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