1. PHYSICS 231 Lecture 35: interference & sound
Remco Zegers
Question hours:Monday 9:15-10:15
Helproom
PHY 231

2. example
A pendulum with a length of 4 m and a swinging mass of 1 kg oscillates with an maximum angle of 10o. What is the gravitational force parallel to the string, perpendicular to the string, the total gravitational force and the centripetal
force when the mass passes through the
equilibrium position and when it reaches its maximum amplitude?
equilibrium
maximum ampl.
gravitation
// to string
1gcos(00)=9.8N
1gcos(100)=9.65N
perpendicular
1gsin(00)=0N
1gsin(100)=1.7N
total gravitational
9.8N
9.8N (vectors!!)
centripetal
0.3N (mv2/L)
0 N
PHY 231

3. describing a traveling wave
: wavelength
distance between
two maxima.
While the wave has traveled one
wavelength, each point on the rope
has made one period of oscillation.
v=x/t=/T= f
On a string: v=(F/)
PHY 231

4. A person is tuning a guitar string. He makes the tension
quiz (extra credit)
A person is tuning a guitar string. He makes the tension
in the string 4 times larger than it originally was. If the
wavelength of the oscillations through the string remains
constant, by what factor does the frequency of the
produced sound change? 1 2 4
v=(F/)
if Fx4 then vx2
v=x/t=/T= f
if vx2 then fx2 if =constant
PHY 231

5. Interference Two traveling waves pass through each other without
affecting each other. The resulting displacement is the
superposition of the two individual waves.
example: two pulses on a string that meet
PHY 231

6. Interference II constructive interference destructive interference
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7. + = + = Interference III   demo: interference
destructive interference
waves ½ out of phase + = + =
constructive interference
waves in phase
PHY 231

8. Interference IV If the two interfering Two interfering waves can
waves always have the
same vertical displacement
at any point along the
waves, but are of opposite
sign: standing waves
Two interfering waves can
at times constructively
interfere and at times
destructively interfere
later more!!!
PHY 231

9. Interference holds for any wave type
The pulses can be sine-waves, rectangular waves
or triangular waves
PHY 231

10. Interference in spherical waves
if r2-r1=n then constructive interference occurs
if r2-r1=(n+½) the destructive interference occurs r1 r2
r1=r2
maximum of wave
minimum of wave
positive constructive interference
destructive interference
negative constructive interference
PHY 231

11. Interference of water waves
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12. Example two speakers separated by 0.7m produce
a sound with frequency 690 Hz (from
the same sound system).
A person starts walking from one of the
speakers perpendicular to the line
connecting the speakers. After what
distance does he reach the first
maximum? And the first minimum?
vsound=343 m/s d1 d2
v=f so =v/f=343/690=0.5m
1st maximum: d1-d2=1=0.5
d2=(0.72+d12) d1-(0.72+d12)= d1=(-)0.24m
1st minimum: d1-d2=½=0.25
d1-(0.72+d12)= d1=(-)0.855
0.7m
direction of
walking person
PHY 231

13. Reflection of waves. Frope on wall= -Fwall on rope demo: rope on wall
FREE END:
no inversion
Frope on wall=
-Fwall on rope
demo: rope on
wall
FIXED END: pulse inversion
PHY 231

14. Connecting ropes If a pulse travels from a light rope to a heavy rope
(light< heavy) the boundary is
nearly fixed. The pulse is
partially reflected (inverted)
and partially transmitted.
If a pulse travels from
a heavy rope to a light rope
(light< heavy) the boundary is
nearly free. The pulse is
partially reflected (not
inverted) and partially
transmitted.
before
before
Ain
Ain AR AT AR AT
after
|AR|<|Ain|
|AT|<|Ain|
after
|AR|<|Ain|
|AT|>|Ain|
PHY 231

15. Sound: longitudinal waves
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16. The speed of sound Depends on the how easy the material is compressed
(elastic property) and how much the material resists
acceleration (inertial property)
v=(elastic property/inertial property)
v=(B/) B: bulk modulus : density
The velocity also depends on
temperature. In air:
v=331(T/273 K)
so v=343 m/s at room temperature
PHY 231

17. Quick question The speed of sound in air is affected in changes in:
(more than one possible)
wavelength
frequency
temperature
amplitude
none of the above
answer c)
PHY 231

18. Intensity Intensity: rate of energy flow through an area Power (P) J/s
A (m2)
I=P/A (J/m2s=W/m2)
example: If you buy a speaker, it gives power output
in Watts. However, even if you put a powerful
speaker in a large room, the intensity of the
sound can be small.
PHY 231

19. Intensity Faintest sound we can hear: I~1x10-12 W/m2 (1000 Hz)
Loudest sound we can
stand: I~1 W/m2
Factor of 1012? Loudness works logarithmic…
PHY 231

20. decibel level  =10log(I/I0) I0=10-12 W/m2
y=log10x inverse of x=10y (y=ln(x) x=ey)
log(ab) =log(a)+log(b)
log(a/b) =log(a)-log(b)
log(an) =nlog(a)
PHY 231

21. decibels =10log(I/I0) I0=10-12 W/m2 2-1=10 An increase of 10 dB:
intensity of the sound is
multiplied by a factor of 10.
2-1=10
10=10log(I2/I0)-10log(I1/I0)
10=10log(I2/I1)
1=log(I2/I1)
10=I2/I1
I2=10I1
Next quiz…
PHY 231

22. Frequency vs intensity
1000 Hz
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23. example A machine produces sound with a level of 80dB. How
many machines can you add before exceeding 100dB?
1 machine
80 dB=10log(I/I0)
8=log(I/I0)=log(I/1E-12)
108=I/1E-12
I1=10-4 W/m2
?? machines
100 dB=10log(I/I0)
10=log(I/I0)=log(I/1E-12)
1010=I/1E-12
I??=10-2 W/m2
I1/I??=10-4/10-2=1/100
The intensity must increase by a factor of 100;
one needs to add 99 machines.
PHY 231