Friday, December 11, 1998

Chapter 14: standing waves superposition interference

REVIEW of Chapters 13 & 14

Our final exam period is scheduled for thevery last possible day of finals’ week:

Friday, December 18, 1998!10:30 am - 12:30 pm

Please, do not leave town early

This 2-hour exam period will contain TWO exams.

The first 50 minutes will be used for an examon the material that we cover Chapters 13 & 14.It will be structured very much like Exam #3(three sections, one of which will be multiplechoice---you do two of the three). You will bepermitted to use the usual 3”X5” note card.

Final Exam: 1 hour plus comprehensive

5 sections, you do 4 (one will be multiple choice)

Each section somewhat shorter (25 points)

8.5”X11” crib sheet + 3”X5” note card fromExam #4

I will give you math & constants but notphysics formulae

Exam #4 (during the first 50 minutes of our final exam period) counts the same as the previous 3 exams (the best 3 scores from Exams #1 - 4 count 15% each).

If you are REALLY happy with your performance on the first 3 exams, simply notify me and you will be permitted to SKIP Exam #4. In this case you may show up 50 minutes into the exam period.

(I’m guessing there won’t be too many of YOU!)

Everyone MUST take the Final Exam.

You cannot pass this class without completing the final exam.

The final exam counts for everyone andis worth 20% of your final grade.

This is best demonstrated with an example.

Notice that all points on the slinky oscillateup and down with the same frequency, butdifferent amplitudes. Some points on theslinky do not move at all.

These points are called NODES.

Notice that the fixed end of the slinky issuch a nodal point---always in the sameplace---never moving.

We also observe nodes at various otherpoints along the slinky, spaced by acharacteristic distance equal to one halfof a wavelength.

In between the nodes are places at whichthe spring oscillates with the greatestamplitudes along its length.

Such places are called ANTI-NODES.

Notice that the end of the slinky from whichI force the waves is such an anti-node.

The lowest frequency (longest wavelength)standing wave that I can produce on theslink is 1/4 of a wavelength.

This lowest frequency is known as thefundamental frequency (a.k.a. the firstharmonic) of the slinky.

And I can produce a whole series of otherwaves, known as harmonics, on the slinky.

Notice that there is a discrete set of possiblewaves that the slinky will support. In fact,for this system (fixed at one end, open at theother), only waves with the followingfrequencies are permitted:

f nv

Lnwave4

L = length of slinkyn = 1, 3, 5, 7, ...

If our slinky were fixed at both ends, onlywaves with integral numbers of half-wavelengths could be supported on theslinky, since both ends would have tobe nodes.

f nv

Lnwave2

L = length of slinkyn = 1, 2, 3, 4, 5, ...

And though hard to imagine with our slinky,if it were free to move at both ends, againonly waves with integral numbers of half-wavelengths could be supported on theslinky’s length. The slinky would requireanti-nodes at both ends!

f nv

Lnwave2

L = length of slinkyn = 1, 2, 3, 4, 5, ...

Rules of thumb:

Open (free) ends have anti-nodes.

Closed (fixed) ends have nodes.

Recall that the speed of the wavespropagating along strings (such as onharps, violins, and pianos) is given by

F

And the speed of the wavespropagating in the air inside pipesare given by the speed of sound.

vsound = 340 m/s

Superposition of WavesSuperposition of Waves

How do I add waves together? Point-by-Point

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

0.0 2.5 5.0 7.5

time

Am

plit

ud

e

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

0.0 2.5 5.0 7.5

time

Am

plit

ud

e

+=

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

0 2.5 5 7.5

timeam

pli

tud

e

constructiveinterference

constructiveinterference

When waves interfere constructively, theanti-nodes become greater in magnitudethan either of the waves of which theresulting wave is composed.

Sounds emanating from stereo speakersare louder in places of constructiveinterference.

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

0.0 2.5 5.0 7.5

time

Am

plit

ud

e

+=

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

0 2.5 5 7.5

time

Am

plit

ud

e

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

0 2.5 5 7.5

time

Am

pli

tud

e

These two waves are 180o out-of-phase. Thatis, if you shifted one by 180o relative to the other,the two waves would appear exactly the same.

destructiveinterference

destructiveinterference

When waves interfere destructively, theanti-nodes precisely cancel one another,resulting in a constant node.

Sounds emanating from stereo speakersare softer in places of destructiveinterference. If the tone is pure, the sounddisappears completely.

Please do your best to fill out these evaluationforms. Your comments are appreciated!

In Part IV of our story, we wave good-byeto our beloved Physics 111 students tothunderous ovations, boos, and hissesas they propagate into the holiday seasonto ring in the New Year.

F = - kx

PE kxspring 1

22 KE PE PEg spring

TA

v

m

k

22

max

fk

m

1

2 2 f

k

m

vk

mA x ( )2 2

Fixedend

x

Tr

v

2

Angular Frequency f

Frequency f = 1/T

FT

Fg

s

F mg sin

TL

g2

Good for SMALLamplitude

oscillations!

Height of Block vs Time

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Time (s)

Hei

gh

t (m

)

Equilibriumposition

Amplitude

period

Amplitude

TRAVELLING WAVES:

Therefore, we note that waves do NOTultimately transport matter. They onlytemporarily displace the matter in whichthey move.

v f

Height of Block vs Time

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Time (s)

Hei

gh

t (m

)

Equilibriumposition

Amplitude

period

Amplitude

Position of Piece of Slinky vs Time

Height of Block vs Time

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Time (s)

Hei

gh

t (m

)

Equilibriumposition

Amplitude

wavelength

Amplitude

Shape of Slinky at Time 1

X-position

Y-p

osi

tio

n

crest

trough

The propagation speed is governed bythe mass per unit length () of the materialand the tension (FT) within the material.

vF

What happens to a wave when it hits afixed boundary?

Bounces back (reflects) inverted.

What happens to a wave when it gets tothe end of the slinky that is not fixed?

Reflects with the same orientation.

mo

lecu

les

rare

fied

mo

lecu

les

com

pre

ssed

t = t3

mo

lecu

les

rare

fied

mo

lecu

les

com

pre

ssed

mo

lecu

les

com

pre

ssed

mo

lecu

les

rare

fied

Crests & Troughs in direction of propagation!

In fact, we find that thespeed of sound is given by:

vB

s

Where B is the bulk modulus and is the equilibrium density of the medium.

In solids, we find that thespeed of sound is given by:

vY

sound

Where Y is the Young’s modulus and is the equilibrium density of the medium.

IP

A

FHGIKJ( ) log10

0

dBI

I

Where I is the intensity of the sound andI0 is the intensity of a reference level, usually

taken to be the threshold of hearing(10-12 W/m2).

Listener Moving f fu

' 0

+ approach

- recede

Source Movingf f

v vsound

'( / )

FHG

IKJ0

1

1

- approach+ recede

v = speedof source

Both moving

Rule of Thumb:

If the objects are getting closer together,the frequency should be higher.

If the objects are separating, the frequencyshould be lower.

f fu v

v vsound

sound

'( / )

( / )

FHG

IKJ0

1

1

u = speed listenerv = speed source

I wish you a happy, peaceful, andrestful holiday season.

It has been my sincere pleasure andprivilege to have the opportunityto explore the world of physics withyou this semester.