STANDING WAVES IN AN AIR COLUMN (Wind Instruments)
Sound waves (longitudinal) reflect from open and closed tube ends
 Twopossibleboundaryconditions
 CanalsodescribewaveintermsofPRESSUREorDISPLACEMENT
AT A CLOSED TUBE END
 Nolongitudinaldisplacementofairnexttoend→DISPLACEMENTNODE
 Largeamplitudepressurechanges→PRESSUREANTINODE
AT AN OPEN END
 Airfreetooscillate→DISPLACEMENTANTINODE
 Airisfreetomovesopressuredoesnotchange→PRESSURENODE
 Realboundaryattheendofanopentubeisoutsideby~0.6xtuberadius
o Endcorrection
We will use DISPLACEMENT to describe normal modes of an air column.
HOW DO WE “REPRESENT” STANDING WAVES IN AN AIR COLUMN?
 RepresentAMPLITUDEOFLONGITUDINALDISPLACEMENTbywidthof
“loops”indiagram
o CAREFUL:thereisnotransversecharactertomodes
o Each“loop”inthemodediagramisλ/2
LONGITUDINAL MODES OF A TUBE OPEN AT BOTH ENDS
 Boundaryconditionisadisplacementantinodeateachend
1  2 L 
2  L 
3 
2L
3

v
f1 
f2 
1
v
2
f3 

v
3

v
2L
v
 2 f1
L

3v
 3 f1
2L
v
f

n
where n  1, 2, 3,   Foratubeopenatbothends n
2L
o ALLharmonicspossible
(fundamental)
(2nd harmonic)
(3rd harmonic)
LONGITUDINAL MODES OF A TUBE WITH ONE OPEN & ONE CLOSED END
 Boundaryconditionisadisplacementnodeattheclosedendanda
displacementantinodeattheotherend
v
1  4 L 
f1 
3 
4L
3

f3 
5 
4L
5

f5 
1

v
3
v
5
v
4L

3v
 3 f1
4L

5v
 5 f1
4L
(fundamental)
(3rd harmonic)
(5th harmonic)
v
f

n
where n  1, 3, 5, n
 Foratubewithoneopenandoneclosedend,
4L
o OnlyODDharmonicspossible
o Fundamentalfrequencyhalfthatforsamelengthtube,bothendsopen
BEATS (14.6 in text)
 Considertwowaveswithslightlydifferentfrequenciesarrivingatobserver
o Willgoinandoutofphase→resultantintensityisperiodic
 To find “BEAT” frequency, find resultant of two waves at x=0
o y1  A cos2 f1t  and y2  A cos2 f 2t 
 Resultant: yR  y1  y2  Acos2 f1t   cos2 f 2t 
 To evaluate, use trigonometric identity
a b ab
cos
a

cos
b

2
cos

 cos

o
2
2

 

  f1  f 2  
  f1  f 2  

y

2
A
cos
2
t
cos


2  2 t  

 Resultant: R
2


 
 
 f1  f 2 

f
o average  2  istheaverageofthetwointerferingfrequencies
  f1  f 2  
2
A
cos
2  2 t  actslikeatime‐dependentamplitude
o

 
 modulateswavewithaveragefrequency
 “envelope”repeatswithfrequency f beat  f1  f 2 NON-SINUSOIDAL WAVES → COMPLEX WAVES
 tonefromatuningfork(orrecorder)ismostlyfundamental→sinusoidal
 cangetnon‐sinusoidalwaves(morecomplexsounds)byaddingharmonics
EXAMPLE: Can make a square wave by adding odd harmonics.
1
y

n n sin 2nf1t 

 Character of sound
depends on harmonics
o Trumpet, clarinet,
etc. act like tubes
open both ends (all
harmonics)
1.0
0.5
IN TEN SITY
 This is a Fourier series
o Get spectrum
(intensity vs
harmonic number)
by Fourier analysis
sin(2f1t)
(1/3)sin(2*3*f1t)
(1/5)sin(2*5*f1t)
(1/7)sin(2*7*f1t)
sum
0.0
-0.5
-1.0
0
20
40
60
TIM E
80
100