21
Ph203:20-6A Waves of small amplitude traveling through the same medium combine, or superpose, by simple addition. Wave Superposition 12a-1
Ph203:20-6A
Waves of small amplitude traveling through the same
medium combine, or superpose, by simple addition.
Wave Superposition
12a-1
Principle of Linear Superposition
fTot = f1(x vt) + f2(xvt)
Ph203:20-7
Wave Interference
12a-2
10-38-2
recall 12a-3
Aside: M. Fourier proved that any (well behaved) wave* can be
written as a sum of simple harmonic waves.
square wave triangle wave
can handle
“anything”
just add up
enough
SH waves
SH waves
are all we need
Flip-side: one can break down any wave into Fourier SH components
# of sine
waves
summed
12a-4
v = wave velocity
0+
-
Add up 2 waves point by point
Interference – partial/total cancelation/increase
Constructive Interference
=+
Destructive Interference
=+ 0
anything in-between
8-3
Interference:
simple harmonic waves
12a-5
2 identical waves traveling in opposite directions
12a-6
t x t x
sin(2π + 2π ) + sin(2π - 2π )T T
A A
Sum=
t -2πx sin(2π ) cos( )
TA
Sum=2
trig. identity
Standing wave
+ vs – only difference
time only space only
/2
0 to 0 |max| to |max|/4/2
Note: same
12a-7
Standing Waves: e.g. string with fixed end points
BOUNDARY CONDITIONS: no amplitude at ends
L
1λL= 1
22λ
L= 223λ
L= 32
nλL= n
2:n =1,2,3,4...
n nv=λ fn
2Lλ =
n
n 1
n
v vf = = n nf
λ 2L
12a-8
L
1λL= 1
2
2λL= 2
2
3λL= 3
2
nλL= n
2
:n =1,2,3,4...
n 1
n
v vf = = n nf
λ 2L
n nv=λ f
n
2Lλ =
n
Standing Sound Waves: e.g. pipe with 2 closed endsBOUNDARY CONDITIONS: no amplitude at ends (again)
12a-9
1λL= 1
2
2λL= 2
2
3λL= 3
2
nλL= n
2:n =1,2,3,4...
n 1
n
v vf = = n nf
λ 2L
n nv=λ f
n
2Lλ =
n
Open-Open end organ pipe
Note: open-open
Same L-f as closed-closed12a-10
Standing Sound Waves: e.g. pipe with 2 open endsBOUNDARY CONDITIONS: max amplitude at both ends
12a-10a
Same situation
as on last slide
Standing Sound Waves: e.g. pipe with 1 closed , 1 open endBOUNDARY CONDITIONS: 0 amplitude one end: max at other
3 3λ λL= 2+
2 4
n nλ λL= (n-1)+
2 4
:n =1,2,3,4...
n
n
vf =
λ
nλL= (2n-1)
4
n
4Lλ
(2n-1)
n
(2n-1)vf
4L
m
mvf
4L
:m =1,3,5,7,9...
or
12a-11
12a-12
Same situation
as on last slide
0pen end
– max. density
oscillation !!
Closed end
– density node !!
a
λL =
2
La
Lb
Lc
b
λL =2
2
c
λL =3
2
/2
Lb=0.48m
La=0.24m
La=72m
v=λf
f=700 Hz
=0.48m
/2=0.24m
=(0.48m)(700 1/s)
v=336 m/s
Literature v=340 m/s
sound
speed in air
!!!
Demonstration- tube with movable speaker-closed-closed
12a-13
a
λL =
4
v=λf
f=700 Hz
La
Lb
Lc
b
λ λL =
2 4
c
λ λL =2
2 4
/2
/4
Lb=0.36m
La=0.12m
Lc=0.6m
=0.48m
/2=0.24m
/4=0.12m
=(0.48m)(700 1/s)
v=336 m/s
Literature v=340 m/s
sound
speed in air
!!!
12a-14
Demonstration- tube with movable speaker-closed-open
12a-15
L =
4f = v
f = = v
v
4L
L smaller f higher
L
wavelength of the fundamental wave (1st harmonic)created in terms of L?
Creating Tones with Bottlestone produced by blowing across a bottle opening
12a-16
L
For a rope of length L show that as the tension is increased that there is a maximum
tension above which no standing waves are possible.
12a-17
max
2
1
LfT
Long period modulationAverage frequency
Not required
' 'sin( ) sin( ' ) 2sin([ ] ) cos([ ] )
2 2t t t t
' 'sin( ) sin( ') 2sin[ ] cos[ ]
2 2
12a-18
Beats
The new frequency is:
where u is source velocity
General case (both source and receiver moving):
Not required: will discuss next semester
Doppler Effect
12a-19
Relation preferred by Croft.
0 wave
Δf v=±
f v
+ coming at you (blue shift)- going away (red shift)
Detecting Tornadoes Doppler Blood Flow Meter
Not required: will discuss next semesterApplications of Doppler
Electro magnetic waves Electro magnetic waves
Electro magnetic waves sound waves 12a-20
