Ch11_HallR

= = toc =**Lesson 0: Vibrations**=

**4/20/12: Lessons A-D**

 * Vibrational Motion:**
 * __Resting/equilibrium position__ is the position assumed by the bobblehead when it is not vibrating.
 * When an object is positioned at its equilibrium position, it is in a state of equilibrium, experiencing balanced forces.
 * All the individual forces - gravity, spring, etc. - are balanced or add up to an overall net force of 0 Newtons.
 * __ Forced vibration __ is the force which sets the otherwise resting object into motion.
 * The extent of displacement from the equilibrium position becomes less and less over time.
 * Because the forced vibration that initiated the motion is a single instance of a short-lived, momentary force, the vibrations ultimately cease.
 * __Damping__ is the tendency of a vibrating object to lose or to dissipate its energy over time.
 * Without a //sustained// forced vibration, the back and forth motion of a bobblehead eventually ceases as energy is dissipated to other objects; sustained input of energy would be required to keep the back and forth motion going.
 * After all, if the vibrating object naturally loses energy, then it must continuously be put back into the system through a forced vibration in order to sustain the vibration.
 * Vibrations repeat themselves over and over, and vibrating objects will move back to (and past) the equilibrium position every time they returns from its maximum displacement to the right or the left (or above or below).
 * So every instant in time that an object is at the equilibrium position, the momentary balance of forces will not stop the motion, it moves past the equilibrium position towards the opposite side of //its swing//.
 * As an object is displaced past its equilibrium position, then a force capable of slowing it down and stopping it exists.
 * This force that slows the object down as it moves away from its equilibrium position is known as a __ restoring force .__
 * The restoring force acts upon the vibrating object to move it back to its original equilibrium position.
 * In translational motion, an object is permanently displaced, and the initial force that is imparted to the object displaces it from its resting position and sets it into motion. Yet because there is no restoring force, the object continues the motion in its original direction.
 * When an object vibrates, it doesn't move permanently out of position; the restoring force acts to slow it down, change its direction and force it back to its original equilibrium position.
 * An object in translational motion is permanently displaced from its original position. But an object in vibrational motion wiggles about a fixed position - its original equilibrium position.


 * Properties of Periodic Motion:**
 * A vibrating object is //moving over the same path// over the course of time; is motion repeats itself over and over again, and if it were not for //damping//, the vibrations would endure forever**.**
 * A motion that is regular and repeating is referred to as a __ periodic motion __.
 * Most objects that vibrate do so in a regular and repeated fashion; their vibrations are periodic.
 * Suppose that a motion detector was placed below a vibrating mass on a spring in order to detect the changes in the mass's position over the course of time:
 * [[image:http://www.physicsclassroom.com/Class/waves/u10l0b1.gif width="467" height="119"]]
 * The graph has the shape of a sine wave.
 * It is also periodic, a //cycle of vibration//might be thought of as the movement of the mass from its resting position (A) to its maximum height (B), back down past its resting position (C) to its minimum position (D), and then back to its resting position (E).
 * A third obvious characteristic of the graph is that damping occurs with the mass-spring system; some energy is being dissipated over the course of time.
 * The extent to which the mass moves above (B, F, J, N, R and V) or below (D, H, L, P, T and X) the resting position (C, E, G, I, etc.) varies over the course of time.
 * __Period__: the time for the mass to complete a cycle.
 * __Amplitude__: the maximum displacement of the mass above (or below) the resting position.
 * The __ frequency __is defined as the number of complete cycles occurring per period of time, and is another quantity that can be used to quantitatively describe the motion of an object is periodic motion.
 * Since the standard metric unit of time is the second, frequency has units of cycles/second, which are equivalent to the units Hertz (abbreviated Hz).
 * Short period = high frequency, long period = low frequency.
 * [[image:http://www.physicsclassroom.com/Class/waves/u10l2b1.gif width="314" height="40" align="center"]]
 * Period and Frequency are //conceptual reciprocals://
 * period = the time for one full cycle to complete itself; i.e., seconds/cycle
 * frequency = the number of cycles that are completed per time; i.e., cycles/second
 * Over the course of time, the amplitude of a vibrating object tends to become less and less.
 * The amplitude of motion is a reflection of the quantity of energy possessed by the vibrating object: an object vibrating with a relatively large amplitude has a relatively large amount of energy.
 * Over time, some of this energy is lost due to damping, and as the energy is lost, the amplitude decreases.
 * If given enough time, the amplitude decreases to 0 as the object finally stops vibrating, when it has lost all its energy.

> alterations in the arc angle have little to no effect upon the period of the pendulum. > where g is a constant known as the gravitational field strength or the acceleration of gravity (9.8 N/kg).
 * Pendulum Motion:**
 * A simple pendulum consists of a relatively massive object hung by a string from a fixed support, and it typically hangs vertically in its equilibrium position.
 * The massive object is affectionately referred to as the //pendulum bob//.
 * When the bob is displaced from equilibrium and then released, it begins its back and forth vibration about its fixed equilibrium position, which is regular and repeating, an example of periodic motion.
 * There are two dominant forces acting upon a pendulum //bob// at all times during the course of its motion: the force of gravity that acts downward upon the bob, and the tension force acting upward and towards the pivot point of the pendulum.
 * [[image:http://www.physicsclassroom.com/Class/waves/u10l0c2.gif width="517" height="234" align="center"]]
 * In the case of a pendulum, it is the gravity force which gets resolved into vectors, since the tension force is already directed perpendicular to the motion.
 * [[image:u10l0c3.gif]]
 * The fact that the tension force (Ftens) is greater than the perpendicular component of gravity (Fgrav-perp) means there will be a net force which is perpendicular to the arc of the bob's motion.
 * It is this tangential component of gravity which acts as the restoring force.
 * As the pendulum bob moves to the right of the equilibrium position, this force component is directed opposite its motion back towards the equilibrium position.
 * [[image:http://www.physicsclassroom.com/Class/waves/u10l0c4.gif width="511" height="178" align="center"]]
 * When the bob is displaced to its maximum displacement to the right of the equilibrium position, it momentarily has a velocity of 0 m/s and is changing its direction.
 * The tension force (Ftens) and the perpendicular component of gravity (Fgrav-perp) balance each other, so there is no net force directed along the axis that is perpendicular to the motion (since the motion of the object is //momentarily paused//, there is no need for a centripetal force).
 * When the bob is at the equilibrium position (the string is completely vertical), there is no component of force along the tangent direction.
 * When moving through the equilibrium position, the restoring force is momentarily absent, since it has been //restored// to the equilibrium position.
 * Also, the tension force (Ftens) is greater than the perpendicular component of gravity (Fgrav-perp) when the bob moves through this equilibrium position (Since the bob is in motion along a circular arc, there must be a net centripetal force at this position).
 * [[image:http://www.physicsclassroom.com/Class/waves/u10l0c8.gif width="399" height="262" align="center"]]
 * [[image:http://www.physicsclassroom.com/Class/waves/u10l0c9.gif width="520" height="278"]]
 * The kinetic energy of the pendulum bob increases as the bob approaches the equilibrium position, and the kinetic energy decreases as the bob moves further away from the equilibrium position.
 * [[image:http://www.physicsclassroom.com/Class/waves/u10l0c10.gif width="511" height="269"]]
 * When the bob is at the equilibrium position (the lowest position), its height is zero and its potential energy is 0 J.
 * __Total Mechanical Energy__ - simply the sum of the two forms of energy – kinetic plus potential energy.
 * [[image:http://www.physicsclassroom.com/Class/waves/u10l0c12.gif width="532" height="124" align="center"]]
 * Although kinetic energy and potential energy change, the total amount of these two forms of energy is remaining constant
 * Yet the total mechanical energy remains constant, and is conserved.
 * In the case of pendulum, its period is the time for the pendulum to start at one //extreme//, travel to the opposite //extreme//, and then return to the original location.
 * The variables that affect period are the mass of the pendulum bob, the length of the string on which it hangs, and the //angular displacement//.
 * The angular displacement or //arc angle// is the angle that the string makes with the vertical when released from rest.
 * Alterations in mass have little effect upon the period of the pendulum, as the string is lengthened, the period of the pendulum is increased (and vice versa), and
 * ** T = 2•π•(L/g)0.5 **


 * Motion of a Mass on a Spring:**
 * __Hooke's Law:__ Fspring= -k•x, where Fspring is the force exerted upon the spring, x is the amount that the spring stretches relative to its relaxed position, and k is the proportionality constant, often referred to as the spring constant.
 * The spring constant is a positive constant whose value is dependent upon the spring which is being studied; a stiff spring would have a high spring constant.
 * The units on the spring constant are Newton/meter (N/m).
 * A negative sign indicates that the direction that the spring stretches is opposite the direction of the force which the spring exerts.
 * For instance, when the spring was stretched below its relaxed position, x is //downward//, and the spring responds to this stretching by exerting an //upward// force (the x and the F are in opposite directions).
 * The amount that the spring extends is proportional to the amount of force with which it pulls.
 * [[image:http://www.physicsclassroom.com/Class/waves/u10l0d3.gif width="385" height="100"]][[image:u10l0d4.gif]]
 * As the air track glider does //the back and forth//, the spring force ( Fspring ) acts as the restoring force: it acts leftward on the glider when it is positioned to the right of the equilibrium position; and it acts rightward on the glider when it is positioned to the left of the equilibrium position.
 * [[image:http://www.physicsclassroom.com/Class/waves/u10l0d5.gif width="425" height="415" align="center"]]
 * [[image:http://www.physicsclassroom.com/Class/waves/u10l0d6.gif width="434" height="117" align="center"]]
 * [[image:http://www.physicsclassroom.com/Class/waves/u10l0d7.gif width="436" height="160" align="center"]]
 * Position C was the equilibrium position and was the position of maximum speed.
 * [[image:Screen_shot_2012-04-23_at_10.17.43_AM.png]]
 * __ Elastic potential energy __ - position refers to the position of the mass on the spring relative to the equilibrium position.
 * For our vibrating air track glider, there is no change in height, so the gravitational potential energy does not change.
 * Every time the spring is compressed or stretched relative to its relaxed position, there is an increase in the elastic potential energy.
 * The amount of elastic potential energy depends on the amount of stretch or compression of the spring.
 * PEspring = ½ • k•x2, where k is the spring constant (in N/m) and x is the distance that the spring is stretched or compressed relative to the relaxed, unstretched position.
 * When the air track glider is at its equilibrium position (position C), it is moving it's fastest, but at this position, the value of x is 0 meter, meaning the amount of elastic potential energy ( PEspring ) is 0 Joules (this is where PE is the lowest).
 * When the glider is at position A, the spring is stretched the greatest distance and the elastic potential energy is a maximum.
 * A similar statement can be made for position E, where the spring is compressed the most and the elastic potential energy at this location is also a maximum.
 * Since the spring stretches as much as compresses, the elastic potential energy at position A (the //stretched// position) is the same as at position E (the //compressed// position); at these two positions - A and E - the velocity is 0 m/s and the kinetic energy is 0 J.
 * [[image:http://www.physicsclassroom.com/Class/waves/u10l0d8.gif width="386" height="108"]]
 * [[image:http://www.physicsclassroom.com/Class/waves/u10l0d9.gif width="428" height="254"]]
 * Again, although KE and PE change, the total amount of these two forms of mechanical energy remains constant, and is conserved,
 * T = 2•π•(m/k).5, where T is the period, m is the mass of the object attached to the spring, and k is the spring constant of the spring.
 * More massive objects will vibrate with a longer period; their greater inertia means that it takes more time to complete a cycle.
 * Springs with a greater spring constant (stiffer springs) have a smaller period; masses attached to these springs take less time to complete a cycle.
 * Their greater spring constant means they exert stronger restoring forces upon the attached mass, and this greater force reduces the length of time to complete one cycle of vibration.

=Lesson 1: The Nature of a Wave=

4/30/12: Lessons A-C

 * Lesson A: Waves and Wavelike Motion**
 * Waves carry energy from one location to another.
 * If the frequency of those waves can be changed, then we can also carry a complex signal that is capable of transmitting an idea or thought from one location to another.

> The medium is a collection of interacting //particles//; in other words, the medium is composed of parts that are capable of interacting with each other.
 * Lesson B: What is a Wave?**
 * A wave is a disturbance that travels through a medium from one location to another location.
 * When the slinky is stretched from end to end and is held at rest, it assumes a natural position known as the **equilibrium or rest position**.
 * The act of moving the first coil of the slinky in a given direction and then returning it to its equilibrium position creates a **disturbance** in the slinky.
 * If the first coil of the slinky is given a single back-and-forth vibration, then we call the observed motion of the disturbance through the slinky a //slinky pulse//.
 * A **pulse** is a single disturbance moving through a medium from one location to another location.
 * A **medium** is a substance or material that carries the wave.
 * The wave medium is not the wave and it doesn't make the wave; it merely carries or transports the wave from its source to other locations.
 * [[image:http://www.physicsclassroom.com/Class/waves/u10l1b2.gif width="251" height="155" align="right"]]
 * The interactions of one particle of the medium with the next adjacent particle allow the disturbance to travel through the medium.
 * As one particle moves, the spring connecting it to the next particle begins to stretch and apply a force to its adjacent neighbor.
 * The individual particles of the medium are only temporarily displaced from their rest position in a wave.
 * Waves are said to be an **energy transport phenomenon** : as a disturbance moves through a medium from one particle to its adjacent particle, energy is being transported from one end of the medium to the other.
 * The first coil receives a large amount of energy that it subsequently transfers to the second coil, so When the first coil returns to its original position, it possesses the same amount of energy as it had before it was displaced.
 * In this manner, energy is transported from one end of the slinky to the other, from its source to another location.
 * In a wave phenomenon, energy can move from one location to another, yet the particles of matter in the medium return to their fixed position, meaning a wave transports its energy without transporting matter.


 * Lesson C: Categories of Waves**
 * While all waves share some basic characteristic properties and behaviors, some waves can be distinguished from others based on some observable (and some non-observable) characteristics.
 * One way to categorize waves is on the basis of the direction of movement of the individual particles of the medium relative to the direction that the waves travel: transverse waves, longitudinal waves, and surface waves.
 * A **transverse wave** is a wave in which particles of the medium move in a direction __perpendicular__ to the direction that the wave moves.
 * [[image:http://www.physicsclassroom.com/Class/waves/u10l1c1.gif width="297" height="132" align="center"]]
 * A **longitudinal wave** is a wave in which particles of the medium move in a direction __parallel__ to the direction that the wave moves.
 * [[image:http://www.physicsclassroom.com/Class/waves/u10l1c2.gif width="261" height="102" align="center"]]
 * A sound wave traveling through air is a classic example of a longitudinal wave.
 * This back and forth motion of particles in the direction of energy transport creates regions within the medium where the particles are pressed together and other regions where the particles are spread apart.
 * Longitudinal waves can always be quickly identified by the presence of such regions.
 * Waves traveling through a solid medium can be either transverse waves or longitudinal waves, but waves traveling through the bulk of a fluid (such as a liquid or a gas) are always longitudinal waves.
 * Transverse waves require a relatively rigid medium in order to transmit their energy; as one particle begins to move it must be able to exert a pull on its nearest neighbor.
 * A **surface wave** is a wave in which particles of the medium undergo a circular motion.
 * Surface waves are neither longitudinal nor transverse.
 * In longitudinal and transverse waves, all the particles in the entire bulk of the medium move in a parallel and a perpendicular direction (respectively) relative to the direction of energy transport.
 * In a surface wave, it is only the particles at the surface of the medium that undergo the circular motion, so the motion of particles tends to decrease as one proceeds further from the surface.
 * [[image:http://www.physicsclassroom.com/Class/waves/u10l1c3.gif width="420" height="96" align="center"]]
 * Any wave moving through a medium has a source, meaning that somewhere along the medium, there was an initial displacement of one of the particles.
 * At the location where the wave is introduced into the medium, the particles that are displaced from their equilibrium position always moves in the same direction as the source of the vibration.
 * [[image:http://www.physicsclassroom.com/Class/waves/u10l1c5.gif width="406" height="160" align="center"]]

=Lesson 3: Behavior of Waves=

5/4/12: Lessons A-D

 * Lesson A: Boundary Behavior**
 * As a wave travels through a medium, it will often reach the end of the medium and encounter an obstacle or perhaps another medium through which it could travel.
 * The behavior of a wave (or pulse) upon reaching the end of a medium is referred to as **boundary behavior**.
 * When one medium ends, another medium begins; the interface of the two media is referred to as the **boundary** and the behavior of a wave at that boundary is described as its boundary behavior.
 * __Fixed End Reflection__:
 * Consider an elastic rope stretched from end to end, with one end attach to a pole.
 * The last //particle// of the rope will be unable to move when a disturbance reaches it, and is referred to as a **fixed end**.
 * If a pulse is introduced at the left end of the rope, it will travel through the rope towards the right end of the medium.
 * This pulse is called the **incident pulse** since it is incident towards (i.e., approaching) the boundary with the pole.
 * When the incident pulse reaches the boundary, two things occur:
 * A portion of the energy carried by the pulse is reflected and returns towards the left end of the rope. The disturbance that returns to the left after bouncing off the pole is known as the **reflected pulse**.
 * A portion of the energy carried by the pulse is **transmitted** to the pole, causing the pole to vibrate.
 * The reflected pulse is **inverted**.
 * Other notable characteristics of the reflected pulse include:
 * The speed of the reflected pulse is the same as the speed of the incident pulse.
 * The wavelength of the reflected pulse is the same as the wavelength of the incident pulse.
 * The amplitude of the reflected pulse is less than the amplitude of the incident pulse.
 * The speed of the incident and reflected pulse are identical since the two pulses are traveling in the same medium (Since the speed of a wave (or pulse) is dependent upon the medium through which it travels, two pulses in the same medium will have the same speed).
 * Since the wavelength of a wave depends upon the frequency and the speed, two waves having the same frequency and the same speed must also have the same wavelength.
 * The amplitude of the reflected pulse is less than the amplitude of the incident pulse since some of the energy of the pulse was transmitted into the pole at the boundary (The reflected pulse is carrying less energy away from the boundary compared to the energy that the incident pulse carried towards the boundary).
 * __Free End Reflection:__
 * Now, consider what would happen if the end of the rope were free to move: instead of being securely attached to a lab pole, suppose it is attached to a ring that is loosely fit around the pole.
 * Because the right end of the rope is no longer secured to the pole, the last //particle// of the rope will be able to move when a disturbance reaches it, and is referred to as a **free end**.
 * This time, when the incident pulse reaches the end of the medium, the last particle of the rope can no longer interact with the first particle of the pole.
 * Since the rope and pole are no longer attached and interconnected, they will slide past each other, so when a crest reaches the end of the rope, the last particle of the rope receives the same upward displacement; only now there is no adjoining particle to pull downward upon the last particle of the rope to cause it to be inverted., so the reflected pulse is not inverted.
 * __Transmission of a Pulse Across a Boundary from Less to More Dense:__
 * [[image:http://www.physicsclassroom.com/Class/waves/u10l3a5.gif width="356" height="51" align="center"]]
 * Consider an incident pulse traveling in the less dense medium (the thin rope) towards the boundary with a more dense medium (the thick rope).
 * Upon reaching the boundary, the usual two behaviors will occur.
 * A portion of the energy carried by the incident pulse is reflected and returns towards the left end of the thin rope. The disturbance that returns to the left after bouncing off the boundary is known as the **reflected pulse**.
 * A portion of the energy carried by the incident pulse is transmitted into the thick rope. The disturbance that continues moving to the right is known as the **transmitted pulse**.
 * The reflected pulse will be found to be inverted in situations such as this.
 * [[image:http://www.physicsclassroom.com/Class/waves/u10l3a6.gif width="366" height="257" align="center"]]
 * The transmitted pulse (in the more dense medium) is traveling slower than the reflected pulse (in the less dense medium).
 * The transmitted pulse (in the more dense medium) has a smaller wavelength than the reflected pulse (in the less dense medium).
 * The speed and the wavelength of the reflected pulse are the same as the speed and the wavelength of the incident pulse.
 * __ Transmission of a Pulse Across a Boundary from More to Less Dense: __
 * Consider an incident pulse traveling in the more dense medium (thick rope) towards the boundary with a less dense medium (thin rope).
 * Once again there will be partial reflection and partial transmission at the boundary, but the reflected pulse in this situation will not be inverted.
 * Similarly, the transmitted pulse is not inverted (as is always the case).
 * Since the incident pulse is in a heavier medium, when it reaches the boundary, the first particle of the less dense medium does not have sufficient mass to overpower the last particle of the more dense medium
 * The transmitted pulse (in the less dense medium) is traveling faster than the reflected pulse (in the more dense medium).
 * The transmitted pulse (in the less dense medium) has a larger wavelength than the reflected pulse (in the more dense medium).
 * The speed and the wavelength of the reflected pulse are the same as the speed and the wavelength of the incident pulse.
 * The boundary behavior of waves in ropes can be summarized by the following principles:
 * The wave speed is always greatest in the least dense rope.
 * The wavelength is always greatest in the least dense rope.
 * The frequency of a wave is not altered by crossing a boundary.
 * The reflected pulse becomes inverted when a wave in a less dense rope is heading towards a boundary with a more dense rope.
 * The amplitude of the incident pulse is always greater than the amplitude of the reflected pulse.
 * Reflection, Refraction, and Diffraction:**
 * [[image:u10l3b1.gif]]
 * If a linear object attached to an oscillator bobs back and forth within the water, it becomes a source of //straight////waves//, which have alternating crests and troughs.
 * The blue arrow is called a **ray** and is drawn perpendicular to the wavefronts.
 * Upon reaching the barrier placed within the water, these waves bounce off the water and head in a different direction
 * Regardless of the angle at which the wavefronts approach the barrier, one general **law of reflection** holds true: the waves will always reflect in such a way that the angle at which they approach the barrier equals the angle at which they reflect off the barrier.
 * Upon reflection off a parabolic barrier, water waves will change direction and head towards the focal poinbt, where all the energy being carried by the water waves converges.
 * After passing through the focal point, the waves spread out through the water.
 * [[image:http://www.physicsclassroom.com/Class/waves/u10l3b6.gif width="265" height="142" align="center"]]
 * **Refraction** of waves involves a change in the direction of waves as they pass from one medium to another (or the bending of the path of the waves)
 * Accompanied by a change in speed and wavelength of the waves, due to change in medium.
 * [[image:u10l3b7.gif]]
 * **Diffraction** involves a change in direction of waves as they pass through an opening or around a barrier in their path.
 * The amount of diffraction (the sharpness of the bending) increases with increasing wavelength and decreases with decreasing wavelength (when the wavelength of the waves is smaller than the obstacle, no noticeable diffraction occurs).
 * [[image:u10l3b8.gif]]
 * Diffraction of water waves is observed in a harbor as waves bend around small boats and are found to disturb the water behind them.
 * The same waves however are unable to diffract around larger boats since their wavelength is smaller than the boat.
 * Diffraction of sound waves is commonly observed; we notice sound diffracting around corners, allowing us to hear others who are speaking to us from adjacent rooms.
 * Diffraction is observed of light waves but only when the waves encounter obstacles with extremely small wavelengths (such as particles suspended in our atmosphere).

__Interference of Waves:__
 * **Wave interference** is the phenomenon that occurs when two waves meet while traveling along the same medium.
 * The interference of waves causes the medium to take on a shape that results from the net effect of the two individual waves upon the particles of the medium.
 * Consider two pulses of the same amplitude traveling in different directions along the same medium, each displaced upward 1 unit at its crest and has the shape of a sine wave.
 * As the sine pulses move towards each other, there will eventually be a moment in time when they are completely overlapped, where the resulting shape of the medium would be an upward displaced sine pulse with an amplitude of 2 units.
 * [[image:http://www.physicsclassroom.com/Class/waves/u10l3c1.gif width="283" height="71" align="center"]]
 * **Constructive interference** is a type of interference that occurs at any location along the medium where the two interfering waves have a displacement in the same direction.
 * Consequently, the medium has an upward displacement that is greater than the displacement of the two interfering pulses.
 * [[image:http://www.physicsclassroom.com/Class/waves/u10l3c2.gif width="286" height="76" align="center"]]
 * **Destructive interference** is a type of interference that occurs at any location along the medium where the two interfering waves have a displacement in the opposite direction: for instance, when a sine pulse with a maximum displacement of +1 unit meets a sine pulse with a maximum displacement of -1 unit.
 * [[image:http://www.physicsclassroom.com/Class/waves/u10l3c3.gif width="277" height="86" align="center"]]
 * The interfering pulses have the same maximum displacement but in opposite directions, so the two pulses completely destroy each other when they are completely overlapped.
 * At the instant of complete overlap, there is no resulting displacement of the particles of the medium, but this "destruction" is not a permanent condition
 * When it is said that the two pulses destroy each other, what is meant is that when overlapped, the effect of one of the pulses on the displacement of a given particle of the medium is destroyed or canceled by the effect of the other pulse.
 * Once the two pulses pass through each other, there is still an upward displaced pulse and a downward displaced pulse heading in the same direction that they were heading before the interference, so destructive interference leads to only a momentary condition in which the medium's displacement is less than the displacement of the largest-amplitude wave.
 * The two interfering waves do not need to have equal amplitudes in opposite directions for destructive interference to occur: For example, a pulse with a maximum displacement of +1 unit could meet a pulse with a maximum displacement of -2 units, where the esulting displacement of the medium during complete overlap is -1 unit.
 * This is still destructive interference since the two interfering pulses have opposite displacements, but in this case, the destructive nature of the interference does not lead to complete cancellation.
 * Interestingly, the meeting of two waves along a medium does not alter the individual waves or even deviate them from their path.
 * The task of determining the shape of the resultant demands that the principle of superposition is applied.
 * The **principle of superposition** is sometimes stated as follows:
 * When two waves interfere, the resulting displacement of the medium at any location is the algebraic sum of the displacements of the individual waves at that same location.
 * [[image:http://www.physicsclassroom.com/Class/waves/u10l3c8.gif width="213" height="194" align="center"]]

>
 * The Doppler Effect:**
 * If disturbances originate at a point, then they would travel outward from that point in all directions.
 * These circles would reach the edges of the water puddle at the same frequency, and have the same speed due being in to the same medium.
 * Suppose that a bug is moving to the right across the puddle of water and producing disturbances at the same frequency of 2 disturbances per second.
 * [[image:http://www.physicsclassroom.com/Class/waves/u10l3d2.gif width="158" height="158" align="right"]]
 * Since the bug is moving towards the right, each consecutive disturbance originates from a position that is closer to observer B and farther from observer A.
 * Subsequently, each consecutive disturbance has a shorter distance to travel before reaching observer B and thus takes less time to reach observer B.
 * Thus, observer B observes that the frequency of arrival of the disturbances is higher than the frequency at which disturbances are produced.
 * On the other hand, each consecutive disturbance has a further distance to travel before reaching observer A, and for this reason, observer A observes a frequency of arrival that is less than the frequency at which the disturbances are produced.
 * The net effect of the motion of the bug (the source of waves) is that the observer towards whom the bug is moving observes a frequency that is higher than 2 disturbances/second; and the observer away from whom the bug is moving observes a frequency that is less than 2 disturbances/second.
 * This effect is known as the Doppler effect.
 * The **Doppler effect** can be described as the effect produced by a moving source of waves in which there is an apparent upward shift in frequency for observers towards whom the source is approaching and an apparent downward shift in frequency for observers from whom the source is receding.
 * It is important to note that the effect does not result because of an __actual__ change in the frequency of the source, it just appears that way.
 * The Doppler effect can be observed for any type of wave - water wave, sound wave, light wave, etc.
 * As a police car approaches with its siren blasting, the pitch of the siren sound (a measure of the siren's frequency) is higher; and then suddenly after the car passes by, the pitch of the siren sound is lower.
 * [[image:http://www.physicsclassroom.com/Class/waves/u10l3d3.gif width="327" height="178" align="center"]]

=Lesson 4: Standing Waves=

5/7/12: Lessons A-E

 * Traveling Waves vs. Standing Waves:**
 * A mechanical wave is a disturbance that is created by a vibrating object and subsequently travels through a medium from one location to another, transporting energy as it moves.
 * The mechanism by which a mechanical wave propagates itself through a medium involves particle interaction; one particle applies a push or pull on its adjacent neighbor, causing a displacement of that neighbor from the equilibrium or rest position.
 * As a wave is observed traveling through a medium, a crest is seen moving along from particle to particle.
 * This sine wave pattern continues to move in uninterrupted fashion until it encounters another wave along the medium or until it encounters a boundary with another medium, and is referred to as a **traveling wave**.
 * Traveling waves are observed when a wave is not confined to a given space along the medium, such as an ocean wave.
 * It is possible to have a wave confined to a given space in a medium and still produce a regular wave pattern that is readily discernible amidst the motion of the medium, such as an elastic rope is held end-to-end and vibrated __at just the right frequency__, where a wave pattern is produced that assumes the shape of a sine wave and is seen to change over time.
 * When the proper frequency is used, the interference of the incident wave and the reflected wave occur in such a manner that there are specific points along the medium that appear to be standing still.
 * Because the observed wave pattern is characterized by points that appear to be standing still, the pattern is often called a **standing wave pattern**.
 * There are other points along the medium whose displacement changes over time, but in a regular manner.
 * These points vibrate back and forth from a positive displacement to a negative displacement; the vibrations occur at regular time intervals such that the motion of the medium is regular and repeating; a pattern is readily observable.
 * Nodes: Points that never move.

> > A standing wave pattern is not actually a wave; rather it is the pattern resulting from the presence of two waves (sometimes more) of the same frequency with different directions of travel within the same medium.
 * Formation of Standing Waves:**
 * A **standing wave pattern** is a vibrational pattern created within a medium when the vibrational frequency of the source causes reflected waves from one end of the medium to interfere with incident waves from the source.
 * This interference occurs in such a manner that specific points along the medium appear to be standing still.
 * Such patterns are only created within the medium at specific frequencies of vibration.
 * These frequencies are known as harmonic frequencies, or merely **harmonics**.
 * At any frequency other than a harmonic frequency, the interference of reflected and incident waves leads to a resulting disturbance of the medium that is irregular and non-repeating.
 * Upon reaching a fixed end, a single pulse will reflect and undergo inversion, so the upward displaced pulse will become a downward displaced pulse.
 * Now suppose that a second upward displaced pulse is introduced at the precise moment that the first crest undergoes its fixed end reflection.
 * If this is done with perfect timing, a rightward moving, upward displaced pulse will meet up with a leftward moving, downward displaced pulse in the exact middle of the snakey. As the two pulses pass through each other, they will undergo destructive interference.
 * Thus, a point of no displacement in the exact middle of the snakey will be produced.
 * [[image:http://www.physicsclassroom.com/Class/waves/di.gif width="70" height="62" align="center" caption="a GIF animation"]]
 * Since in a wave, the introduction of a crest is followed by the introduction of a trough, if two waves are in the same medium at the precise time, every crest and trough will destructively interfere in such a way that the middle of the medium is a point of no displacement.
 * The waves are interfering in such a manner that there are points of no displacement produced at the same positions along the medium, called **nodes**, that are labeled with an **N**.
 * There are also points along the medium that vibrate back and forth between points of large positive displacement and points of large negative displacement, known as **antinodes** and labeled with an **AN**.
 * [[image:http://www.physicsclassroom.com/Class/waves/swf.gif width="205" height="147" align="center"]]
 * The main idea behind the timing is to introduce a crest at the instant that another crest is either at the halfway point across the medium or at the end of the medium.
 * Regardless of the number of crests and troughs that are in between, if a crest is introduced at the instant another crest is undergoing its fixed end reflection, a node (point of no displacement) will be formed in the middle of the medium
 * The number of other nodes that will be present along the medium is dependent upon the number of crests that were present in between the two //timed////crests//: If a crest is introduced at the instant another crest is at the halfway point across the medium, then an antinode (point of maximum displacement) will be formed in the middle of the medium by means of constructive interference.
 * In such an instance, there might also be nodes and antinodes located elsewhere along the medium.


 * Nodes and Anti-nodes:**
 * One characteristic of every standing wave pattern is that there are points along the medium that appear to be standing still, called **nodes**.
 * There are other points along the medium that undergo vibrations between a large positive and large negative displacement.
 * These are the points that undergo the maximum displacement during each vibrational cycle of the standing wave.
 * These points are the opposite of nodes, and so they are called **antinodes**.
 * A standing wave pattern always consists of an alternating pattern of nodes and antinodes.
 * [[image:http://www.physicsclassroom.com/Class/waves/h4.gif width="272" height="128" align="center"]]
 * The positioning of the nodes and antinodes in a standing wave pattern can be explained by focusing on the interference of the two waves.
 * The nodes are produced at locations where destructive interference occurs, such as where a crest of one wave meets a trough of a second wave; or a //half-crest// of one wave meets a //half-trough// of a second wave; or a //quarter-crest// of one wave meets a //quarter-trough//  of a second wave; etc.
 * Antinodes, on the other hand, are produced at locations where constructive interference occurs. For instance, if a crest of one wave meets a crest of a second wave, a point of large positive displacement results.
 * Similarly, if a trough of one wave meets a trough of a second wave, a point of large negative displacement results.
 * Antinodes are always vibrating back and forth between these points of large positive and large negative displacement; this is because during a complete cycle of vibration, a crest will meet a crest; and then one-half cycle later, a trough will meet a trough.
 * Because antinodes are vibrating back and forth between a large positive and large negative displacement, a diagram of a standing wave is sometimes depicted by drawing the shape of the medium at an instant in time and at an instant one-half vibrational cycle later. This is done in the diagram below.
 * [[image:http://www.physicsclassroom.com/Class/waves/u10l4c5.gif width="213" height="93" align="bottom"]]
 * Crests and throughs represent points __of the disturbance__ that travel from one location to another through the medium.
 * An antinode on the other hand is a point __on the medium__ that is staying in the same location.
 * Furthermore, an antinode vibrates back and forth between a large upward and a large downward displacement.
 * And finally, nodes and antinodes are not actually part of a wave.
 * The nodes and antinodes are merely unique points on the medium that make up the wave pattern.
 * The nodes and antinodes are merely unique points on the medium that make up the wave pattern.

> **Second Harmonic Standing Wave Pattern** > || 1st || 2 || 1 || || > || 2nd || 3 || 2 || || > || 3rd || 4 || 3 || || > || 4th || 5 || 4 || || > || 5th || 6 || 5 || || > || 6th || 7 || 6 || || > || nth || n + 1 || n || -- ||
 * Harmonics and Patterns:**
 * The waves reflect off of a fixed end and interfere with the waves introduced by the source to produce this regular and repeating pattern known as a standing wave pattern.
 * A variety of actual wave patterns could be produced, with each pattern characterized by a distinctly different number of nodes.
 * Such standing wave patterns can only be produced within the medium when it is vibrated at certain frequencies.
 * There are several frequencies with which the snakey can be vibrated to produce the patterns, and ech frequency is associated with a different standing wave pattern.
 * These frequencies and their associated wave patterns are referred to as **harmonics**.
 * The production of standing wave patterns demand that the introduction of crests and troughs into the medium be precisely timed, and if f the timing is not precise, then a regular and repeating wave pattern will not be discerned within the medium - a harmonic does not exist at such a frequency.
 * With precise timing, reflected vibrations from the opposite end of the medium will interfere with vibrations introduced into the medium in such a manner that there are points that always appear to be standing still, called nodes.
 * Positioned in between every node is a point that undergoes maximum displacement from a positive position to a negative position, referred to as antinodes.
 * [[image:http://www.physicsclassroom.com/Class/waves/u10l4d1ani.gif width="266" height="122" align="bottom"]]
 * The above standing wave pattern is known as the **first harmonic** . It is the simplest wave pattern produced within the snakey and is obtained when the teacher introduced vibrations into the end of the medium at low frequencies.
 * Other wave patterns can be observed within the snakey when it is vibrated at greater frequencies; for instance, if the teacher vibrates the end with twice the frequency as that associated with the first harmonic, then a second standing wave pattern can be achieved.
 * This standing wave pattern is characterized by nodes on the two ends of the snakey and an additional node in the exact center of the snakey, and as in all standing wave patterns, every node is separated by an antinode.
 * This pattern with three nodes and two antinodes is referred to as the **second harmonic** and is depicted in the animation shown below.
 * If the frequency at which the teacher vibrates the snakey is increased even more, then the third harmonic wave pattern can be produced within the snakey.
 * The standing wave pattern for the third harmonic has an additional node and antinode between the ends of the snakey.
 * **Third Harmonic Standing Wave Pattern** [[image:http://www.physicsclassroom.com/Class/waves/u10l4d3ani.gif width="266" height="122" align="bottom"]]
 * Each consecutive harmonic is characterized by having one additional node and antinode compared to the previous one.
 * **Harmonic** || **# of Nodes** || **# of Antinodes** || **Pattern** ||
 * As one studies harmonics and their standing wave patterns, it becomes evident that there is a predictability about them., which expresses itself in a series of mathematical relationships that relate the wavelength of the wave pattern to the length of the medium.
 * Additionally, the frequency of each harmonic is mathematically related to the frequency of the first harmonic.

> > > > > > > > > || 1st || || 1 || **L** = 1 / 2 • > || > || 2nd || || 2 || **L** = 2 / 2 • > || > || 3rd || || 3 || **L** = 3 / 2 • > || > || 4th || || 4 || **L** = 4 / 2 • > || > || 5th || || 5 || **L** = 5 / 2 • > || > || 6th || || 6 || **L** = 6 / 2 • > || > || nth || -- || n || **L** = n / 2 • > || > >
 * Mathematics of Standing Waves:**
 * Wave patterns produced in a medium when two waves of identical frequencies interfere in such a manner to produce points along the medium that always appear to be standing still, and such standing wave patterns are produced within the medium when it is vibrated at certain frequencies.
 * Each frequency is associated with a different standing wave pattern, and are referred to as harmonics.
 * There is a mathematical relationship between the wavelength of the wave that produces the pattern and the length of the medium in which the pattern is displayed.
 * Furthermore, there is a predictability about this mathematical relationship that allows one to generalize and deduce a statement concerning this relationship.
 * [[image:http://www.physicsclassroom.com/Class/waves/u10l4e1.gif width="95" height="44" align="center"]]
 * The pattern for the first harmonic reveals a single antinode in the middle of the rope.
 * This antinode position along the rope vibrates up and down from a maximum upward displacement from rest to a maximum downward displacement as shown.
 * The vibration of the rope in this manner creates the appearance of a **loop** within the string.
 * A complete wave in a pattern could be described as starting at the rest position, rising upward to a peak displacement, returning back down to a rest position, then descending to a peak downward displacement and finally returning back to the rest position.
 * One complete wave in a standing wave pattern consists of two //loops//, so, one loop is equivalent to one-half of a wavelength.
 * [[image:http://www.physicsclassroom.com/Class/waves/u10l4eani1.gif width="458" height="187" align="center"]]
 * In comparing the standing wave pattern for the first harmonic with its single loop to the diagram of a complete wave, it is evident that there is only one-half of a wave stretching across the length of the string, so the length of the string is equal to one-half the length of a wave.
 * Now consider the string being vibrated with a frequency that establishes the standing wave pattern for the second harmonic.
 * The second harmonic pattern consists of two anti-nodes. Thus, there are two loops within the length of the string. Since each loop is equivalent to one-half a wavelength, the length of the string is equal to two-halves of a wavelength.[[image:http://www.physicsclassroom.com/Class/waves/u10l4e8.gif width="179" height="29" align="bottom"]]
 * The same reasoning pattern can be applied to the case of the string being vibrated with a frequency that establishes the standing wave pattern for the third harmonic.
 * The third harmonic pattern consists of three anti-nodes, so there are three loops within the length of the string.
 * Since each loop is equivalent to one-half a wavelength, the length of the string is equal to three-halves of a wavelength.
 * When inspecting the standing wave patterns and the length-wavelength relationships for the first three harmonics, a clear pattern emerges.
 * The number of antinodes in the pattern is equal to the **harmonic number** of that pattern.
 * The first harmonic has one antinode; the second harmonic has two antinodes; and the third harmonic has three antinodes.
 * Thus, it can be generalized that the **n** th harmonic has **n** antinodes where **n** is an integer representing the harmonic number.
 * Furthermore, one notices that there are **n** halves wavelengths present within the length of the string.
 * **Harmonic** || **Pattern** || **# of Loops** || **Length-Wavelength** **Relationship** ||
 * For standing wave patterns, there is a clear mathematical relationship between the length of a string and the wavelength of the wave that creates the pattern.
 * The mathematical relationship simply emerges from the inspection of the pattern and the understanding that each loop in the pattern is equivalent to one-half of a wavelength.
 * The general equation that describes this length-wavelength relationship for any harmonic is: