Time As we discussed in relativity, time in physics is treated as a dimen- sion. 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:
1 Frequency = 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 mo- tion’ 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 equi- librium 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, asso- ciated with the motion which are related by: 1 f= 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: 2π ω =2πf or ω = T The spring constant is related to ω by: ω k = sm The units of ω must be rad/s. The position at any time t is given by: x =Acosωt 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 am- plitude) 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 ω2 cosωt When cosωt =1 the acceleration is at it’s maximum value: a=-A ω2 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 string. θ
θ W = mg
mg sinθ
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 = g rL 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 L s 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 me- chanical 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 vibra- tion. 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: 1 Frequency = period For example, a wave with a frequency of 10 Hz (10 cycles/second) 1 has a period of 10 sec. 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: λ v= T = λ f 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. Rarefaction V Compression Music When a musician speaks of pitch or different ’notes’. the are describ- ing different frequencies. The musical notes produced by an instru- ment 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· 262Hz (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 f = n( ) n 2L where n = 1, 2, 3 ... and v is the velocity of the wave. The different frequencies corresponding to different n are called harmonics.
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: v f = n( ) n 2L If only one end is open the frequencies are given by: v f = n( ) n 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. λ The discussion above is for a wave in deep water (greater than 2 ) 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 ex- perience 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.