Principle of Linear Superposition and Wave Interference Phenomena

1
Principle of Linear SuperpositionandWave
Interference Phenomena

• Chapter 17

2
Linear Superposition
When two waves overlap, the resulting total wave
form is just the sum of the two individual waves.
3
Constructive Interference
When two waves always meet crest-to-crest or
trough-to-trough, so that they are exactly in
phase at one point.
4
Destructive Interference
When two waves always meet crest-to- trough,
so that they are exactly OUT of phase at one
point.
5
Interference and Path Lengths
Whether a wave will interfere constructively or
destructively depends on the difference in path
lengths. Constructive Interference (between two
sources emitting the same frequency of sound)
occurs when there is a difference in path lengths
of a whole number of wavelengths (0, 1, 2, 3
...). This means that the two waves will both be
at a crest, or both be at a trough. Destructive
Interference occurs when there is a difference in
path lengths of a half number of wavelengths
(1/2, 1 1/2, 2 1/2, 3 1/2 ...). This
means that the two waves will both be at a crest,
or both be at a trough.
6
Example
Two speakers emit identical frequencies. What
sort of interference will occur at the points
shown? A
B C
7
Diffraction
Diffraction describes the process where a wave
bends around a barrier.
8
Dispersion
The sound is loudest directly in front of the
door, and decreases as you move to one side. The
width of the loud part is called the dispersion,
and is related to the wavelength and the size
and shape of the opening. For a thin slit (like
an open door), we can derive where D is the
width of the slit
9
Dispersion
For a circular opening (like a speaker), we can
derive where D is the diameter. A speaker
with a diameter of 30 cm will have a dispersion
of 70 degrees for 1500 Hz 9
degrees for 8500 Hz To get a good dispersion
for the higher frequency, we need a smaller
speaker. For a 5 cm speaker we have 70
degrees for 8500 Hz
10
Beats
If two sources have slightly different
frequencies, we will get an alternating pattern
of constructive and destructive interference,
called beats. The beat frequency is the number
of times per second that the loudness rises and
falls.
11
How to tune an instrument
If you have one tuned instrument, you can use
the concept of beats to tune another. For a
guitar, you just play the same note on both
instruments, then adjust the tension in the
string (thereby adjusting the speed of the
wave, and hence the frequency) until the beats
vanish.
12
Example
A musician is tuning a guitar string to a tuning
fork. By striking the string and the fork
together, she can listen to the beat frequency of
the two sources. At one point, she hears a beat
frequency of 4 Hz. (a) Which has the higher
frequency, the guitar string or the tuning
fork? (b) As she tightens the string, the beat
frequency increases. What must she do to get the
string at the same frequency as the fork, tighten
further or loosen the string?
13
Group Problem Solving
At an open-air rock concert the music is played
through speakers that are on the stage, facing
straight forward (not toward the sides). As you
walk around, you notice that the sounds of the
female vocalists can be heard in front of the
stage but not off to the sides. The rhythmic
bass, however, can be heard both in front of the
stage and off to the sides. How can we explain
this? (Explain in detail)
14
Transverse Standing Waves
A standing wave is the result of an interference
effect. Vocabulary node a place that does
not vibrate. antinode a place where the
maximum vibration occurs. fundamental
frequency the smallest frequency that will
produce a standing wave (one loop, half
wavelength) harmonics higher frequencies that
produce standing waves (2nd harmonic 2 loops
3rd harmonic 3 loops) overtones the
harmonics above the fundamental frequency
15
Transverse Standing Waves
Why do we get interference? When the wave hits
the wall, it reflects back on itself. Successive
reflections add together with new waves to
produce the standing wave pattern.
16
Harmonic Frequencies (String)
The frequency of the harmonics depends on the
length, L, of the string the speed, v, of the
wave First harmonic reinforces every new
cycle of the wave Second harmonic reinforces
every other new cycle Third harmonic reinforces
every third new cycle
17
Example
A guitar string has a fundamental frequency of
100 Hz. What are the frequencies of the first
two overtones?
18
Example
An instrument maker selects a heavy guitar string
with a mass density of 5.28 x 10-3 kg/m, and a
tension of 226 N (when tuned). He would like to
use this string to play a low E (with a
fundamental frequency of 164.8 Hz) when plucked.
(a) What length of string does the craftsman
need to select? (b) To play an E one octave
higher (with a frequency of 2 x 164.8 Hz 329.6
Hz) on the same string, the strings needs to be
shorter. Where should the craftsman place the
fret so that this is possible?
19
Longitudinal Standing Waves (air)
A standing wave is the result of an interference
effect. Standing waves in a tube that is open at
both ends The standing wave will have an antinode
at both ends. Standing waves in a tube that
is closed at one end The standing wave will have
an antinode at the open end, and a node at the
closed end.
20
Example

• An instrument maker is designing a pipe organ.
  For the pipe that will play the low E note
  (fundamental frequency of 164.8 Hz), how long
  must the pipe be if
• the pipe is open at both ends?
• the pipe is closed at the bottom end?

21
Group Problem Solving
The standing wave frequencies of a resonant
vibrating system are determined by the boundary
conditions. Any place in a system where the
vibrating medium is prevented from moving must be
a node. An end where the medium is free to move
must be an antinode. Sketch each of the
following vibrating systems as simply as possible
and mark the nodes and antinodes for the
fundamental standing wave. (a) a metal bar
clamped at one end (b) a guitar string (c) a
flagpole (d) an organ pipe closed at one end and
open at the other (e) an organ pipe open at both
ends