Time
As we discussed in relativity, time in physics is treated as a dimension. Measuring time is fundamental to science and technology. A
device like an hourglass gives us a simple clock. From ancient times
people have used the constant motion of the earth as a ’clock’ to
measure time interval like days and years. Until the last hundred
years, time was defined by the motion of the earth around the sun.
To make a good clock, we need something that repeats in a regular
way. This is called periodic motion. Practical clocks use some
natural resonance. A natural resonance is a repetitive motion
due to energy conservation in an isolated system. To be useful for a
clock, we have to be able to maintain the motion (overcome friction
in some way) and the period of the motion must be constant.
A motion that repeat in time is called periodic. The period is
the time required for one complete cycle of the motion in seconds.
The frequency is the number of complete vibrations in one second.
The unit for frequency is second−1 which is defined for complete
oscillations (or cycles) as the Hertz (Hz). The frequency and period
are related by:
Frequency =
1
period
For example, a wave with a frequency of 10 Hz (10 cycles/second)
1
has a period of 10
sec.
Simple Harmonic Motion
The most basic form of harmonic motion is ’simple harmonic motion’ or the motion of a mass bouncing on a spring. A pendulum
under certain circumstances is another example of simple harmonic
motion. In fact any periodic motion can be ’decomposed’ into a
sum of simple harmonic motions. Since simple harmonic motion is
so basic, I will describe this type of motion in more detail than the
book.
If we place a mass on a spring, displace the mass from the equilibrium position and let it go, the mass moves back and forth in a
simple harmonic motion. We know the force is given by F =
m a = k x. The acceleration is not constant so we can not use our
normal equations of motion to describe the motion.
The equation which describes the motion:
ma=F=kx
related the (non-constant) acceleration and the displacement. This
is actually a differential equation. The solution of this differential
equation is beyond the scope of this course so we will only discuss
the results of the solution.
We have something which looks like the following.
v=0 x=max
Position
v=max x=0
speed
v=0 x=max
Since the motion repeats, there is a period,T, and frequency, f, associated with the motion which are related by:
f=
1
T
The period, T, is the time it takes to complete one complete cycle of
the motion. The unit for the period is seconds.
The frequency is the number of oscillations (complete cycles) the
motion makes in 1 second. The units for frequency is s−1 = 1 Hertz
(Hz).
We normally define the angular frequency, ω as:
ω = 2πf or ω =
2π
T
The spring constant is related to ω by:
ω=
s
k
m
The units of ω must be rad/s.
The position at any time t is given by:
x = A cosωt
where A is the Amplitude of the motion. The unit for A is meter.
It is the maximum displacement that occurs during each oscillation.
(Note that other types of ’motion’ may have different units for amplitude)
The velocity is given by:
v = -A ω sinωt
The maximum velocity occurs when the position is at the equilibrium
point so that (sinωt =1):
vmax = -A ω
The acceleration of the mass is given by:
a = -A ω 2 cosωt
When cosωt =1 the acceleration is at it’s maximum value:
a = -A ω 2
Pendulums
Another example of simple harmonic motion is the pendulum.
Instead of linear motion with a spring, in this case, the motion is
rotational. We can resolve the weight of the pendulum mass into a
component perpendicular to the pendulum and along the supporting
θ
1
0
0
1
θ
W = mg
mg sin θ
We could analyze this motion in terms of the torque. If the angle of
oscillation is small (less than 5o) the result is an equation which looks
just like the mass on the spring. In this case, the angular frequency
is:
ω=
r
g
L
where g is the acceleration of gravity (9.8 m/s2) and L is the length
of the pendulum.
The period, T, is then:
T = 2π
s
L
g
If the angle is larger than 5o, the restoring force can no longer
be treated as constant and the equations become more ’ugly’. The
oscillations are still periodic but are ’anharmonic’.
Of course, unless we have a frictionless bearing for the pendulum
arm, friction will eventually bring the pendulum to a stop. Even air
resistance would eventually stop the pendulum. To overcome this
and keep the pendulum swinging, a weight or spiral spring is used to
give the pendulum a little ’kick’ on each oscillation to compensate
for the friction.
Balance Clocks and Electronic Clocks
It is hard to carry a pendulum around in your watch. Most mechanical watches use a small ’balance wheel’ with good bearings to
reduce friction. The balance wheel has a coil spring attached to it.
It is rotational version of a mass of a spring. The wheel turns part
of circle, stops and rotates back the other direction.
Most mechanical clocks are limited by friction and changes due to
thermal expansions or contractions. The best mechanical clocks can
keep time to a fraction of a minute per year if properly maintained.
With the advent of electronics, quartz crystals have become common
for time keeping.
In an electronic (quartz) watch, a quartz tuning fork is used. The
small sealed metal cylinder contains the quartz tuning fork. The
tuning fork oscillates because of a small electrical current. The tuning
fork vibrates at a frequency of 32,768 (215) Hz. These vibrations are
counted and ’divided down’ by factors of 2 to give a very accurate
time. The hands to the watch are then moved by a small electrical
motor.
Watch crystal are formed to have a natural oscillation at 32,768
Hz (215Hz). These oscillations generate small electrical signals which
are used by the circuits in the watch to ’count’ the time.
Sound Waves
An oscillation (vibration or wiggle) is something that repeat in
time. An oscillation that repeats in space and time is a wave. Water
waves, sound and electromagnetic waves (radio, light, X-rays etc) are
all forms of waves. What is ’wiggling’ e.g., water, air pressure, the
electromagnetic fields, determines the type and characteristics of the
wave. However, there are certain characteristics which are common
to all waves.
λ
Velocity
Amplitude
Distance
The amplitude is the maximum displacement from the equilibrium
position. It is the ’distance’ from the mid-line to the top of the ’crest’.
Note that amplitude is measured in units of the displacement. For a
water wave it is distance (meters).
For sound it is pressure (Newton/meter2). We will focus on sound.
The wavelength (λ) is the distance to complete one complete vibration. It is the distance between crests (or troughs). It is always a
length (meters).
Frequency is the number of complete vibrations in one second. For
sound, we sometimes call frequency pitch. The unit of frequency is
the Hertz (cycles/sec). The period is the time in seconds for one
complete vibration. The frequency and period are related by:
Frequency =
1
period
For example, a wave with a frequency of 10 Hz (10 cycles/second)
1
sec.
has a period of 10
The velocity of the wave is the speed (meters/second) of a crest or
trough. For sound the speed in air at 1 atmosphere of pressure is
330 m/s.
The wave speed is the distance it travels in a given time:
v=
λ
T
=λf
For example if the wavelength is 1 meter and 10 complete cycles
pass a given point in 1 second, the speed is 10 meters/second.
There are different ways the medium can be disturbed during a
vibration. The two types we are concerned with are transverse and
longitudinal. When the motion is at right angles to the medium,
the wave is transverse. This is the wave form we normal think about
like a string vibrating, or a water wave.
A longitudinal wave is when the motion is along (parallel) to the
medium.
Sound is a longitudinal (compression/decompression) wave.
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Compression
Music
When a musician speaks of pitch or different ’notes’. the are describing different frequencies. The musical notes produced by an instrument are just different frequencies. Notes are not linear. ’Middle C’
on the piano keyboard has a frequency of 262 Hz.
Octave
An octave is a doubling of the frequency. Between ’Middle C’ and
the next ’C’ on the piano keyboard, the frequency is doubled. There
are 12 notes in this range including the black keys. You can express
the octaves mathematically as:
Frequency = 2n· 262 Hz
(n= ... -2, -1, 0, 1, 2 ...)
The 262 Hz is the frequency of middle ’C’ on the piano. The
notes (A, A’, B etc) are evenly divided fractions of powers of
two. The difference in frequency between notes is not just a constant
frequency. For example, the difference in frequency between E and
F in the ’middle C’ octave is 20 Hz while in the top octave on the
piano, the difference is 160 Hz.
What makes one instruments sound different from another? A
flute does not sound like a piano? This is called ’Timbre’.
Violin String
Something else
Standing Waves - Strings on a Violin
Many musical instruments produce sound by vibrating a string. The
frequency of vibration depends on the ’stiffness’ of the string (the
tension or the force holding the string), the thickness of the string
and especially, the length. On most instruments, you don’t change
the length of the string to tune the ’pitch’ but change the tension in
the string.
When a string is ’plucked’ or set in motion by a bow for a violin,
a wave is sent down the string. This wave will reflect off the far end
and set up a standing wave. A standing wave is an interference effect
that can occur when two waves overlap.
For a string of length L fixed at both ends, the amplitude is zero
at both ends. Each standing wave pattern is produced by a unique
frequency. This frequency corresponds to an integer number of half
wavelengths that will fit into the length d. The positions where
there is no motion on the string is called a node. The positions
with maximum motion are called anti-notes. The frequency that
produces the ’one loop’, ’two loop’, etc patterns is given by:
v
)
2L
where n = 1, 2, 3 ... and v is the velocity of the wave. The different
frequencies corresponding to different n are called harmonics.
fn = n(
Standing Longitudinal Waves - Pipe Organ
In a flute or a pipe organ, it is not a string that vibrates but the air
in a tube which vibrates. The frequency of vibration depends on one
or both ends being open.
Standing waves can be established with longitudinal waves. If a
sound wave of a certain frequency is established in a tube of length
L where both ends are open the frequencies are given by:
fn = n(
v
)
2L
If only one end is open the frequencies are given by:
fn = n(
v
)
4L
In both cases, n is again a integer.
Forced Vibration and Resonance
An electric guitar without the amplifier is very quiet. The sound is
produced by electrically amplifying a small electrical signal generated
by the vibrating strings.
What about an acoustic instrument. How can an acoustic guitar or
violin produce a loud sound?
If you strike a tuning fork, the sound is not that loud. If you place
the tuning fork in contact with a table, the sound will be louder
because the table will be forced to vibrate also. If the table is just
the right size such as an exact fractional part of the wavelength
generated by the tuning fork, the sound will be even louder. This is
called the natural frequency of the object. For example the box
connected to this tuning fork is just the right size to vibrate at the
same frequency of the tuning fork. The box and the tuning fork are
in resonance.
Tides
When the moon’s gravity pulls on the earth, it distorts the earth’s
shape to stretch out from the sphere we normally think of for the
earth. This distortion is about 1-2 m. Why are there two tides 12
hrs apart? The moon’s pull is weaker on the opposite side from
the moon, ’normal’ in the middle and strongest on the side nearest
the moon. The earth gets elongated into a spheroidal shape. The
’bumps’ on the deformed earth raise the sea level.
earth
moon
The sun also contributes to tides but the sun is much further away
so the effect is smaller. Sometimes the sun adds to the moon’s tidal
effect. Other times it cancels out some of the moon’s effect.
Some places have tides greater than 1-2 meters. In some places
like the bay of Fundy, the tides can be as high as 15 m. These large
tides are due to resonance effects of the channel. The flow of water in
a large body of water is controlled by gravity and can have a natural
resonance with the tides unlike a small amount of water where the
surface tension controls the flow. If the channel or estuary has a
natural resonance and the tides are in ’phase’ with this resonance
the tides can reach heights larger than then the normal 1-2 m. In
effect, the tides push the water up when the natural oscillations of the
water are raising and pull the water out when the natural oscillation
is falling. This is called a forced oscillation. An example of a forced
oscillation is pushing a child on a swing. Each time the child starts
down on a swing, you push a little bit and the child goes higher and
higher if you push ’in phase’.
Water Waves
When you watch a wave on a large body of water like a lake or
ocean, the surface of the water oscillates up and down. The water
may appear to move but it only oscillates up and down since it is a
transverse wave.
If you see a beach ball floating on the surface, it down not move with
the wave but ’bobs’ up and down as the wave crests pass.
Actually, the beach ball moves in a small circle as viewed from the
side.
As you move down from the surface, the circular motion decreases
with depth.
The discussion above is for a wave in deep water (greater than λ2 )
and without any boundary like the shore. When a wave encounters
the shore, the shallow water forces the water up to maintain the
circular motion. This leads to the wave ’breaking’.
Interference
When two waves cross, they add together at the point where they
cross. The waves then separate an go on their separate ways.
All type of waves (sound, water wave, electromagnetic etc) can experience interference.
a
b
c
The principle of superposition states:
• When two or more waves are present simultaneously at the same
place, the resultant wave is the sum of the individual waves.
When the waves are distributed in two or three dimensions, the
interference ’pattern’ can be very complex.
This can happen for periodic waves as well and not just the single
pulses shown above. Furthermore, if one wave is oscillating ’down’
and encounter another wave oscillating ’up’ the two waves can cancel
out. When the two waves are both oscillating in the save direction
and encounter one another, it is called constructive interference.
These waves are said to be in phase. If they are ’out of phase’ (
one up and one down) and cancel out, this is called destructive
interference.
Noise suppression systems use this principle to eliminate unwanted
sound. ’Road noise’ in high-end car stereo systems can eliminate the
unwanted noise from outside the car. If you have every ’phased’ your
speakers wrong (red and black wires reversed on one speaker) on your
stereo, you may have noticed for a (non-stereo) source the sound is
reduced. If you place the speakers face-to-face there would be almost
no sound because the two identical sound waves would completely
cancel out.