Reflection
At an abrupt boundary between one medium and another, reflection occurs.
If the end of the string is a fixed point, the reflected wave is inverted.
If the speed of the wave decreases (a light string is tied to a heavy string) the reflected
wave is inverted.
If the speed of the wave increases (a heavy string is tied to a light string) the reflected
wave has the same orientation as the incident wave
Mathematically complicated but with good animations:
http://physics.usask.ca/~hirose/ep225/animation/reflection/anim-reflection.htm
Below is an illustration of the first bullet point.
When the medium changes, the speed of the wave will change. Since
v
f
what changes, the frequency of the wave, its wavelength or both? The frequency is a measure of
the up and down motion of the wave. The up and down motion in one media causes up and
down motion in the other media. Hence, the frequency is the same from one media into another.
Therefore, the wavelength must change when the wave’s speed changes when it passes into
another medium.
Refraction
When the wave travels from one medium into another, the direction of the wave will change. In
the picture below, the direction of the wave is given by the black arrow. Going from higher
speed to lower speed, the wave bends towards the normal to the boundary. The speed of the
wave is implied by the spacing between the wavefronts.
Interference
Interference is a consequence of the principle of superposition.
The waves need to be coherent – same frequency and maintain a constant phase relationship.
(For incoherent waves the phase relation varies randomly.)
Constructive interference occurs when the waves are in phase with each other. The
amplitude of the resulting wave is the sum of the amplitudes of the two waves, |A1 + A2|
Destructive interference occurs when the waves are 180o out of phase with each other.
The amplitude of the resulting wave is the difference of the amplitudes of the two waves,
|A1 – A2|
Otherwise the wave has amplitude between |A1 – A2| and (A1 + A2).
Diffraction
Diffraction is the spreading of waves around an obstacle. The obstacle must be similar in size to
the wavelength of the wave for the effect to be noticeable.
Many animations are on the web. Here is one
http://www.youtube.com/watch?v=uPQMI2q_vPQ
Standing Waves
Standing waves occur when a wave is reflected at a boundary and the reflected wave interferes
with the incident wave so that the wave appears not to propagate. A wave propagating in the +xdirection is described by
y( x, t )
A sin( t kx)
The inverted reflected wave is
y( x, t )
A sin( t kx)
(Why are there sines and not cosines?) The waves interfere and
y( x, t ) 2 A cos t sin kx
This looks like
The places that are stationary are called nodes. Midway between the nodes are antinodes.
Suppose a string is held at both ends.
The first four possible patterns are given. Higher orders are possible, but become less
important. (Bending the string takes energy. More bending requires more energy.)
For the top pattern = 2L and the frequency is
f
The second pattern
=L
v
v
2L
f1
v
f2
v
L
2
v
2L
2 f1
The third pattern 1.5 = L and
f3
v
1.5v
L
3
v
2L
3 f1
The possible frequencies are multiples of the lowest frequency f1 which is called the fundamental
frequency. These are the natural frequencies or resonant frequencies of the string. We will
find a similar situation when discussing standing sound waves in a pipe.
Chapter 12 Sound Waves
We study the properties and detection of a particular type of wave – sound waves.
A speaker generates sound. The density of the air changes as the wave propagates.
The range of frequencies that can be heard by humans is typically taken to be between 20
Hz and 20,000 Hz. Most people struggle to hear the highest frequencies and that ability lessens
with age.
Speed of Sound
Recall
v
Restoring Force
Inertia
In fluids
B
v
The speed of the wave in a fluid (especially air) depends on temperature. In solids
Y
v
SKIP Equations 12-6 and 12-7 on pressure and sound intensity.
Decibel Scale
The perception of hearing is roughly proportional to the logarithm of the intensity. The lowest
intensity of sound that can be heard by most people is
I0
1.0 10
12
W/m2
I0 is called the threshold of hearing. It is used as the reference level for measuring sound
intensity. The sound intensity level in decibels is defined as
(10 dB) log 10
I
I0
(Be sure to practice with the decibel scale. Logarithms can be tricky.) An intensity level of 0 dB
corresponds to the threshold of hearing.
For incoherent sound waves with intensities I1 and I2, the total intensity is
I
I1 I 2
If the sound waves are coherent, the waves can interfere and the intensity is between |I1 – I2| and
I1 + I2, depending on the phase relationship between the two waves.
Decibels can be used in a relative sense. The difference in two dB readings
2
1
(10 dB) log 10
I2
I0
(10 dB) log 10
I2
I1
(10 dB) log 10
I1
I0
is related to the ratio of the intensities.
Standing Sound Waves
Recall that a standing wave is the superposition of two traveling waves. The wave reflects at the
boundary of the wave.
Pipe open at Both Ends
The boundary conditions are the same at both ends. Since the end is open to the atmosphere, the
pressure at the ends can not deviate much from atmospheric pressure. The ends are pressure
nodes. Pressure nodes are displacement antinodes.
From the diagram, the wavelengths satisfy
n
2L
n
The frequencies
fn
v
n
n
v
2L
nf1
The index n is an integer and it can vary from 1, 2, etc.
Pipe Open at One End
The situation is different from the pipe opened at both ends. The closed end is a pressure
antinode. The air at the closed end is isolated from the atmosphere and the pressure can deviate
far from atmospheric. The air at the closed end is a displacement node since the rigid wall
prevents the air from moving.
From the diagram, the wavelengths satisfy
n
4L
n
The frequencies
fn
v
n
n
v
4L
nf1
This time n has odd values only (1, 3, 5, etc.)
Problem 27 Two tuning forks A and B, excite the next-to-lowest resonant frequencies in two air
columns of the same length, but A’s column is closed at one end and B’s column is open at both
ends. What is the ratio of A’s frequency to B’s frequency.
Since A excites the pipe open at one end, only the odd harmonics are possible
v
fn
n
n
v
4L
nf1
Where n = 1, 3, 5, etc. Next to lowest resonant frequency refers to the second frequency. Here
that mean n = 3 and
fA
n
v
4L
3
v
4L 1
For B, all the harmonics are possible since it is exciting a pipe open at both ends.
v
fn
n
n
v
2L
nf1
n = 1, 2, 3, etc. Next to lowest in this sequence corresponds to n = 2,
fB
n
v
2L
2
v
2L
Forming a ratio
fA
fB
v
4L
v
2
2L
3
3v 2 L
4 L 2v
3
4