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Frequency: The rate at which crests pass a point in space.
Period is the number of seconds per cycle and frequency is the number of cycles per second. The
period and the frequency are reciprocals of each other:
There is a relationship between the wavelength, frequency, and speed of a wave.
Recall that the wave moves a distance equal to one wavelength in a time equal to one period.
Speed is distance traveled divided by time to travel the distance. The speed of a wave is the ratio
of its wavelength to its period:
Because the frequency is the reciprocal of the period, we can write this as
Interference of Waves:
Based on the Principle of Superposition — Suppose we have two waves traveling on the same
string. The net effect of both waves is the sum of the effects of each individual wave.
Standing Waves
A standing wave is produced when two identical waves travel along the string but in opposite
directions.
One wave carries energy at a given rate to the right on the string, while the second wave carries
energy at the same rate to the left on the string.
No net energy transport along the string — but there must energy on the string; it can’t just
disappear (conservation of energy!).
The string must be oscillating which stores energy
on the string — but there must be points on the
string that remain at rest — no energy is carried
past these points.
At nodes, the string does not move. Between nodes
— antinodes — the string oscillates in place.
Energy doesn’t move along the string; the oscillating string stores energy.
One way to get waves moving in opposite directions is to have them reflect off a termination
point.
Note that music is produced by standing waves:
Vibrating strings fixed at both ends or vibrating columns of air.
For a string fixed at both ends, the ends of the string must be nodes. This limits the wavelengths
and frequencies of the waves produced on the string — standing waves.
For the longest wavelength standing wave
for this string,
This follows because a single antinode is ½
wavelength.
The next standing wave configuration adds a node halfway between the nodes at the end of the
string with now two antinodes. This wavelength is now
In this case, the two antinodes form one wavelength.
The next standing wave pattern — add another node and get
This follows because three antinodes form 3/2 wavelengths
In general
We are interested in frequencies:
From here we can see why shorting the string — putting finger on the fingerboard of a musical
instrument — increases the frequency. Divide by a smaller number giving larger frequency.
To understand other aspects of musical instruments, we need to know about the speed of a wave
on a string.
Send a pulse down the string — larger tension means larger force pulling the string back to
equilibrium — string returns to equilibrium more rapidly and the pulse moves away more
quickly — larger speed.
If the string has a larger mass per unit length — larger inertia — tension produces a smaller
acceleration — string returns to equilibrium more slowly and the pulse leaves more slowly —
smaller speed.
Musical/Physics Terminology
Physics: Harmonics.
Each mode of vibration produces a harmonic. n = 1 is the first harmonic, n = 2, the second and so
on.
Music — Fundamental and Overtones — The fundamental frequency is the first harmonic. The
first overtone is the second harmonic. The second overtone is the third harmonic and so on.
Musical Instruments based on Vibrating Columns of Air — organ pipes.
Two types of organ pipes:
Open — open at both ends
Closed — open at one end and closed at the other.
The open organ pipe has the same frequency
structure as the string fixed at both ends.
We will do a closed organ pipe.
Note that the closed end is a displacement
node and the open end is (roughly) a
displacement antinode.
Longest possible wavelength is 1/4
wavelength in the pipe.
Next longest wavelength
Next:
Closing the end of an organ pipe increase the wavelength and eliminates even harmonics.
In general
The frequencies:
where v is the speed of sound.
We don’t have the luxury of changing the speed of sound — for air column instruments, we
change the length of the air column.
Quality of Sound
Two musical instruments sound different even when playing the same note because they
generally generate different amounts of different harmonics.
If we take sound from an oboe and clarinet and filter out all but the fundamental, they will
sound the same.
Doppler Effect — apparent change in frequency of a wave due to motions of listener and source.
Consider the following analogy: drop a pebble into a smooth pond. A ripple moves out from the
point where the pebble hits the water. Drop a pebble a second, and a series of uniformly spaced
ripples will be produced. The wavelength of the disturbance is the distance between the ripples,
and the period is the one-second time between the consecutive pebbles hitting the water. We will
use a bug in the water as a “listener” and imagine
that he “hears” the ripples.
Source and listener both at rest—the bug
intercepts the ripples at the same rate they are
emitted (that is, the same rate at which the
pebbles strike the pond) and “hears” the same
frequency has the source.
Listener moving, source at rest
If the bug moves toward a source, it intercepts the
ripples at a greater rate than the source emits
them and “hears” a higher frequency.
If the bug moves away from the source, it will
intercept the ripples at a lower rate than the
source emits and will “hear” a lower frequency.
Source moves and listener is at
rest.
In this case, the pebbles are
not dropped into the same
point on the surface of the
pond. Each pebble is dropped
at a point displaced from the
point at which the previous
pebble hit the pond. In the
picture, S1 is the location of the
source when the ripple R1 was
emitted. The same goes for the
other two source, ripple pairs.
Note that the three S’s are all
the same source; the subscripts
indicate the position of the
source when its associated
ripple was emitted.
We see that the ripples are
“scrunched up” ahead (to the
left) of the source and spread out behind the source.
If the bug is in front of the source, it will intercept the ripples at a greater rate because they are
closer together —it will “hear” a higher frequency.
If the bug is behind the source, it will intercept the ripples at a lower rate because they are
farther apart—it will “hear” lower frequency.
The amount of shift in frequency depends on the speeds of the source and listener.
Measuring the frequency shift
allows us to determine the
speed of the source.
What if the speed of the source
is the same as the speed of
propagation in the medium?
Source experiences a large
wave disturbance due to
constructive interference of the
ripples at its location. This is
why it was difficult to build
supersonic aircraft—the
aircraft would shake to pieces
when it reached the speed of
sound.
When an aircraft exceeds the
speed of sound, sonic boom!
You hear the sonic boom as the
shock wave passes. In the
picture, person P2 has heard
the sonic boom but P1 has not.
It is impossible to exceed the
speed of light in a vacuum, but
it is possible for and electron in
water to exceed the speed of
light in the water — Cerenkov
radiation.
Water — bow wave
Constructive Interference
occurs when the crest of one
wave arrives at a point in
space at the same time the
crest of a second arrives. Their
effects add for a larger
amplitude.
Destructive Interference occurs when the crest of one wave arrives at a point in space at the
same time the trough of a second wave arrives. Their effects subtract to make a smaller
amplitude.
Beats — interference between two waves that have slightly different frequencies — or slightly
different periods.
Assume at one time at your ear both sounds are max—sound loud After one short period the
short period sound is max but long period wave is not — sound is quieter. After yet another short
period, long period wave even smaller and sound even quieter. Eventually, one sound is max
while other is min, and they cancel out to give zero sound. And then the sound starts getting
louder again.
No beats if the two frequencies are the same.
Used for tuning musical instruments — piano tuner adjusts the tension in the string until beats
with a tuning fork disappear.
Pitch and Loudness
Pitch is roughly frequency
Loudness is roughly intensity — square of the amplitude of the sound wave.
Physiological differences because the human ear hears some frequencies better than others.
The human ear is amazing — it can hear sounds as quiet as one-trillionth of a W/m2.
As loud as 1 W/m2 without pain.
The first of these is called the threshold of hearing and the second the threshold of pain.
We developed a decibel scale for sound — the threshold of hearing is 0 dB
The threshold of pain — 120 dB
Decibel scale is similar to the Richter scale for earthquakes. Increasing the intensity level by 10
dB increases the intensity of the sound by a factor of 10.
Note: the dB is the smallest intensity level change that can be detected by the human hearing
apparatus.
Be very careful of loud sounds — they can harm your hearing.
Tiny hairs in your ears vibrate at specific frequencies — if one breaks, you lose your ability to
hear that frequency. A loud enough sound at that frequency can cause that hair to break.
Note — the brain will turn down the sound without protecting your ears — you shouldn’t listen
to music louder than your brain wants to hear — don’t destroy your ears.
Ear doesn’t hear high frequencies as well as low frequencies. If you increase the intensity of a
high-frequency sound, it’s pitch decreases.
By the same token, keep a sound at the same intensity but lower the frequency and it will sound
louder.
Chapter 22 - Electrostatics
Observations:
Glass rod rubbed with silk.
If put close to pepper flakes, rod will pick them up.
Same thing happens with little flakes of metal — but after a short period of time, the metal
flakes will fly off the rod.
Benjamin Franklin developed a hypothesis to explain these observations — two fluid model of
electricity.
In this model — all bodies contain both positive and negative fluids.
Most bodies contain equal amounts of positive and negative fluids and are electrically neutral.
When the glass rod is rubbed with silk, the cloth rubs some of one of these fluids off the rod
leaving it either positive or negative.
Franklin made the unfortunate choice that the glass rod rubbed with silk is left positive.
Modern Model — We know that bodies are made of atoms. Recall that atoms have a positive
nucleus with negative electrons in orbit.
Since the proton and electron have the same size charge, opposite signs, atoms with equal
numbers of protons and electrons are neutral — a body composed of atoms is neutral.
It is the electrons that move when the glass rod is
rubbed with silk. When rubbed, electrons are
transferred from the glass rod to the silk, leaving
the glass rod positive and the cloth, negative.
Franklin’s choice leads to electrons being
negative.
Positive Rod Attracting a Pepper Flake
Electric forces between charged particles —
unlike charges attract, like charges repel.
Like gravity, electric forces get weaker with
distance.
The top figure shows the atoms in the flake
before the rod is brought near.
When positive rod is brought near the pepper flake, electrons in atoms are distorted so that they
spend more time on the side of the atom nearest the rod and less time on the side of the atom
farthest from the rod. This is shown in the
bottom figure.
This makes the side of the atom nearest the rod
negative and the side of the atom farthest from
the rod, positive.
The positive and negative sides of adjacent
atoms cancel.
There are no atoms to cancel charges at the
edges of the flake nearest and farthest from the
rod — effectively, the near side of the flake is
negatively charged and the far side, positively
charged.
Recall that like charges repel and unlike charges
attract. Thus, the positive rod attracts the
negative charges on the near side of the flake
and repels the positive charges on the far side of the flake.
The force gets weaker with distance so the attraction of the near negative charge is greater than
the repulsion of the far positive charge, and the flake is attracted to the rod.
Note: this process of distorting the electrons in the atom is called polarization.
To understand why the metal flakes eventually fly off the rod, we must understand how metal
flakes differ from pepper flakes.
Recall from the periodic table — the last column of noble gases have especially stable electronic
structures — they don’t want to take on or give up electrons and don’t interact strongly.
First column — alkali metals — have one electron outside the stable structure. This electron is
loosely held, which means that these metals are highly reactive.
If we bring together a large number of these atoms to form a macroscopic body, the outer electron
become shared among all the atoms in the body — so called free electrons.
It is easy to move electrons around in a metal.
For an element such as sulfur that needs two electrons to form a stable structure, no free
electrons are produced when sulfur atoms are brought together — each atom holds tightly to the
electrons it has. Electrons don’t move freely through sulfur.
Metals are conductors — they can conduct electricity.
Sulfur, for instance, is an insulator — it can’t conduct electricity — more commonly: glass,
plastic, rubber, wood.
Note that good thermal conductors are good electrical conductors and poor thermal conductors
are poor electrical conductors.