CHAPTER 17 NOTES - LINEAR SUPERPOSITION
AND INTERFERENCE
When two or more sound waves are present at the same place at
the same time we experience linear superposition. The principle of
linear superposition states that the resultant effect of two or more
concurrent waves is equal to the sum of the disturbances created by
each of the individual waves.
In the diagram above, the two waves produce a resultant
disturbance that is found by adding the two individual
displacements from equilibrium.
When two waves add together to produce a disturbance that is
larger than that of either individual wave, we call this constructive
interference.When two waves add together to produce a
disturbance that is smaller than either individual wave, we call this
destructive interference.
Waves travelling in the same direction can also exhibit
constructive interference. When crest corresponds to crest and
trough to trough the two waves are said to be in phase and interfere
constructively.
If crests match up with troughs, the waves are said to be 180° out
of phase and destructive interference results.
When two sources of sound produce the same frequency and
vibrate in phase, constructive interference occurs if the observer's
position is an integral number of wavelengths farther from one
source than the other. If the difference in path lengths is a half
integer number of wavelengths, destructive interference occurs.
The red dots show locations where constructive interference
occurs. The white dots indicate where destructive interference
occurs.
Example
Loudspeakers A and B are set up on the X axis some distance
apart. Both speakers vibrate in phase at 68.6 hz and the speed of
sound is 343 m/s. Point C is 1.0 meter from speaker A and the line
connecting A with C makes a 60 degree angle with the X axis.
Find the minimum distance between the speakers that will produce
destructive interference at point C.
Diffraction
When a wave encounters an opening or an obstacle, the wave
spreads out in all directions after passing through the opening or by
the edge of the obstacle. This bending is called diffraction.
Bending occurs because each vibrating molecule in the opening
acts as a new source for a wave. Using the principle of linear
superposition, we can see that the waves from all of these
molecules add together to produce a resultant wave that spreads
out in all directions. Constructive and destructive interference
occur at specific directions of travel depending on the size of the
opening and the wavelength of the wave. The strongest
constructive interference occurs directly in front of the opening.
Other areas of constructive and destructive interference occur but
become increasingly weaker as the direction moves away from
directly in front of the opening.
The angle θ in the drawing indicates the direction of the first
minimum intensity due to destructive interference. It is measured
relative to the direction straight through the opening.
The equation used to find θ is:
sin θ = λ/D
where λ is the wavelength and D is the width of the opening. This
is called a single slit minimum since only one opening is involved.
If the opening in round(like a loudspeaker) instead of rectangular,
the equation becones:
sin θ = 1.22λ/D
If λ/D is small, the angle is small and the effect is called narrow
dispersion. If λ/D is relatively large, we get wide dispersion. Low
frequencies have long wavelengths and are heard over a large area
around a speaker. High frequencies have short wavelengths and
often are only heard directly in front of the speaker.
Example
The entrance to a large room consists of two doors that cover one
large opening. Each door is 0.700 m wide. If a sound with a
frequency of 607 hz passes through the doors when the speed of
sound is 343 m/s, find the diffraction angle with (a) one door open
and (b) both doors open.
Beat frequencies occur when two sounds interfere first
constructively then destructively producing alternating loud and
quiet sounds. The rate at which the beat frequency occurs is equal
to the frequency difference between the two sounds. For 440 Hz
and 438 Hz we would hear 2 beats per second. This is how many
stringed instruments are tuned. When the beat frequency is zero,
both the instrument and the tuner are playing the same frequency.
Standing wave patterns result from interference of waves that have
the same wavelength and are in a medium whose length is a
multiple of one half of that wavelength.
Places within the medium which show little or no movement are
called nodes. Places which show maximum movement are called
antinodes.
Harmonics are the frequencies associated with the wavelengths
that can produce standing waves in a medium. The first harmonic
is called the fundamental tone and is the lowest pitched sound that
can be produced by a particular set of conditions. Whole number
multiples of this frequency are called overtones and are labeled
second harmonic, third harmonic, etc.
When musical instruments are played, it is possible to distinguish
one instrument from another because of the harmonic content that
is different for each instrument. Today that distinction is being
reduced by the use of synthesizers to produce waveforms that very
closely mimic any type of instrument.
p506 Questions 3, 4, 5, 8, 15
P506 Problems 1,3,5,7,10,11,16,20,24,25