cdot (f\mathbf{A}) = f(\nabla \cdot \mat... | {
"Header 1": "**Product Rules**",
"token_count": 400,
"source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf"
} |
In general:
In matter:
$$\begin{cases} \nabla \cdot \mathbf{E} = \frac{1}{\epsilon_0} \rho \\ \nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t} \\ \nabla \cdot \mathbf{B} = 0 \\ \nabla \times \mathbf{B} = \mu_0 \mathbf{J} + \mu_0 \epsilon_0 \frac{\partial \mathbf{E}}{\partial t} \end{cases}$$
$$\... | {
"Header 1": "BASIC EQUATIONS OF ELECTRODYNAMICS",
"Header 2": "**Maxwell's Equations**",
"token_count": 251,
"source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf"
} |
Definitions:
Linear media:
$$\begin{cases} \mathbf{D} = \epsilon_0 \mathbf{E} + \mathbf{P} \\ \mathbf{H} = \frac{1}{\mu_0} \mathbf{B} - \mathbf{M} \end{cases}$$
$$\begin{cases} \mathbf{P} = \epsilon_0 \chi_e \mathbf{E}, & \mathbf{D} = \epsilon \mathbf{E} \\ \mathbf{M} = \chi_m \mathbf{H}, & \mathbf{H} = \frac{1}{... | {
"Header 1": "**Auxiliary Fields**",
"token_count": 598,
"source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf"
} |
$$\begin{cases} x = r \sin \theta \cos \phi \\ y = r \sin \theta \sin \phi \\ z = r \cos \theta \end{cases}$$
$$\begin{cases} \hat{\mathbf{x}} = \sin \theta \cos \phi \, \hat{\mathbf{r}} + \cos \theta \cos \phi \, \hat{\boldsymbol{\theta}} - \sin \phi \, \hat{\boldsymbol{\phi}} \\ \hat{\mathbf{y}} = \sin \theta \sin ... | {
"Header 1": "**Spherical**",
"token_count": 422,
"source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf"
} |
$$\begin{cases} x = s \cos \phi \\ y = s \sin \phi \\ z = z \end{cases} \begin{cases} \hat{\mathbf{x}} = \cos \phi \, \hat{\mathbf{s}} - \sin \phi \, \hat{\boldsymbol{\phi}} \\ \hat{\mathbf{y}} = \sin \phi \, \hat{\mathbf{s}} + \cos \phi \, \hat{\boldsymbol{\phi}} \\ \hat{\mathbf{z}} = \hat{\mathbf{z}} \end{cases}$$
$$... | {
"Header 1": "Cylindrical",
"token_count": 277,
"source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf"
} |
#### **SECOND EDITION**
#### **Daniel R. Raichel**
CUNY Graduate Center and School of Architecture, Urban Design and Landscape Design The City College of the City University of New York
With 253 Illustrations
Daniel R. Raichel 2727 Moore Lane Fort Collins, CO 80526 USA draichel@com... | {
"Header 1": "**THE SCIENCE AND APPLICATIONS OF ACOUSTICS**",
"token_count": 392,
"source_pdf": "datasets/websources/Physics_v1/Physics/pdf_esp_198.pdf"
} |
The science of acoustics deals with the creation of sound, sound transmission through solids, and the effects of sound on both inert and living materials. As a mechanical effect, sound is essentially the passage of pressure fluctuations through matter as the result of vibrational forces acting on that medium. Sound pos... | {
"Header 1": "Preface",
"token_count": 1948,
"source_pdf": "datasets/websources/Physics_v1/Physics/pdf_esp_198.pdf"
} |
| Preface | | vii |
|---------|----------------------------------------------------------------------|-----|
| 1. | A<br>Capsule<br>History<br>of<br>Acoustics | 1 |
| 2. | Fundamentals<br>of<br>Acoustics ... | {
"Header 1": "Preface",
"Header 2": "Contents",
"token_count": 650,
"source_pdf": "datasets/websources/Physics_v1/Physics/pdf_esp_198.pdf"
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Of the five senses that we possess, hearing probably ranks second only to sight in regular usage. It is therefore with little wonder that human interest in acoustics would date to prehistoric times. Sound effects entailing loud clangorous noises were used to terrorize enemies in the course of heated battles; yet the ge... | {
"Header 1": "1 A Capsule History of Acoustics",
"token_count": 2010,
"source_pdf": "datasets/websources/Physics_v1/Physics/pdf_esp_198.pdf"
} |
A Capsule History of Acoustics
Robert Boyle (1626–1691) with the help of his assistant Robert Hooke (1635– 1703) performed a classic experiment (1660) on sound by placing a ticking watch in a partially evacuated glass chamber. He proved that air is necessary, either for the production or emission of sound. In this re... | {
"Header 1": "1 A Capsule History of Acoustics",
"token_count": 2002,
"source_pdf": "datasets/websources/Physics_v1/Physics/pdf_esp_198.pdf"
} |
And the next generation provided a further flowering of genius: Joseph Louis Lagrange (1736–1813), Pierre Simon Laplace (1749–1827), Adrian Marie Legendre (1736–1833), Jean Baptiste Joseph Fourier (1768–1830), and Sim´eon Denis Poisson (1781–1840). The nineteenth century was also dominated by discoveries in electricity... | {
"Header 1": "1 A Capsule History of Acoustics",
"token_count": 2026,
"source_pdf": "datasets/websources/Physics_v1/Physics/pdf_esp_198.pdf"
} |
Harvey Fletcher (1884–1990) led the Bell Telephone Laboratories in describing and quantifying the concepts of loudness and masking, and there, many of the determinants of speech communication were also established (1920–1940). Fletcher, now regarded as "the father of psychoacoustics," worked with the physicist Robert M... | {
"Header 1": "1 A Capsule History of Acoustics",
"token_count": 2047,
"source_pdf": "datasets/websources/Physics_v1/Physics/pdf_esp_198.pdf"
} |
Bethesda, MD: David Taylor Basin publication DTNSRDC-84/010, June 1964.
Bobber, Robert J. 1970. *Underwater Electroacoustic Measurements*. Washington, DC: Naval Research Laboratory.
Chladni, E. F. F. 1802. *Die Acustik*. Leipzig: Breitkopf & Hartel.
Clay, Clarence S. and Medwin, Herman. 1977. *Acoustical Oceanogr... | {
"Header 1": "1 A Capsule History of Acoustics",
"token_count": 1274,
"source_pdf": "datasets/websources/Physics_v1/Physics/pdf_esp_198.pdf"
} |
#### 2.1 Wave Nature of Sound and the Importance of Acoustics
Acoustics refers to the study of sound, namely, its production, transmission through solid and fluid media, and any other phenomenon engendered by its propagation through media. Sound may be described as the passage of pressure fluctuations through an elas... | {
"Header 1": "1 A Capsule History of Acoustics",
"Header 2": "Fundamentals of Acoustics",
"token_count": 2004,
"source_pdf": "datasets/websources/Physics_v1/Physics/pdf_esp_198.pdf"
} |
At a temperature of 20°C sound moves at the rate of 344 m/s (1127 ft/s) through air at the normal atmospheric pressure of
$$mv = \frac{hv}{c}$$
where mv represents the moment of the particle, h Planck's constant = $6.625 \times 10^{-27}$ erg s, c the velocity of light = $3 \times 10^8$ m/s, and v the radial fre... | {
"Header 1": "1 A Capsule History of Acoustics",
"Header 2": "Fundamentals of Acoustics",
"token_count": 2017,
"source_pdf": "datasets/websources/Physics_v1/Physics/pdf_esp_198.pdf"
} |
In vector terminology
$$\mathbf{V} = u\mathbf{i} + v\mathbf{j} + w\mathbf{k}$$
<sup>2</sup> It can be argued that because the equation of state derives from the principles of conservation of momentum and energy in classic kinetic theory, it effectively becomes the equivalent of the energy conservation principle in ... | {
"Header 1": "1 A Capsule History of Acoustics",
"Header 2": "Fundamentals of Acoustics",
"token_count": 2010,
"source_pdf": "datasets/websources/Physics_v1/Physics/pdf_esp_198.pdf"
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It is represented by the symbol $\tau_{mn}$ , where the force produced by the shear is normal to coordinate m and parallel for coordinate n, and either m or n may represent the principal coordinate x, y, or z, provided that $m \neq n$ . If m = n, then $\tau_{mm}$ really represents the normal force $\sigma_{mm}$ a... | {
"Header 1": "1 A Capsule History of Acoustics",
"Header 2": "Fundamentals of Acoustics",
"token_count": 2020,
"source_pdf": "datasets/websources/Physics_v1/Physics/pdf_esp_198.pdf"
} |
- (2) the passage of an acoustic signal through the fluid results in small perturbations of pressure, temperature, density, and velocity. These perturbations are expressed as $p_0 + p$ , $\rho_0 + \rho$ , u, and so on. The unperturbed velocity $u_0$ is set to zero; the unperturbed fluid does not undergo macroscop... | {
"Header 1": "1 A Capsule History of Acoustics",
"Header 2": "Fundamentals of Acoustics",
"token_count": 2051,
"source_pdf": "datasets/websources/Physics_v1/Physics/pdf_esp_198.pdf"
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(Sir James Lighthill compared the significance of this four-volume compilation with that of Lord Rayleigh's *The Theory of Sound*, which is not at all far-fetched considering that this encyclopedia contains contributions from an editorial board whose members constitute a veritable *Who's Who in Acoustics. Handbook of A... | {
"Header 1": "1 A Capsule History of Acoustics",
"Header 2": "Fundamentals of Acoustics",
"token_count": 1150,
"source_pdf": "datasets/websources/Physics_v1/Physics/pdf_esp_198.pdf"
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#### 3.1 The Nature of Sound Propagation
When energy passes through a medium resulting in a wave-type motion, several different types of waves may be generated, depending upon the motion of a particle in the medium. A *transverse* wave occurs when its amplitude varies in the direction normal to the direction of the p... | {
"Header 1": "3 Sound Wave Propagation and Characteristics",
"token_count": 1916,
"source_pdf": "datasets/websources/Physics_v1/Physics/pdf_esp_198.pdf"
} |
To demonstrate this, let us solve Equation (3.4) for those times $t_n$ when the amplitude of the superimposed sound pressure is zero,
$$t_n = \frac{2n-1}{2(f_1 - f_2)}$$
$(n = 1, 2, 3, ...)$
Now consider in a general fashion the time difference between two consecutive beats, namely, the nth and the (n + 1)th:
$... | {
"Header 1": "3 Sound Wave Propagation and Characteristics",
"token_count": 2033,
"source_pdf": "datasets/websources/Physics_v1/Physics/pdf_esp_198.pdf"
} |
#### Example Problem 2
A train emits a 250-Hz signal while traveling at the rate of 200 km/h. What are the apparent frequencies in approaching the observer and retreating from the observer at the railroad crossing?
#### Solution
From Equation (3.10)
$$v = (200,000 \text{ m/h})/(3600 \text{ s/h}) = 55.6 \text{... | {
"Header 1": "3 Sound Wave Propagation and Characteristics",
"token_count": 2020,
"source_pdf": "datasets/websources/Physics_v1/Physics/pdf_esp_198.pdf"
} |
Application of Huygens' principle leads to
Figure 3.9. A sound wave passing from medium 1 to medium 2. In this case the speed of sound *c*<sup>2</sup> in medium 2 is greater than the speed of sound *c*<sup>1</sup> in medium 1.
the basic laws of refraction, the most useful of which is ... | {
"Header 1": "3 Sound Wave Propagation and Characteristics",
"token_count": 1599,
"source_pdf": "datasets/websources/Physics_v1/Physics/pdf_esp_198.pdf"
} |
| Lower Band Limit<br>fL<br>(Hz) | Center Frequency<br>fC<br>(Hz) | Upper Band Limit<br>fU<br>(Hz) |
|--------------------------------|--------------------------------|--------------------------------|
| 18.0 | 20 | 24.4 |
| 22.4a ... | {
"Header 1": "3 Sound Wave Propagation and Characteristics",
"token_count": 1922,
"source_pdf": "datasets/websources/Physics_v1/Physics/pdf_esp_198.pdf"
} |
Because
$$\left(\frac{p_i}{p_{\text{ref}}}\right)^2 = \log^{-1}\left(\frac{L_{p_i}}{10}\right)$$
the total sound pressure level $L_{pt}$ becomes
$$L_{pt} = 10\log\left[\sum_{i=1}^{n} \left(\frac{p_i}{p_{\text{ref}}}\right)^2\right]$$
(3.23)
or, in terms of the sound pressure levels,
$$L_{pt} = 10 \log \left... | {
"Header 1": "3 Sound Wave Propagation and Characteristics",
"token_count": 1511,
"source_pdf": "datasets/websources/Physics_v1/Physics/pdf_esp_198.pdf"
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| Frequency | A weighting | B weighting | C weighting |
|-----------|-------------|-------------|-------------|
| (Hz) | (dB) | (dB) | (dB) |
| 10 | -70.4 | -38.2 | -14.3 |
| 12.5 | -63.4 | -33.2 | -11.2 |
| 16 | -56.7 | -28.5 ... | {
"Header 1": "3 Sound Wave Propagation and Characteristics",
"token_count": 2044,
"source_pdf": "datasets/websources/Physics_v1/Physics/pdf_esp_198.pdf"
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A histogram showing probability of exceedance for plant noise.
#### Example Problem 7
From perusal of Figure 3.14, estimate the sound levels that are exceeded 10%, 50%, and 90% of the time. Also establish the percentage of the total observation time that the sound was between 65 and 67 dB(A).
#### Solution
The ... | {
"Header 1": "3 Sound Wave Propagation and Characteristics",
"token_count": 1727,
"source_pdf": "datasets/websources/Physics_v1/Physics/pdf_esp_198.pdf"
} |
Consider two sound waves which are detected at a point in space:
$$p_1 = P_1 \cos(\omega_1 t + \phi_1) \tag{3.33a}$$
$$p_2 = P_2 \cos(\omega_2 t + \phi_2)$$
(3.33b)
or in terms of complex exponential functions
$$p_1 = \mathbf{Re} \left[ P_1 e^{i(\omega_1 t + \phi_1)} \right]$$
$$p_2 = \mathbf{Re} \left[ P_2... | {
"Header 1": "3 Sound Wave Propagation and Characteristics",
"token_count": 1807,
"source_pdf": "datasets/websources/Physics_v1/Physics/pdf_esp_198.pdf"
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From Equation (2.22) the velocity is
$$u(r,t) = -\frac{1}{\rho} \int \left[ \frac{A}{r} k \sin k(r - ct) + \frac{A}{r^2} \cos k(r - ct) \right] dt$$
$$= -\frac{1}{\rho} \left[ -\frac{kA}{kcr} \cos k(r - ct) - \frac{A}{r^2 kc} \sin k(r - ct) \right]$$
or
$$u(r,t) = \frac{A}{\rho cr} \cos k(r - ct) \left[ 1 + \frac... | {
"Header 1": "3 Sound Wave Propagation and Characteristics",
"token_count": 1969,
"source_pdf": "datasets/websources/Physics_v1/Physics/pdf_esp_198.pdf"
} |
Combining Equations (3.51a) and (3.51b) with the definition of sound pressure level given by Equation (3.22), we express the sound pressure Lp in terms of sound power W of the source and distance r from the source:
$$L_p = 10 \log \left(\frac{p_{\text{rms}}^2}{p_{\text{ref}}^2}\right) = 10 \log \left(\frac{\rho c W}{... | {
"Header 1": "3.19 The Spherical Wave: Sound Pressure Level and Sound Intensity Level",
"token_count": 2059,
"source_pdf": "datasets/websources/Physics_v1/Physics/pdf_esp_198.pdf"
} |
(A classic text which probably contains more details than necessary, with the result that the mathematics tends to obscure the physics of acoustics.)
Pierce, Alan D. 1981. *Acoustics: An Introduction to Its Physical Principles and Applications*. New York: McGraw-Hill, Chapters 1 and 2. (An excellent modern text by a ... | {
"Header 1": "3.19 The Spherical Wave: Sound Pressure Level and Sound Intensity Level",
"token_count": 1519,
"source_pdf": "datasets/websources/Physics_v1/Physics/pdf_esp_198.pdf"
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#### 4.1 Introduction
In dealing with vibrating systems it is commonly assumed that the entire mass of the system is concentrated at a single point and the motion of the system can be described by giving the displacement as a function of time. This rather simplified approach yields approximations rather than accurate... | {
"Header 1": "4 Vibrating Strings",
"token_count": 2045,
"source_pdf": "datasets/websources/Physics_v1/Physics/pdf_esp_198.pdf"
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The shape of the pulse after its second reflection is identical with that of the original pulse. This periodicity has resulted from the specified boundary conditions, i.e., fixed points at *x* = 0 and *x* = *L*.
#### 4.6 Simple Harmonic Solutions of the Wave Equation
Simple harmonic vibrations frequently occur in n... | {
"Header 1": "4 Vibrating Strings",
"token_count": 1884,
"source_pdf": "datasets/websources/Physics_v1/Physics/pdf_esp_198.pdf"
} |
The actual amplitude of the *n*th mode is
$$a_n = \sqrt{A_n^2 + B_n^2}$$
Consider the initial condition at t = 0 at the time when the string is displaced from its normal linear configuration so that the displacement y(x,t) at each point of the string is given by the function
$$y(x,0) = y_0(x)$$
The correspondin... | {
"Header 1": "4 Vibrating Strings",
"token_count": 2025,
"source_pdf": "datasets/websources/Physics_v1/Physics/pdf_esp_198.pdf"
} |
where **A** is a complex constant of which magnitude equals the displacement amplitude of the wave motion and whose phase angle renders the difference in phase between the motion of the string and the driving force.
In complex format the harmonic driving force can be written as
$$\mathbf{f} = \mathbf{F}e^{i\omega... | {
"Header 1": "4 Vibrating Strings",
"token_count": 1985,
"source_pdf": "datasets/websources/Physics_v1/Physics/pdf_esp_198.pdf"
} |
For extremely low frequencies the input impedance has the limits
$$\mathbf{Z}_{\mathrm{s}} = -\frac{i\delta c}{kL} = -\frac{iT}{\omega L}$$
#### 4.12 Real Strings: Free Vibration
Real strings manifest some degree of stiffness, causing the observed frequencies to be higher than the theoretical values for idealized... | {
"Header 1": "4 Vibrating Strings",
"token_count": 2029,
"source_pdf": "datasets/websources/Physics_v1/Physics/pdf_esp_198.pdf"
} |
#### 5.1 Introduction
The theory underlying the physical operation of vibrating bars is of great interest to acousticians, because a number of acoustic devices employ longitudinal vibrations in bars and frequency standards are established by producing sounds of specific pitches in circular rods of different lengths. ... | {
"Header 1": "5 Vibrating Bars",
"token_count": 2022,
"source_pdf": "datasets/websources/Physics_v1/Physics/pdf_esp_198.pdf"
} |
We now assume that the bar is rigidly fixed at both ends; the boundary conditions $\xi(x, t)$ becomes $\xi = 0$ at x = 0 and at x = L at all times t. Applying the condition $\xi(0, t) = 0$ yields $\mathbf{A} = -\mathbf{B}$ , and Equation (5.10) revises to
$$\xi = \mathbf{A}e^{i\omega t(e^{-ikx} - e^{ikx})} \ta... | {
"Header 1": "5 Vibrating Bars",
"token_count": 1923,
"source_pdf": "datasets/websources/Physics_v1/Physics/pdf_esp_198.pdf"
} |
The appropriate boundary condition for this case would be
$$-F_x(0,t) = m \left(\frac{\partial^2 \xi}{\partial t^2}\right)_{x=0}$$
(5.18)
Incorporating the boundary condition Equation (5.18) into Equation (5.16) results in
$$-E\hat{A}e^{i\omega t}(-ike^{-ikt}+ike^{ikt})=mAe^{i\omega t}(-\omega^2)(e^{-ikL}+e^{ikL}... | {
"Header 1": "5 Vibrating Bars",
"token_count": 1248,
"source_pdf": "datasets/websources/Physics_v1/Physics/pdf_esp_198.pdf"
} |
Consider a freely vibrating bar with arbitrary loadings at each end. We shall establish the normal modes of vibration in terms of the mechanical impedances at both ends of the bar. Designating the mechanical impedance of the support at x = 0 by $\mathbf{Z}_{mi0}$ we express the force acting at this support due to the... | {
"Header 1": "5.6 General Boundary Conditions for a Freely Vibrating Bar",
"token_count": 1778,
"source_pdf": "datasets/websources/Physics_v1/Physics/pdf_esp_198.pdf"
} |
But a bending moment M occurs in the bar, i.e.,
$$M = \int r \, df = -\frac{E}{R} \int r^2 \, d\hat{A}$$
From classical mechanics we recognize the radius of gyration $\kappa$ of the cross-sectional area $\hat{A}$ , as defined by
$$\kappa^2 = \frac{\int r^2 d\hat{A}}{\hat{A}}$$
We now obtain
$$M = \frac{\ka... | {
"Header 1": "5.6 General Boundary Conditions for a Freely Vibrating Bar",
"token_count": 1965,
"source_pdf": "datasets/websources/Physics_v1/Physics/pdf_esp_198.pdf"
} |
We make use of the following hyperbolic and trigonometric identities:
$$\sin y = (e^{iy} - e^{-iy})/2, \qquad \cos y = (e^{iy} + e^{-iy})/2$$
$$\sin(iy) = i \sinh y, \qquad \sinh(iy) = i \sin y$$
$$\cos(iy) = \cosh y, \qquad \cosh(iy) = \cos y$$
to recast Equation (5.33) as
$$y = \cos(\omega t + \phi) \left( ... | {
"Header 1": "5.6 General Boundary Conditions for a Freely Vibrating Bar",
"token_count": 2018,
"source_pdf": "datasets/websources/Physics_v1/Physics/pdf_esp_198.pdf"
} |
In the free–free bar the modes correspond to the fundamental frequency *f*<sup>1</sup> and all additional odd-numbered frequencies. The odd-numbered frequencies *f*3, *f*5, and so on, are symmetric about the center of the bar. The slope ∂*y*/∂*x* is always zero at the center, which is a true antinode. But the even-numb... | {
"Header 1": "5.6 General Boundary Conditions for a Freely Vibrating Bar",
"token_count": 1594,
"source_pdf": "datasets/websources/Physics_v1/Physics/pdf_esp_198.pdf"
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#### 6.1 Introduction
In this chapter we are now applying two-dimensional wave equations to membranes and plates. In this group of physical applications, three dimensions are really involved in the theory that applies to two-dimensional surfaces such as drumheads and diaphragms of microphones or loudspeakers. Two spa... | {
"Header 1": "6 Membrane and Plates",
"token_count": 1987,
"source_pdf": "datasets/websources/Physics_v1/Physics/pdf_esp_198.pdf"
} |
Therefore, the normal modes occur from
$$\mathbf{z}(x, y, t) = \alpha e^{i\omega t} \sin k_x x \sin k_y y \tag{6.12}$$
for which $k_x$ and $k_y$ turn out to be discrete values established by
$$k_x = n\pi/L_x$$
$n = 1, 2, 3, ...$
$k_y = m\pi/L_y$ $m = 1, 2, 3, ...$
The frequencies of the allowed modes of v... | {
"Header 1": "6 Membrane and Plates",
"token_count": 2032,
"source_pdf": "datasets/websources/Physics_v1/Physics/pdf_esp_198.pdf"
} |
A number of the first few (and simpler) modes of vibration with ε*mn* = 0 are shown in Figure 6.3. Each mode is designated by the ordered pair of integers (*m*, *n*). Integer *m* governs the number of *radial nodal lines*, whereas integer *n* determines the number of *azimuthal nodal circles*. Since mode (0, 0) would o... | {
"Header 1": "6 Membrane and Plates",
"token_count": 608,
"source_pdf": "datasets/websources/Physics_v1/Physics/pdf_esp_198.pdf"
} |
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