PHY 2311

PHYSICS 231Lecture 34: standing waves &

harmonics

Remco ZegersLast lecture: Friday (Review)

PHY 2312

doppler effect: general

source you

source

observer

vv

vvff

vobserver: positive if moving towards to sourcevsource: positive if moving towards the observer

PHY 2313

quiz

An ambulance is moving towards you with its sirens on. Thefrequency of the sound you here is …… than the frequencyyou would hear if the ambulance were not moving at all.

source

observer

vv

vvff

a) higherb) the samec) lower

vobserver=0vsource= positive

so f’>f

PHY 2314

standing waves

Two interfering waves canat times constructivelyinterfere and at times destructively interfere

If the two interferingwaves always have thesame vertical displacementat any point along thewaves, but are of oppositesign: standing waves

PHY 2315

How to create standing waves: a rope

The oscillations in the rope are reflected from the fixedend (amplitude is reversed) and create a standing wave.

demo

PHY 2316

we can produce different wave lengths

1=2L 2=L 3=2L/3

4=2L/4 5=2L/5

both ends fixed n=2L/n or L=nn/2

PHY 2317

standing waves

both ends fixed n=2L/n or L=nn/2

F

L

n

L

nvvf

nn 22

F: tension in rope: mass per unit length

1

2

1

2

2

22

nfL

nvf

L

vf

L

vf

n

nth harmonics

f1: fundamental frequency

PHY 2318

example: the guitar

F

L

nfn 2

nth harmonics: depends where and how the string is strucknote that several harmonics can be present and that non-harmonics are washed out

length can be chosen by placing fingers

changes from string to string:bass string is very heavy

tension can be varied by stretchingthe wire

PHY 2319

example

A guitar string is struck. Assume that the first harmonicis only excited. What happens to the frequency if:a) The player put a finger at half the length of the string?b) The player makes the tension 10% larger (by turning the tuning screw)?c) A string is struck in the same way, but its mass is 3 times higher?

F

L

nfn 2

a) L x 0.5 then f x 2b) F x 1.1 then f x 1.1=1.05c) x 3 then f x (1/3)=0.58

PHY 23110

Standing waves in air columns

Just like standing waves in transverse oscillations, one canmake standing waves in longitudinal oscillations as well.

PHY 23111

An air column (pipe)

A pipe can be open or closed on either or both sides.

For now, let’s consider the air-displacements (anti-)nodes

PHY 23112

Both ends open

3,2,12 1 nnfL

nvfn

PHY 23113

One end open, one end closed

5,3,14 1 nnfL

nvfn

even harmonics are missing!!!

PHY 23114

exampleA simple flute is played by blowing air in on one side considered to be open and the other end is closed. The length of the tube can be variedmanually (like a trombone). What are the frequencies of the first two possible harmonics if L=0.5m? If the length is made half of the original length, how will these changev=343m/s?

f1=343/(4*0.5)=172 Hz f3=3*343/(4*0.5)=514 Hz

f1=343/(4*0.25)=343 Hz f3=3*343/(4*0.25)=1028 Hz

PHY 23115

example

A simple flute is played by blowing air in on one side considered to be open and the other end is open as well. The length of the tube can be variedmanually (like a trombone). What are the frequencies of the first two possible harmonics if L=0.5m? If the length is made half of the original length, how will these changev=343m/s?

f1=343/(2*0.5)=343 Hz f2=2*343/(2*0.5)=686 Hz

f1=343/(2*0.25)=686Hz f2=2*343/(2*0.25)=1372 Hz

PHY 23116

beatsSuperposition of 2 waves with slightly different frequency

The amplitude changes as a function of time, so the intensityof sound changes as a function of time. The beat frequency (number of intensity maxima/minima per second): fbeat=|fa-fb|

DEMO

PHY 23117

example

Someone is trying to tune a guitar. One of the strings issupposed to have a frequency of 500 Hz. The person isusing a tuning fork which produces a sound of exactlythis frequency, but while sounding the fork and the playingthe guitar, hears a beat in the sound with a frequency of3 Hz (3 beat per second). a) What is the real frequency ofthe guitar string? b) By what fraction does the person need tochange the tension of the guitar string to tune it properly?

a) fb=|ffork-fguitar| 3=|500-fguitar| fguitar=497 or 503 Hz

b)

F

L

nfn 2

so f~F

fcurrent/fideal= (Fcurrent/Fideal)497/500=0.954 or 503/500=1.006Fideal=Fcurrent/(0.994)2=1.012Fcurrent

or Fideal=Fcurrent/(1.006)2=0.988Fcurrent

PHY 23118

Resonances

Realistically, oscillations are damped due to frictional forces.However, we can drive the oscillation via an external source.Example: mass on a spring: natural frequency f=1/(2)(k/m)

If the frequency of the driving force equals the naturalfrequency: large oscillations occur: Resonance demo

Resonances occur in many daily situations:•shock absorber in car•playing basketball•resonating lecture room!!

Famous example: Tacoma bridge