1. PHYSICS 231 Lecture 37: standing waves,harmonics & resonances
Remco Zegers
PHY 231

2. standing waves 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
PHY 231

3. How to create standing waves: a rope
demo
The oscillations in the rope are reflected from the fixed
end (amplitude is reversed) and create a standing wave.
PHY 231

4. we can produce different wave lengths
both ends fixed n=2L/n or L=nn/2
PHY 231

5. standing waves both ends fixed n=2L/n or L=nn/2 F: tension in rope
: mass per unit length
nth harmonics
f1: fundamental frequency
PHY 231

6. example: the guitar
nth harmonics: depends where and how the string is struck
note that several harmonics can be present and that
non-harmonics are washed out
tension can be varied by stretching
the wire
changes from string to string:
bass string is very heavy
length can be chosen by placing fingers
PHY 231

7. example A guitar string is struck. Assume that the first harmonic
is only excited. What happens to the frequency if:
The player put a finger at half the length of the string?
The player makes the tension 10% larger (by turning
the tuning screw)?
A string is struck in the same way, but its mass is
3 times higher?
PHY 231

8. Standing waves in air columns
Just like standing waves in transverse oscillations, one can
make standing waves in longitudinal oscillations as well.
PHY 231

9. 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
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10. Both ends open
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11. One end open, one end closed
even harmonics are missing!!!
PHY 231

12. example A simple flute is played by blowing air in on one side and the
other end is open. The length of the tube can be varied
manually (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 change
v=343m/s?
PHY 231

13. example A simple flute is played by blowing air in on one side and the
other end is closed. The length of the tube can be varied
manually (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 change
v=343m/s?
PHY 231

14. harmonics
Generally speaking, many harmonics with different intensities
can be present at the same time. + L
PHY 231

15. beats Superposition of 2 waves with slightly different frequency DEMO
The amplitude changes as a function of time, so the intensity
of sound changes as a function of time.
The beat frequency (number of intensity maxima/minima
per second): fbeat=|fa-fb|
PHY 231

16. example Someone is trying to tune a guitar. One of the strings is
supposed to have a frequency of 500 Hz. The person is
using a tuning fork which produces a sound of exactly
this frequency, but while sounding the fork and the playing
the guitar, hears a beat in the sound with a frequency of
3 Hz (3 beat per second). a) What is the real frequency of
the guitar string? b) By what fraction does the person need to
change the tension of the guitar string to tune it properly?
PHY 231

17. 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 natural
frequency: large oscillations occur: Resonance
demo
Resonances occur in many daily situations:
shock absorber in car
playing basketball
resonating lecture room!!
Famous example: Tacoma bridge
PHY 231

18. review: vote ! chapter 1 intro 2 1D motion 3 2D motion
4 Laws of motion
5 Energy
6 momentum/collisions
7 Rotational motion
8 Rotational equilibrium/Dynamics
9 Solids & Fluids
10 Thermal Physics
11 Energy in thermal processes
12 Laws of thermodynamics
13Vibrations & waves
14 Sound
PHY 231