Chapter 17 Waves — I

Wave is everywhere in nature!

Why can you hear me?

Why can you receive a TV signal emitted from a TV station?

§1 Types of Waves

Classification According to natures of matter

1. Mechanical Waves.

Examples: water waves, sound waves, and seismic waves( 地震波 ).

They are governed by Newton’s laws, and they can existonly within a material medium, such as water, air, and rock.

2. Electromagnetic Waves.

These waves have certain central features:

Examples: Visible and ultraviolet light, radio and television waves, microwaves, x rays, and radar waves.They don’t require material medium to exist. These waves have a limited wave speed in vacuum,

. m/s 458 792299c

3. Matter Waves.

Examples: Electrons, protons, and other fundamentalparticles; ultracold atoms.

The most remarkable property of the matter waves isthat wave functions of matter waves are referred toas probability amplitudes of waves.

Classification According to Oscillation Types

1. Transverse Waves.

The direction of oscillation of medium elements is perpendicular to the direction of travel of the wave.

2. Longitudinal Waves. The direction of oscillation of medium elements is parallel to the direction of the wave’s travel, the motionis said to be longitudinal, and the wave is said to be a longitudinal wave.

Both a transverse wave and a longitudinal wave are said to be traveling waves because they both travelfrom one point to another, as from one end of the string to the other end.

sparse and dense

§2 Creation of Waves and Propagation

For a source of simple harmonic oscillation with angularfrequency , the traveling in the positive direction of x axis and oscillating parallel to the y axis, the displacementy of the element located at position x at time t is given by

Conditions of a Mechanical wave

1. A Wave Source

2. Medium in which a mechanical wave propagates.

A wave means that the oscillation state propagates withtime. It does not mean the particles of medium move forward with wave. Particles of the medium oscillate onlyaround their corresponding equilibrium positions.

).sin(),( kxtytxy m

Displacement Phase

Angularwave number

Angular frequency

Amplitude

ym is the amplitude ofa wave.

The phase of the waveis the argumentt-kx)

).sin(),( kxtytxy m

Space Periodicity and Time Periodicity

Wavelength and Angular Wave Number

The wavelength of a wave is the distance (parallel to the direction of the wave’s travel) between repetitions ofthe shape of the wave (or wave shape).

If we have a snapshot at t=0, the oscillation at x = x1 is.sin)0,( 1kxyxy m

By definition, the displacement y is the same at both ofthis wavelength — that is at x = x1 and x =x1 + Thus,

kkxykxy mm 11 sinsin

According to the property of sine function, it begins to repeat itself when its angle is increased by 2 rad,

That is

2k (Angular wave number)

Period, Frequency, and Angular Frequency

Considering the displacement y of the wave at the pointx=0,

.sin),0( tyty m

You can choose any time t1, through a period, the oscillation is repeated.

The period of oscillation T of a wave is defined to be the time any mass element takes to moves through one fulloscillation.

).sin(sin 11 Ttyty mm

This can be true only ifT=2or

.2T

Angular frequency

Frequency f is defined by

.2

1

Tf

Wavefronts

Wavefronts are surfaces over which the oscillations ofwave have the same value; usually such surfaces are represented by whole or partial circles in a two-dimensional drawing for a point source.

Rays

Rays are directed lines perpendicular to the wavefrontsthat indicate the direction of travel of the wavefronts.

Wavefronts of plane waveand rays

If wavefronts of a wave are planes, it is called the planewave; if wavefronts of a wave are spherical, the wave is called spherical wave.

The Speed of a Traveling Wave

Assuming that the wave travels along x axis. Forexample, at t=0 we have a snapshot, at t = t, we have another snapshot, meanwhile the wave moves a distance x. The ratio x / t (or dx/dt) is the wave speed v.

If point A retains its displacement as it moves, its phase must remain a constant, that is

constant. a kxt

Taking a derivative, we have

,0dtdx

k

or.

kv

dtdx

By using definitions of and k, we can rewrite v as

fT

v

Phase speed

Question:

How to describe a wave traveling in negativedirection of x?

Similarly to the discussion of positive direction of x,the wave speed of a wave traveling in the negative xdirection should be have a form

kdtdx

v

This corresponding to the condition

constant a kxt

This means that the wave function should have a form

)sin(),( kxtytxy m

It seems that k may be have a property of vector, its magnitude is

vk

and its direction is in that of wave traveling, or wave velocity.

For a traveling wave in positive x, this means the oscillation at x is

kxtytxy m sin),(

For a traveling wave in negative x, this means the oscillation at x is

)sin(),( kxtytxy m

Since x is negative here, so the phase delay is also -|kx|.In general, wave function of an arbitrary shape of traveling wave has a form

).(),( kxthtxy

Wave Speed on a Stretched String

Consider a small stringelement within the pulse, oflength l, forming an arc ofa circle of radius R and subtendingan angle 2 at the center of that circle.

If a wave is to travel through amedium such as water, air, steel,or a stretched string, it must causethe particles of that medium to oscillate as it passes. Youcannot send a wave along a string unless the string is under tension of its two ends.

Assuming that the linear density of the string is =m/l,The tension in the string is equal to the common magnitude of two forces at two ends.

There are only vertical components of to form a radial restoring force . In magnitude,

F

.)2(sin2Rl

F

The mass of the element is given by

.lm

The centripetal acceleration of the element toward thecenter of that circle is given by

.2

Rv

a

From the Newton’s Second Law, we have

,2

Rv

lRl

v

The speed of a wave along a stretched ideal string depends only on the tension and linear density of the string and not on the frequency of the wave.

§3 Energy and Power of a Traveling String Wave

Kinetic Energy

An element of the string of mass dm, oscillating transversely in simple harmonic motion, its kinetic associates with its transverse velocity .u

Element b maximum kineticenergy

Element a zero kinetic energy

For mass element dm, the kineticenergy is given by

,22

1dmudK

where)cos( kxty

ty

u m

and dm = dx. So we have

).(cos)( 222

2

1 kxtydxdK m

Dividing the equation by dt, and by using v = dx/dt, we have

)(cos222

2

1 kxtyvdtdK

m

This is actually the rate of energy transmission. In general we consider the average energy transmissionin a period, that is making an integral from 0T for theargument t, one can have

22

0222

4

1

2

1 )(cos1

m

Tm

avg

yv

dtkxtT

yvdtdK

This is the average kinetic energy transmission in oneperiod on per second a string.

It can be shown that, for the potential energy, it has asame expression,

),(cos222

2

1 kxtyvdtdU

m

because it needs other concepts to show the formula, wedo not show that further more.

So the average power, which is the average rate at which energy of both kinds is transmitted by the wave, is given by

22

2

12 mavg

avg yvdtdK

P

In general, the average energy flow through an unit area of perpendicular to the propagating direction, or density of energy flow, or intensity of the wave, is given by

22

21

myvI

where is the density of wave in volume.

v

§4 Superposition for Waves

The Principle of Superposition for Waves

Suppose that two waves travel simultaneously along thesame stretched string. Let y1(x,t) and y2(x,t) be the displacements that the string would experience if eachwave traveled alone. The displacement of the string whenthe waves overlap is then the algebraic sum

).,(),(),( 21 txytxytxY

It means

Overlapping waves algebraically add to produce a resultant wave (or net wave).

Overlapping waves do not in any way alter the travel of each other.

simulation 17-8 CD Med

Interference of Waves

Suppose that there are two sinusoidalwaves of the same wavelength and amplitude in the same direction alonga stretched string.

)sin(),(1 kxtytxy m

)sin(),(2 kxtytxy m

From the principle of superposition, the resultant wave is

)sin(]cos2[

)sin()sin(

),(),(),(

2

1

2

1

21

kxty

kxtykxty

txytxytxY

m

mm

The new resultant amplitude depends on phase :

1. If = 0 rad, the two interfering waves are exactly in phase.

2

1cos2 mm yY

)sin(2),( kxtytxY m

In this case, the interference produces the greatestpossible amplitude which is called fully constructiveinterference.

2. If =0),( txY

Although we send two waves along the string, we see nomotion of the string. This type of interference is calledfully destructive interference (see fig. (e)).

3. When interference is neither fully constructive nor fully destructive, it is called intermediate interference.Figures (c) and (f) for =2/3.

Two waves with the same wavelength are in phase if their phase difference is zero or any integer number of wavelengths. Thus, the integer part of any phase difference expressed in wavelengths may be discarded.

Phasors

The same as in oscillation, a wave can be representedvectorially by a phasor. A phasor is a vector that has a magnitude equal to the amplitude of the wave and that rotates around an origin; the angular speed of the phasoris equal to the angular frequency of the wave.

)sin(),( 11 kxtytxy m

kxtytxy mm sin),( 22

)sin(),( kxtYtxY m

(a)

(b)

(c)

where and b are given as in previous discussions.To find the values of and , we have to sum the twocombining waves as we did previously.

mY mY

In conclusion we can use phasors to combine waveseven if their amplitude are different.

§5 Interference

We consider two point sources S1 and S2 emit wavesthat are in phase and of identical wave length .Thus the source themselves are said to be in phase; thatis, as the waves emerge from the sources, their displacements are always identical.

Point Pto S1

to S2

L1

L2

Their phase difference at Pdepends on their path lengthdifference L = |L2-L1| .

L

;

The phase difference between L1 and L2 is

.2

L (Why?)

According to the previous section, the condition for fully constructive interference is

.210,2 ,,,nn for

This occurs when

)(or

nLL

,2,1,0

Similarly, fully destructive interference occurs when

.,2,1,0,)12( nn for

This occurs when

)(or ,,,2)12(

25

23

21

nLL

Example :

Two point sources S1 and S2, which are in phase and and separated by distance D=1.5, emit identical soundwaves of wavelength .

Standing Waves

Let’s consider two waves with the same oscillating frequencies, but traveling in opposite directions.

To analyze a standing wave, we represent the two waveswith the equations

)sin(),(1 kxtytxy m )sin(),(2 kxtytxy m

The principle of superposition gives, for the combinedwaves,

,sin]cos2[

)sin()sin(),(

tkxy

kxtykxtytxY

m

mm

Question:

Does above expression be exactly agreement with the previous figures and simulation? Are there any differences?

From the expression and the simulation, we find thatunlike a traveling wave, the amplitude of the wave is the same for all string elements, the amplitude in a standing wave here varies with position. For example,the amplitude is zero for values of kx that give coskx = 0.Those values are

The wave like this — two sinusoidal waves of the same amplitude and wavelength travel in opposite directions,their interference with each other produces a standing wave.

)( ,2,1,0,)12(2

1 nnkx

(Pay your attention to the difference between this equation and the corresponding equation in the textbook (equation 17-48).)

22

1)( nx

The positions satisfied this condition are called nodes.The interval between pairs of nodes is given by

or

.1 2

nn xx

The amplitude of the standing wave has a maximumvalue 2ym, which occurs for values of kx given by coskx=1.

These points satisfy the conditions)( ,...2,1,0, nnkx

or 2nx

The positions satisfied these condition are called antinodes. The interval between pairs of two antinodes isalso /2.

Reflection at a Boundary

A standing wave can be set up in a stretched string byallowing a traveling wave to be reflected from the far end of the string so that it travels back through itself.The incident (original) wave and the reflected wave cancombine to form a pattern of standing wave.

Questions:

If a string is fixed at its end, how about the reflection at this end? Is it a node or an antinode?

If one end of a string is fastened to a light ring that is free to slide without friction along a rod. When an incident wave is reflected at this “soft” reflection end,is it a node or an antinode?

For a string of both fixed ends, we note that a node mustexist at each ends, because each end is fixed and cannotoscillate. For a standing wave of string of length L, there may be one antinode, two antinotes, … n antinodes, as shown in following figures.

It is easy to see the relation

2nL

ornL2

For a wavelength which satisfiesthis relation, a standing wave can be set up on the string.

The resonant frequencies are correspondingly

,3,2,1,2

nLv

nv

f for

where v is the speed of traveling wave on the string.

For n=1, the frequency f=v/2L. This Lowest frequency is called the fundamental mode or the first harmonic .For n=2, we have the second harmonic oscillation mode.For n=3, we have the third harmonic oscillation mode.

Assignments 1:17 — 917 — 1917 — 21

Assignments 2:17 — 2817 — 4017 — 45