SUMMARY
The goal of Chapter 21 has been to understand and use the idea of superposition.
General Principles
Principle of Superposition
The displacement of a medium when more than one wave is present is the sum of the
displacements due to each individual wave.
Important Concepts
Standing waves are due to the superposition of
two traveling waves moving in opposite directions.
Antinodes
Nodes
Node spacing is 12 l.
The amplitude at position x is
A (x) = 2a sin kx
where a is the amplitude of each wave.
The boundary
conditions determine
which standing-wave
frequencies and
wavelengths are
allowed. The allowed
standing waves are
modes of the system.
m
1
m
2
m
3
Interference
Antinodal lines, constructive
In general, the superposition of two or more waves interference. A 2a
into a single wave is called interference.
Maximum constructive interference occurs
where crests are aligned with crests and troughs
with troughs. These waves are in phase. The
maximum displacement is A = 2a.
Perfect destructive interference occurs where
crests are aligned with troughs. These waves are
out of phase. The amplitude is A = 0.
Interference depends on the phase difference f
between the two waves.
r
Constructive: f = 2p
+ f0 = m # 2p
l
Nodal lines, destructive
1 #
r
interference. A 0
+ f0 = m +
2p
Destructive: f = 2p
l
2
1
2
r is the path-length difference of the two waves, and f0 is any phase
difference between the sources. For identical sources (in phase, f0 = 0 ) :
Interference is constructive if the path-length difference
Interference is destructive if the path-length difference
Standing waves on a string
The amplitude at a point where the phase difference is
r = ml.
r=
1 m + 12 2 l.
f is A = ` 2a cos
Applications
Strings, electromagnetic waves, and sound waves in closedclosed tubes must have nodes at both ends:
2L
m
fm = m
4L
m
fm = m
two waves of slightly different frequency are superimposed.
D
v
= mf1
2L
where m = 1, 2, 3, p .
The frequencies and wavelengths are the same for a sound wave
in an open-open tube, which has antinodes at both ends.
A sound wave in an open-closed tube must have a node at the
closed end but an antinode at the open end. This leads to
lm =
f
`.
2
Beats (loud-soft-loud-soft modulations of intensity) occur when
Boundary conditions
lm =
1 2
v
= mf1
4L
t
0
Soft Loud Soft Loud Soft
The beat frequency between waves of frequencies f1 and f2 is
fbeat = f1 - f2
where m = 1, 3, 5, 7, p .
Terms and Notation
principle of superposition
standing wave
node
antinode
amplitude function, A(x)
boundary condition
fundamental frequency, f1
harmonic
mode
interference
in phase
constructive interference
out of phase
destructive interference
phase difference, f
path-length difference, x or
thin-film optical coating
antinodal line
nodal line
beats
modulation
beat frequency, fbeat
r