《振动与波》——光与物质系列物理教材

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"3.vibrations_and_waves_crowell.pdf 是一本由 Benjamin Crowell 编写的关于振动与波的物理学教材，属于 Light and Matter 系列免费初级物理教科书的一部分。该系列还包括牛顿物理学、守恒定律、电磁学、光学以及现代物理学革命等内容。这本书采用创作共享署名-相同方式共享许可协议1.0版，并在某些情况下，使用了非作者所有的照片和图纸，具体在图片信用中列出。读者可以按照许可协议复制、下载电子版或根据 GNU 自由文档许可协议1.2版进行复制。" 在 "3.Vibrations and Waves" 这一章中，Benjamin Crowell 将深入探讨物理学中的关键概念，主要包括以下几个方面： 1. **振动基础**：振动是物体往复运动的形式，书中可能涵盖简谐振动、阻尼振动和受迫振动等。简谐振动是最基本的振动类型，如弹簧振子和摆的振动，其中力与位移成正比。阻尼振动涉及到振动系统能量的损失，而受迫振动则涉及到物体在外力作用下发生的振动。 2. **波动理论**：波动是能量传递的方式，包括机械波（如声波和水波）和电磁波（如光波）。书中可能讲解波的基本性质，如波长、频率、速度和相位，以及波动方程的建立和解析。 3. **波的类型**：书中可能区分纵波（粒子沿波的传播方向振动）和横波（粒子垂直于波的传播方向振动），并讨论表面波和内部波。 4. **波的干涉和衍射**：干涉是两列或多列波相遇时产生的现象，可能导致振幅增强或减弱。衍射是波遇到障碍物或通过孔径时，波纹向所有方向扩散的现象，这两者都是波的重要特性。 5. **共振**：当一个振动系统的自然频率接近外部驱动力的频率时，会发生共振，此时振动幅度显著增加。共振在许多工程和物理问题中都至关重要。 6. **多普勒效应**：多普勒效应描述了波源和观察者相对移动时，接收到的波的频率变化。这在声波和光波中都有应用，如医疗超声和红移蓝移现象。 7. **波动的应用**：书中可能会介绍振动与波在实际生活和科学中的应用，如地震波的探测、声纳技术、光纤通信以及光的波动理论在光学仪器中的应用。通过深入学习这些内容，读者将能够理解振动与波的基本原理，以及它们在物理学和工程技术中的广泛影响。此外，书中可能还包含丰富的实例、习题和实验设计，以帮助学生巩固理论知识并提高实践能力。

d / 1. The amplitude of the

vibrations of the mass on a spring

could be deﬁned in two different

ways. It would have units of

distance. 2. The amplitude of a

swinging pendulum would more

naturally be deﬁned as an angle.

Period and frequency of a ﬂy’s wing-beats example 2

A Victorian parlor trick was to listen to the pitch of a ﬂy’s buzz, reproduce

the musical note on the piano, and announce how many times the ﬂy’s

wings had ﬂapped in one second. If the ﬂy’s wings ﬂap, say, 200 times in

one second, then the frequency of their motion is f = 200/1 s = 200 s

−1

The period is one 200th of a second, T = 1/f = (1/200) s = 0.005 s.

Units of inverse second, s

−1

, are awkward in speech, so an abbre-

viation has been created. One Hertz, named in honor of a pioneer

of radio technology, is one cycle per second. In abbreviated form,

1 Hz = 1 s

−1

. This is the familiar unit used for the frequencies on

the radio dial.

Frequency of a radio station example 3

. KKJZ’s frequency is 88.1 MHz. What does this mean, and what period

does this correspond to?

. The metric preﬁx M- is mega-, i.e., millions. The radio waves emitted

by KKJZ’s transmitting antenna vibrate 88.1 million times per second.

This corresponds to a period of

T = 1/f = 1.14 × 10

−8

s .

This example shows a second reason why we normally speak in terms

of frequency rather than period: it would be painful to have to refer to

such small time intervals routinely. I could abbreviate by telling people

that KKJZ’s period was 11.4 nanoseconds, but most people are more

familiar with the big metric preﬁxes than with the small ones.

Units of frequency are also commonly used to specify the speeds

of computers. The idea is that all the little circuits on a computer

chip are synchronized by the very fast ticks of an electronic clock, so

that the circuits can all cooperate on a task without getting ahead

or behind. Adding two numbers might require, say, 30 clock cycles.

Microcomputers these days operate at clock frequencies of about a

gigahertz.

We have discussed how to measure how fast something vibrates,

but not how big the vibrations are. The general term for this is

amplitude, A. The deﬁnition of amplitude depends on the system

being discussed, and two people discussing the same system may

not even use the same deﬁnition. In the example of the block on the

end of the spring, d/1, the amplitude will be measured in distance

units such as cm. One could work in terms of the distance traveled

by the block from the extreme left to the extreme right, but it

would be somewhat more common in physics to use the distance

from the center to one extreme. The former is usually referred to as

the peak-to-peak amplitude, since the extremes of the motion look

like mountain peaks or upside-down mountain p e aks on a graph of

position versus time.

In other situations we would not even use the same units for am-

plitude. The amplitude of a child on a swing, or a pendulum, d/2,

would most conveniently be measured as an angle, not a distance,

16 Chapter 1 Vibrations

e / Sinusoidal and non-sinusoidal

vibrations.

since her feet will move a greater distance than her head. The elec-

trical vibrations in a radio receiver would be measured in electrical

units such as volts or amperes.

1.2 Simple Harmonic Motion

Why are sine-wave vibrations so common?

If we actually construct the mass-on-a-spring system discusse d

in the previous section and measure its motion accurately, we will

ﬁnd that its x−t graph is nearly a perfect sine-wave shape, as shown

in ﬁgure e/1. (We call it a “sine wave” or “sinusoidal” even if it is

a cosine, or a sine or cosine shifted by some arbitrary horizontal

amount.) It may not be surprising that it is a wiggle of this general

sort, but why is it a speciﬁc mathematically perfect shape? Why is

it not a sawtooth shape like 2 or some other shape like 3? The mys-

tery deepens as we ﬁnd that a vast number of apparently unrelated

vibrating systems show the same mathematical feature. A tuning

fork, a sapling pulled to one side and released, a car bouncing on

its shock absorbers, all these systems will exhibit sine-wave motion

under one condition: the amplitude of the motion must be small.

It is not hard to see intuitively why e xtremes of amplitude would

act diﬀerently. For example, a car that is bouncing lightly on its

shock absorbers may behave smoothly, but if we try to double the

amplitude of the vibrations the bottom of the car may begin hitting

the ground, e/4. (Although we are assuming for simplicity in this

chapter that energy is never dissipated, this is clearly not a very

realistic assumption in this example. Each time the car hits the

ground it will convert quite a bit of its potential and kinetic en-

ergy into heat and sound, so the vibrations would actually die out

quite quickly, rather than repeating for many cycles as shown in the

ﬁgure.)

The key to understanding how an object vibrates is to know how

the force on the object depends on the object’s position. If an object

is vibrating to the right and le ft, then it must have a leftward force

on it when it is on the right side, and a rightward force when it is on

the left side. In one dimension, we can represent the direction of the

force using a positive or negative sign, and since the force changes

from positive to negative there must be a point in the middle where

the force is zero. This is the equilibrium point, where the object

would stay at rest if it was release d at rest. For convenience of

notation throughout this chapter, we will deﬁne the origin of our

coordinate system s o that x equals zero at equilibrium.

Section 1.2 Simple Harmonic Motion 17

g / Seen from close up, any

F − x curve looks like a line.

f / The force exerted by an

ideal spring, which behaves

exactly according to Hooke’s law.

The simplest example is the mass on a spring, for which force

on the mass is given by Hooke’s law,

F = −kx .

We can visualize the behavior of this force using a graph of F versus

x, as shown in ﬁgure f. The graph is a line, and the spring constant,

k, is equal to minus its slope. A stiﬀer spring has a larger value of

k and a steeper slope. Hooke’s law is only an approximation, but

it works very well for most springs in real life, as long as the spring

isn’t compressed or stretched so much that it is permanently bent

or damaged.

The following important theorem, whose proof is given in op-

tional section 1.3, relates the motion graph to the force graph.

Theorem: A linear force graph makes a sinusoidal motion

graph.

If the total force on a vibrating object depends only on the

object’s position, and is related to the objects displacement

from e quilibrium by an equation of the form F = −kx, then

the object’s motion displays a sinusoidal graph with period

T = 2π

m/k.

Even if you do not read the proof, it is not too hard to understand

why the equation for the period makes sense. A greater mass causes

a greater period, since the force will not be able to whip a massive

object back and forth very rapidly. A larger value of k causes a

shorter period, because a stronger force can whip the object back

and forth more rapidly.

This may seem like only an obscure theorem about the mass-on-

a-spring system, but ﬁgure g shows it to b e far more general than

that. Figure g/1 depicts a force curve that is not a straight line. A

system with this F − x curve would have large-amplitude vibrations

that were complex and not sinusoidal. But the same syste m would

exhibit sinusoidal small-amplitude vibrations. This is because any

curve looks linear from very close up. If we magnify the F − x

graph as shown in ﬁgure g/2, it becomes very diﬃcult to tell that

the graph is not a straight line. If the vibrations were conﬁned to

the region shown in g/2, they would be very nearly sinusoidal. This

is the reason why sinusoidal vibrations are a universal feature of

all vibrating systems, if we restrict ourselves to small amplitudes.

The theorem is therefore of great general signiﬁcance. It applies

throughout the universe, to objects ranging from vibrating stars to

vibrating nuclei. A sinusoidal vibration is known as simple harmonic

motion.

Period is approximately independent of amplitude, if the

amplitude is small.

Until now we have not even mentioned the most counterintu-

itive aspect of the equation T = 2π

m/k: it does not depend on

18 Chapter 1 Vibrations

h / The object moves along

the circle at constant speed,

but even though its overall

speed is constant, the x and y

components of its velocity are

continuously changing, as shown

by the unequal spacing of the

points when projected onto the

line below. Projected onto the

line, its motion is the same as

that of an object experiencing a

force F = −kx.

1.3 ? Proofs

In this section we prove (1) that a linear F − x graph gives

sinusoidal motion, (2) that the period of the motion is 2π

m/k,

and (3) that the perio d is independent of the amplitude. You may

omit this section without losing the continuity of the chapter.

The basic idea of the proof can be understood by imagining

that you are watching a child on a merry-go-round from far away.

Because you are in the same horizontal plane as her motion, she

appears to be moving from side to side along a line. Circular motion

viewed edge-on doesn’t just look like any kind of back-and-forth

motion, it looks like motion with a sinusoidal x−t graph, because the

sine and cosine functions can be deﬁned as the x and y coordinates

of a point at angle θ on the unit circle. The idea of the proof, then,

is to show that an object acted on by a force that varies as F = −kx

has motion that is identical to circular motion projected down to

one dimension. The equation will also fall out nicely at the end.

For an object performing uniform circular motion, we have

|a| =

The x component of the acceleration is therefore

cos θ ,

where θ is the angle measured counterclockwise from the x axis.

Applying Newton’s second law,

= −

cos θ , so

= −m

cos θ .

Since our goal is an equation involving the period, it is natural to

eliminate the variable v = circumference/T = 2πr/T , giving

= −

4π

cos θ .

The quantity r cos θ is the same as x, so we have

= −

4π

x .

Since everything is constant in this equation except for x, we have

proved that motion with force proportional to x is the same as circu-

lar motion projected onto a line, and therefore that a force propor-

tional to x gives sinusoidal motion. Finally, we identify the constant

factor of 4π

m/T

with k, and solving for T gives the desired equa-

tion for the p eriod,

T = 2π

Since this equation is independent of r, T is independent of the

amplitude, subject to the initial assumption of perfect F = −kx

behavior, which in reality will only hold approximately for small x.

20 Chapter 1 Vibrations

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