Waves and Sound
An Introduction to Waves and
Wave Properties
Wednesday, November 19, 2008
Mechanical Wave
A mechanical wave is a disturbance which
propagates through a medium with little or no net
displacement of the particles of the medium.
Water Waves
Wave “Pulse”
People Wave
Animation courtesy of Dr. Dan Russell, Kettering University
Parts of a Wave
λ: wavelength
3
equilibrium
2
-3
y(m)
crest
A: amplitude
4
trough
6
x(m)
Speed of a wave
The speed of a wave is the distance traveled
by a given point on the wave (such as a crest)
in a given interval of time.
v = d/t
d: distance (m)
t: time (s)
v
= λƒ
v : speed (m /s)
λ : wavelength (m)
ƒ : frequency (s–1, Hz)
Period of a wave
T =
1/ƒ
T : period (s)
-1
ƒ : frequency (s , Hz)
Problem: Sound travels at approximately 340 m/s, and
light travels at 3.0 x 108 m/s. How far away is a lightning
strike if the sound of the thunder arrives at a location 2.0
seconds after the lightning is seen?
Problem: The frequency of an oboe’s A is 440 Hz. What
is the period of this note? What is the wavelength?
Assume a speed of sound in air of 340 m/s.
Types of Waves
Refraction and Reflection
Thursday, November 20, 2008
Announcements
Pass Post Test toward the back of the
room.
Calendar adjustments
CPS
Wave Types
A transverse wave is a wave in which particles of
the medium move in a direction perpendicular to
the direction which the wave moves.
Example: Waves on a String
A longitudinal wave is a wave in which particles
of the medium move in a direction parallel to the
direction which the wave moves. These are also
called compression waves.
Example: sound
http://einstein.byu.edu/~masong/HTMstuff/WaveTrans.html
Wave types: transverse
Wave types: longitudinal
Longitudinal vs Transverse
Other Wave Types
Earthquakes: combination
Ocean waves: surface
Light: electromagnetic
Reflection of waves
• Occurs when a wave strikes a medium
boundary and “bounces back” into original
medium.
• Completely reflected waves have the
same energy and speed as original wave.
Reflection Types
Fixed-end reflection: The
wave reflects with inverted
phase.
Open-end reflection: The
wave reflects with the
same phase
Animation courtesy of Dr. Dan Russell, Kettering University
Refraction of waves
• Transmission of wave
from one medium to
another.
• Refracted waves may
change speed and
wavelength.
• Refraction is almost
always accompanied by
some reflection.
• Refracted waves do
not change frequency.
Animation courtesy of Dr. Dan Russell, Kettering University
Sound is a longitudinal wave
Sound travels through the air at approximately 340
m/s.
It travels through other media as well, often much
faster than that!
Sound waves are started by vibration of some other
material, which starts the air moving.
Animation courtesy of Dr. Dan Russell, Kettering University
Hearing Sounds
We hear a sound as “high” or “low” depending on its
frequency or wavelength. Sounds with short wavelengths
and high frequencies sound high-pitched to our ears, and
sounds with long wavelengths and low frequencies sound
low-pitched. The range of human hearing is from about 20
Hz to about 20,000 Hz.
The amplitude of a sound’s vibration is interpreted as its
loudness. We measure the loudness (also called sound
intensity) on the decibel scale, which is logarithmic.
© Tom Henderson, 1996-2004
Doppler Effect
The Doppler Effect is
the raising or lowering
of the perceived pitch of
a sound based on the
relative motion of
observer and source of
the sound. When a car
blowing its horn races
toward you, the sound
of its horn appears
higher in pitch, since the
wavelength has been
effectively shortened by
the motion of the car
relative to you. The
opposite happens when
the car races away.
Doppler Effect
Stationary source
Moving source
http://www.kettering.edu/~drussell/Demos/doppler/mach1.mpg
Animations courtesy of Dr. Dan Russell, Kettering University
http://www.lon-capa.org/~mmp/applist/doppler/d.htm
Supersonic source
Wave Demo Day
November 21, 2008
Pure Sounds
Sounds are longitudinal waves, but if we graph
them right, we can make them look like
transverse waves.
When we graph the air motion involved in a pure
sound tone versus position, we get what looks
like a sine or cosine function.
A tuning fork produces a relatively pure tone. So
does a human whistle.
Later in the period, we will sample various pure
sounds and see what they “look” like.
Graphing a Sound Wave
Complex Sounds
Because of the phenomena of “superposition”
and “interference” real world waveforms may not
appear to be pure sine or cosine functions.
That is because most real world sounds are
composed of multiple frequencies.
The human voice and most musical instruments
produce complex sounds.
Later in the period, we will sample complex
sounds.
The Oscilloscope
With the Oscilloscope we can view waveforms in the “time
domain”. Pure tones will resemble sine or cosine functions, and
complex tones will show other repeating patterns that are formed
from multiple sine and cosine functions added together.
The Fourier Transform
We will also view waveforms in the “frequency
domain”. A mathematical technique called the Fourier
Transform will separate a complex waveform into its
component frequencies.
Principle of Superposition
When two or more waves pass a particular
point in a medium simultaneously, the
resulting displacement at that point in the
medium is the sum of the displacements
due to each individual wave.
The waves interfere with each other.
Types of interference.
If the waves are “in phase”, that is crests
and troughs are aligned, the amplitude is
increased. This is called constructive
interference.
If the waves are “out of phase”, that is
crests and troughs are completely
misaligned, the amplitude is decreased
and can even be zero. This is called
destructive interference.
Interference
Let’s watch some exciting Physics Movies!
Constructive Interference
crests aligned with crest
waves are
“in phase”
Constructive Interference
Destructive Interference
crests aligned with troughs
waves are
“out of
phase”
Destructive Interference
Sample Problem: Draw the waveform from
its two components.
Sample Problem: Draw the waveform from
its two components.
Standing Wave
A standing wave is a wave which is
reflected back and forth between fixed
ends (of a string or pipe, for example).
Reflection may be fixed or open-ended.
Superposition of the wave upon itself
results in constructive interference and an
enhanced wave.
Let’s see some “wavy train”
demonstrations.
Fixed-end standing waves
(violin string)
1st harmonic
2nd harmonic
Animation available at:
3rd harmonic
http://id.mind.net/~zona/mstm/physics/waves/standingWaves/standingWaves1/StandingWaves1.html
Fixed-end standing waves
(violin string)
L
Fundamental
First harmonic
λ = 2L
First Overtone
Second harmonic
λ=L
Second Overtone
Third harmonic
λ = 2L/3
Open-end standing waves
(organ pipes)
L
Fundamental
First harmonic
λ = 2L
First Overtone
Second harmonic
λ=L
Second Overtone
Third harmonic
λ = 2L/3
Mixed standing waves
(some organ pipes)
L
First harmonic
λ = 4L
Second harmonic
λ = (4/3)L
Third harmonic
λ = (4/5)L
Sample Problem
How long do you need to make string that produces a high C (512 Hz)?
The speed of the waves on the string 1040 m/s.
A) Draw the situation.
B) Calculate the pipe length.
C) What is the wavelength and frequency of the 2nd harmonic?
Sample Problem
How long do you need to make an organ pipe whose fundamental
frequency is a middle C (256 Hz)? The pipe is closed on one end,
and the speed of sound in air is 340 m/s.
A) Draw the situation.
B) Calculate the pipe length.
C) What is the wavelength and frequency of the 2nd harmonic?
Resonance
Resonance occurs when a vibration from
one oscillator occurs at a natural
frequency for another oscillator.
The first oscillator will cause the second to
vibrate.
Demonstration.
Another exciting physics movie.
Beats
“Beats is the word physicists use to
describe the characteristic loud-soft
pattern that characterizes two nearly (but
not exactly) matched frequencies.
Musicians call this “being out of tune”.
Let’s hear (and see) a demo of this
phenomenon.
What word best describes this to
physicists?
Amplitude
Answer: beats
What word best describes this to
musicians?
Amplitude
Answer: bad intonation
(being out of tune)
Lunch Bunch organ pipe lab
Create an organ pipe that will resonate as loudly
as possible with your tuning fork.
a)
b)
c)
Predict the length of your organ pipe. Draw the first
harmonic for a standing wave in your pipe, and
determine what fraction of a wavelength it is. Use 340
m/s as the speed of sound in air to get the wavelength
from the frequency. Estimate the length of the pipe.
Construct your organ pipe. Does it resonate loudly?
Do “fine tuning” to adjust the length of the pipe to
produce the loudest possible sound.
At the end of the period we will sound all the
organ pipes at once to create a cord. The class
grade depends upon the loudness of the sound.
After you get a loud sound in your pipe, help
somebody else!
Diffraction
The bending of a wave around a barrier.
Diffraction of light combined with
interference of diffracted waves causes
“diffraction patterns”.
More exciting movies
Diffraction around obstacles in a ripple
tank.
Diffraction and interference in a ripple
tank.
Double-slit or multi-slit diffraction
n=2
n=1
θ
n=0
n=1
nλ = dsinθ
Diffraction of light
More exciting movies
Laser demonstrations
Double slit diffraction
Single slit diffraction
Determine the wavelength of the laser light
from the diffraction pattern.
Double slit diffraction
nλ = d sinθ
n: bright band number (n = 0 for central)
λ: wavelength (m)
d: space between slits (m)
θ: angle defined by central band, slit, and
band n
This also works for diffraction gratings.
Single slit diffraction
nλ = s sinθ
n: dark band number
λ: wavelength (m)
s: slit width (m)
θ: angle defined by central band, slit, and
dark band n
Sample Problem
Light of wavelength 360 nm is passed through a diffraction
grating that has 10,000 slits per cm. If the screen is 2.0 m from
the grating, how far from the central bright band is the first order
bright band?
Sample Problem
Light of wavelength 560 nm is passed through two slits. It is
found that, on a screen 1.0 m from the slits, a bright spot is
formed at x = 0, and another is formed at x = 0.03 m? What is
the spacing between the slits?
Sample Problem
Light is passed through a single slit of width 2.1 x 10-6 m. How
far from the central bright band do the first and second order
dark bands appear if the screen is 3.0 meters away from the
slit?
Periodic Motion
Motion that repeats itself over a fixed and
reproducible period of time is called
periodic motion.
The revolution of a planet about its sun is
an example of periodic motion. The highly
reproducible period (T) of a planet is also
called its year.
Mechanical devices on earth can be
designed to have periodic motion. These
devices are useful timers. They are called
oscillators.
Oscillator Demo
Let’s see demo of an oscillating spring using
DataStudio and a motion sensor.
Simple Harmonic Motion
You attach a weight to a spring, stretch the spring
past its equilibrium point and release it. The weight
bobs up and down with a reproducible period, T.
Plot position vs time to get a graph that resembles a
sine or cosine function. The graph is “sinusoidal”, so
the motion is referred to as simple harmonic motion.
Springs and pendulums undergo simple harmonic
motion and are referred to as simple harmonic
oscillators.
Analysis of graph
Equilibrium is where kinetic energy is maximum and
potential energy is zero.
3
equilibrium
2
-3
x(m)
4
6
t(s)
Analysis of graph
Maximum and minimum positions
3
2
-3
x(m)
4
6
t(s)
Maximum and minimum positions have maximum
potential energy and zero kinetic energy.
Oscillator Definitions
Amplitude
Maximum displacement from equilibrium.
Related to energy.
Period
Length of time required for one oscillation.
Frequency
How fast the oscillator is oscillating.
f = 1/T
Unit: Hz or s-1
Sample Problem
Determine the amplitude, period, and
frequency of an oscillating spring
using DataStudio and the motion
sensors. See how this varies with the
force constant of the spring and the
mass attached to the spring.
Thursday, November 29, 2007
Springs
Springs
Springs are a common type of simple
harmonic oscillator.
Our springs are “ideal springs”, which
means
They are massless.
They are both compressible and extensible.
They will follow Hooke’s Law.
F = -kx
Review of Hooke’s Law
Fs
m
mg
Fs = -kx
The force constant of a
spring can be determined
by attaching a weight and
seeing how far it
stretches.
Period of a spring
m
T = 2π
k
T: period (s)
m: mass (kg)
k: force constant (N/m)
Sample Problem
Calculate the period of a 300-g mass attached to an
ideal spring with a force constant of 25 N/m.
Sample Problem
Clicker
A 300-g mass attached to a spring undergoes
simple harmonic motion with a frequency of 25 Hz.
What is the force constant of the spring?
Clicker
Sample Problem
An 80-g mass attached to a spring hung vertically causes
it to stretch 30 cm from its unstretched position. If the
mass is set into oscillation on the end of the spring, what
will be the period?
Spring combinations
Parallel combination: springs work
together.
Series combination: springs work
independently
Question?
Does this combination act as parallel
or series?
Clicker
Sample Problem
You wish to double the force constant of
a spring. You
A.
B.
C.
D.
Double its length by connecting it to another
one just like it.
Cut it in half.
Add twice as much mass.
Take half of the mass off.
Conservation of Energy
Springs and pendulums obey conservation of
energy.
The equilibrium position has high kinetic
energy and low potential energy.
The positions of maximum displacement have
high potential energy and low kinetic energy.
Total energy of the oscillating system is
constant.
Sample problem.
• A spring of force constant k = 200 N/m is attached to a 700-g
mass oscillating between x = 1.2 and x = 2.4 meters. Where is
the mass moving fastest, and how fast is it moving at that
location?
Friday, November 30, 2007
Pendulums
Clicker
Sample problem.
• A spring of force constant k = 200 N/m is attached to a 700-g
mass oscillating between x = 1.2 and x = 2.4 meters. What is
the speed of the mass when it is at the 1.5 meter point?
Clicker
Sample problem.
• A 2.0-kg mass attached to a spring oscillates with an amplitude
of 12.0 cm and a frequency of 3.0 Hz. What is its total energy?
Pendulums
The pendulum can be thought of as a simple
harmonic oscillator.
The displacement needs to be small for it to
work properly.
Pendulum Forces
θ
T
mg sinθ
θ mg
Period of a pendulum
l
T = 2π
g
T: period (s)
l: length of string (m)
g: gravitational acceleration (m/s2)
Pendulum
Number of
oscillations
Elapsed
time (s)
Period
(s)
Length
(m)
Sample problem
• Predict the period of a pendulum consisting of a
500 gram mass attached to a 2.5-m long string.
Sample problem
• Suppose you notice that a 5-kg weight tied to a
string swings back and forth 5 times in 20
seconds. How long is the string?
Sample problem
• The period of a pendulum is observed to be T. Suppose
you want to make the period 2T. What do you do to the
pendulum?
Conservation of Energy
Pendulums also obey conservation of energy.
The equilibrium position has high kinetic
energy and low potential energy.
The positions of maximum displacement have
high potential energy and low kinetic energy.
Total energy of the oscillating system is
constant.