# 13-3 harmonics

```13-3 harmonics
1. Understand standing wave and harmonics
2. Understand Interference and Beats
Standing Wave Patterns and harmonics
• All objects have a frequency or set of frequencies with which they
naturally vibrate when struck, plucked, strummed or somehow
disturbed. Each of the natural frequencies at which an object vibrates
is associated with a standing wave pattern.
• A standing wave pattern is created within a medium when the
reflected waves from one end of the medium interfere with incident
waves from the source, The result of the interference is that specific
points along the medium appear to be standing still while other points
vibrated back and forth. Such patterns are only created within the
medium at specific frequencies of vibration. These frequencies are
known as harmonic frequencies or merely harmonics. At any
frequency other than a harmonic frequency, the interference of
reflected and incident waves results in a disturbance of the medium
that is irregular and non-repeating.
The natural frequencies of an object are merely the harmonic
frequencies at which standing wave patterns are established within
the object. These standing wave patterns represent the lowest
energy vibrational modes of the object.
• Harmonics are integral multiples of the fundamental
frequency.
– Fundamental frequency is the lowest frequency of vibration
of a standing wave.
– Harmonic series is a series of frequencies that includes the
fundamental frequency and integral multiples of the
fundamental frequency.
• For example, the harmonic series can include: 64 Hz, 128 Hz,
256 Hz, 516 Hz …
• Fundamental frequency determines pitch
• Harmonic series depends on the length of the string:
fn  n
v
2L
Sound Reflection - echo or a
reverberation.
• A reverberation often occurs in a small room with height,
width, and length dimensions of approximately 17 meters or less.
The reflected sound wave has a very short delay, it seems to the
person that the sound is prolonged. You might observe
reverberations when talking in an empty room, when honking the
horn while driving through a highway tunnel or underpass, or
when singing in the shower.
• Echoes occur when a reflected sound is perceived as a second
sound rather than the prolonging of the first sound.
http://paws.kettering.edu/~drussell/Demos/RT60/RT60.html
Sound Interference and Beats
• When sound waves meet, interference occurs. The interference
causes the medium to take on a shape which results from the net
effect of the two individual waves upon the particles of the
medium.
• Constructive interference occurs if compression meets up with
compression and rarefaction meets up with rarefaction.
Constructive interferences produce a anti-node, results a louder
sound.
• Destructive interference occurs if compression of one wave meets
the rarefaction of another wave. Destructive interference produce a
node, results no sound at all.
• ..\..\sound_en.jar
• Destructive interference of sound waves is an important issue in the
design of concert halls and auditoriums. One means of reducing the
severity of destructive interference is by the design of walls, ceilings,
and baffles that serve to absorb sound rather than reflect it.
• The destructive interference of sound waves can also be used
advantageously in noise reduction systems. Ear phones have been
produced which can be used by factory and construction workers to
reduce the noise levels on their jobs. Such ear phones capture sound
from the environment and use computer technology to produce a
second sound wave which one-half cycle out of phase. The
combination of these two sound waves within the headset will result
in destructive interference and thus reduce a worker's exposure to
loud noise.
Musical Beats
• When sound waves with slightly different frequencies traveling in
the same direction, the effect of interference is perceived as a
variation in loudness, called beats.
• Note: the diagrams represents a sound wave by a sine wave.
Because the variations in pressure with time take on the
pattern of a sine wave. Sound is not a transverse wave,
sound is a longitudinal wave.
• The beat frequency refers to the number of beats per second.
For example, if two complete cycles of high and low volumes
are heard every second, the beat frequency is 2 Hz. The beat
frequency equals to the difference in frequencies of the two
interfering notes.
• For example, if two sound waves with frequencies of 256 Hz
and 254 Hz are played simultaneously, a beat frequency of 2
Hz will be detected.
• http://www.phys.unsw.edu.au/jw/beats.html#sounds
• http://www.acoustics.salford.ac.uk/feschools/waves/super3.ht
m#beats
Lab - Harmonic Frequencies
Overview: in order to produce a standing wave, certain conditions must be
satisfied. For example, to produce a standing wave in a given string with
length l, certain wavelength must be fulfilled.
Purpose (5 pt):
To determine the mathematical relationship between the wavelengths of
various harmonics and the length of the string in a standing wave.
Material (5 pt): Computer, physics classroom website
Procedure: go to http://www.physicsclassroom.com/shwave/harmonic.cfm
set length = 200 cm; amplitude = 20 cm
change the harmonic # and record the # of harmonics, stand wave pattern,
length of the string and wavelength
Data (20 pt): includes a table with column headings: harmonic number, standing
wave pattern, and frequency.
Conclusion (20 pt) includes a couple of sentences and an equation which
responds to the purpose of the lab; symbols used within the equation are
clearly defined. The Discussion of Results should explain how the collected
data are consistent with the equation written in the Conclusion. Specific rows
of the table should be referenced and discussed in an effort to show how the
equation fits the data.
# of
harmonics
1
2
3
4
5
6
stand wave
pattern
length of the
string (cm)
Wavelength
(m)
Frequency
(Hz)
Speed
(m/s)
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