Vibrations And Waves Ap French Pdf
Note: You can click on any unit title to go directly to the unit. Topics: Periodic Phenomena (Oscillations, Waves), SHO, Complex Notation, Differential Equations, Physical Pendulum Topics: Beats, Damped Free Oscillations (Under- Over- and Critically Damped), Quality Q Topic: Forced Oscillations with Damping Topics: Forced Oscillations, Power at Resonance (Resonance Absorption, Resonance Width, Quality Q). Study Guide Unit 1: Introduction and Tools This first unit introduces periodic motion (in a general sense).
Great book for anyone interested in classical wave phenomena or is planning on studying quantum waves in the future. Word of caution, it does get very math/trig heavy in certain proofs and there were multiple times where i was expected to know trig identities in order to follow along with the proofs. Unit 1: Introduction and Tools. Reading Assignment. Unit 1 Reading Notes (PDF). Vibrations and Waves. New York: W.W. Norton & Company, 1971. Chapter 1: Periodic Motions (pp. Chapter 3: The Free Vibrations of Physical Systems (pp. 41-62, up to “The Decay of Free Vibrations”).
It is recommended that you view the lecture first, and then complete the Reading Assignment. Review the rest of the Unit 1 material presented here, including the important part about wave polarization. Topics. Periodic Phenomena (Oscillations, Waves).
SHO. Complex Notation. Differential Equations. Physical Pendulum Learning Objectives After completing this unit, you should be able to. identify everyday examples of periodic motion. identify examples of complicated oscillatory motion and sound produced through oscillatory motion. define the frequency range of pressure oscillations in air that create sound.
explain simple harmonic oscillation (SHO) through its basic equation, relation to circular motion, and complex exponential form. describe the relationships between phase, angular velocity, frequency, and period. explain the qualitative and quantitative dynamics of springs. discuss the relationship of complex numbers to circular motion.
explain the quantitative dynamics of a pendulum in the small (angle) amplitude approximation. explain the concept of wave polarization. Study Guide Unit 1: Introduction and Tools Exercises. Find a small spring or elastic band.
Stretch it and pluck it and estimate (very roughly) the vibration period. Next, find a car, which has stiff springs but a large mass. Note: If the car isn’t yours, please acquire permission from the owner before you take the next step. Try to move the car up and down by pushing down on it gently but firmly with your hand.
What is the frequency of the vibration? Although we are not taking into account the effects of varying k, you can confirm that the frequency is longer for this larger mass than it was for the spring or elastic band. (For pendulums, which we will cover later, this is not the case: the period does not vary with mass). The video lecture for this unit discusses error.
Vibration And Waves Pdf
Based on the discussion in the video, figure out why 1% uncertainty becomes 0.5% error due to the square root (use the binomial theorem). The moment of inertia for a mass M at the end of a massless rod of length L is ML 2.
Following the method shown in the video lectures, treat the simple pendulum as a physical pendulum, using this information. Show that you get the expected result for the period of a simple pendulum. Download Problem Set #1 and solutions from MIT from the links below. Attempt all problems except 1.5. Once you have attempted these problems, review the solutions provided.
Note that problem 1.2 is problem 1-10 and problem 1.3 is problem 1-11 from the French textbook. Similarly, problem 1.4 is problem 3-4 from the French textbook. (PDF) (PDF).
Do Problems 1-5, 1-6, 1-8, and 1-12 in the French textbook. Answers for Problem 1-12 are provided near the end of the book. Do Problem 3-1 in the French textbook. Note the use of “cgs” units. The unit “dyne” (abbreviated dyn) in the answer provided is a unit of force equal to 1 g cm s -2. Do Problems 3-2 and 3-3 in the French textbook. Answers are provided near the end of the book.
Do Problems 3-6 and 3-9 in the French textbook. Answers are provided near the end of the book. If you experience any difficulty in doing any problem, please discuss it with your tutor. Study Guide Unit 1: Introduction and Tools Wave Polarization The concept of polarization is very important in discussing general wave motion. Polarization means that the displacement of waves occurs in a particular direction. In most types of waves in which polarization is relevant, the displacement of the wave is at right angles to the direction of wave propagation.
For example, with waves on a string, polarization depends on how the string is agitated. When a string is agitated by moving one end of it in a straight line perpendicular to the string, waves propagate along the string in a plane containing the string and that line. If the string is moved up and down, the waves will also move up and down. While this situation is easiest to demonstrate, one can imagine swinging the end of a string side to side, in which case the waves would also move side to side. If the string was lying on a table, this would be the only way it could easily move.
Without the action of gravity, no wave direction would be preferred—it would be easy to set up waves in any plane containing the string and the direction of shaking. The direction the waves move to displace the string (perpendicular to its rest position), is called the direction of linear polarization.
Practically, the direction of linear polarization may be vertical or horizontal. On a table, for example, the polarization would be horizontal. In space (without gravity), any direction would be easily possible. We can also imagine more complex polarization, such as that in which circular motion at the end of a string creates the wave. Polarization can also be filtered. The waves of a vertically oscillating string can pass through a vertical slit, but a horizontal slit would stop such waves. Some types of waves do not have this type of directional polarization.
Sound waves, for example, are compressive, and any motion associated with a sound wave is parallel to its direction of propagation. In any case, when sound waves are discussed in terms of pressure, it is unnecessary to discuss the motion of the medium. Polarization is an important aspect of electromagnetic waves. It is very evident with these waves largely because the electric field in them interacts with matter. Electromagnetic waves have both electric and magnetic fields, which in empty space (and in many media) are perpendicular to the direction of propagation.
The electric field is also perpendicular to the magnetic field in the wave. By convention, the direction of the electric field is defined as the direction of polarization of the wave. This makes sense, since interaction of electromagnetic waves with most media is due to the effect of the wave’s electric field on charges in the medium. We discussed above that waves on strings can pass through a slot parallel to the waves, but not one perpendicular to the waves. Similarly, media can be created that interact strongly with light of one polarization, yet not with the perpendicular polarization.
These media are called polarizers. For reasons not discussed in this course, reflected light from water or smooth surfaces or interfaces is often polarized. By use of a polarizer, polarized light can be filtered. Glare from water or road surfaces is usually polarized parallel to the surface. Given that these common reflecting surfaces are usually horizontal, they create horizontally polarized light.
Use of a vertical polarizer removes such light. Polaroid® glasses, for example, consist of a vertical polarizer. By removing the horizontally polarized light that is created by reflections, Polaroid® glasses prevent glare from reaching the eyes.
Study Guide Unit 1: Introduction and Tools Other Notes Oscillatory behaviour can arise in other ways than those described in this course. An interesting example is the paired equations for predators and prey: the. Basically, these equations represent the tendency of prey populations to increase when predators are few, the tendency of predators to increase when prey is plentiful, and the inverse of each of these statements.
Under certain conditions, rather than attaining a steady balance of predators and prey, the balance oscillates, usually with a period of a few years. Another interesting oscillation, familiar to Canadians, is that of daily and seasonal temperature cycles.
Each day, the warmest time is well past noon (on average), while solar heating is maximal around noon. Similarly, the coldest part of winter is often well after the shortest day (with the least energy input). These cyclical temperature changes clearly depend on local conditions involving heat stored in the air, in the ground, and in nearby water bodies. There are specific and recently developed mathematical techniques that can be applied to these more complex systems that involve oscillation.
For the purposes of this course, try to think of things these complex systems may have in common with the simple systems we study. Study Guide Unit 2: Oscillations Topics. Beats. Damped Free Oscillations (Under- Over- and Critically Damped).
Quality Q Learning Objectives After completing this unit, you should be able to. describe how to combine (superpose) two vibrations and show that beats arise. identify the sound of beats and how they look on an oscilloscope or graph. use a viscous frictional force in Newton’s second law to solve for damped oscillatory motion. outline how a viscous force causes an exponential decay and a slightly lower oscillation frequency than a smaller viscous force. describe the quality factor. draw a simple RLC circuit and outline the physics behind its components.
solve an RLC circuit without necessarily understanding the derivation. graph the behaviours of underdamping, overdamping, and critical damping.
Study Guide Unit 2: Oscillations Exercises. Attempt Problems 1.5 and 1.7 in Problem Set #1, downloaded in Unit 1. Note: Problem 1.5 is problem 3-14 in the French textbook. Review the solutions provided after you have made a serious attempt to solve the problems on your own. Download Problem Set #2 and solutions from MIT from the links below. You will also need to download Take Home Experiment #1 and provide the simple materials described therein.
Do the experiment described in Problem 1.2 of Problem set #2. Compare your results to those given in the solutions.
(PDF) (PDF). Do Problems 2-1, 2-3, and 2-4 in the French textbook. Use suitable trigonometric identities to solve these problems. Answers are provided near the end of the book.
Do Problem 3-12 in the French textbook. This problem introduces the concept of phase space, which is important in sophisticated approaches to classical mechanics, in particular in problems involving cyclical motion. Do Problems 3-13 and 3-15 in the French textbook.
Answers are provided near the end of the textbook. Do Problem 3-16 in the French textbook. Be aware that although interesting results come from this exercise (see p.
310), the failure of electrons to radiate as expected in the classical formula led to quantum mechanics, in which an electron in an atom does not radiate when it is simply in a stable state in an atom. Study Guide Unit 3: Forced Oscillations Topic. Forced Oscillations with Damping Learning Objectives After completing this unit, you should be able to.
write Newton’s second law for a driven, damped, harmonic oscillator and transform the variables to the complex plane. determine and understand the steady state solution for a driven, damped, harmonic oscillator.
outline (by setting up a force diagram) how a pendulum can be driven at the suspension point. demonstrate aspects of resonant motion of a pendulum and generalize the results to similar systems.
give examples of the selection of a resonant frequency from a broad spectrum of input driving frequencies. Study Guide Unit 3: Forced Oscillations Exercises. Find the ω max through the procedure described in the video lecture (23:00). Swing and drive a pendulum, and estimate the Q as done in the video lecture. The Tacoma Narrows Bridge Collapse movie was removed from the RealMedia version of the MIT videos for copyright reasons. However, it can easily be found online through a simple search using “Tacoma Narrows Bridge Collapse.” Find the movie online and watch it (footage from this film is included in Supplemental Material in Unit 4 as well). You can find a high quality video of the shattering of a wine glass through sound and resonance online.
It has been debated whether this can actually be done using an unamplified human voice—you may wish to research that. However, please don’t try this potentially dangerous experiment at home!
(Material about this topic may also be found in the Supplemental Material in Unit 4.). In Sample Problem Set #2 (downloaded in Unit 2), do Problems 2.2, 2.3, and 2.4. Note: Problems 2.3 and 2.4 correspond to problems 4-6 and 4-10 in the French textbook. Check your solutions against those provided. (PDF) (PDF).
Do problems 4-3, 4-4, 4-5, and 4-9 in the French textbook. Answers are provided near the end of the book. For problem 4-9, an answer is only provided for (a). Study Guide Unit 4: Forced Vibrations and Resonance Topics. Forced Oscillations. Power at Resonance: Resonance Absorption, Resonance Width, and Quality Q. Transient Phenomena: General Solutions Including Initial Conditions Learning Objectives After completing this unit, you should be able to.
calculate the general solution of a differential equation as the sum of homogeneous and inhomogeneous equation solutions. identify that a damped system needs energy input to attain a steady state solution.
calculate the average power input into a steady state damped oscillation and determine its phase. describe power input into a damped oscillator, using the limiting and maximum power cases (in particular). outline the resonance and power in an RLC circuit. define electromagnetic radiation, and describe how it relates to atomic energy level transitions,. identify that red light is of low relative frequency, while blue/violet light is of higher frequency.
describe the production of an absorption spectrum by resonant absorption of light. Study Guide Unit 4: Forced Vibrations and Resonance Other Notes RLC Circuits In this unit we saw that a tuned electronic circuit can allow selection of a single broadcasting station from among many. Tuned electronic circuits have resistance (as do all circuits), inductance (in the image here, provided by the big red coil), and capacitance (in this image, provided by the variable capacitor made up of closely-spaced, mobile, metal plates, directly in front of the coil).
Tuned electronic circuits are known as RLC circuits. This example, corresponding to the radio technology of the early twentieth century, also shows a vacuum tube amplifier (back right).
If you want to examine “old school” circuitry up close, you can buy a hobby kit (like this one) from. Modern electronic circuits are much more compact than the one above. However, if you take apart an old transistor radio, you should be able to recognize the little coils that are in the RLC circuit and maybe the tuning capacitor (usually attached to the tuning dial).
Infrared Absorption Spectra The image shown here, from Chartres Cathedral in France, shows its blue stained glass. Light absorption due to the presence of chemical substances produces the rich colour hues. Does knowing this detract from (or enhance) its beauty? (Note of interest: the formula (recipe) for making the glass that gives the Chartres blue colour has been lost; we are no longer capable of reproducing it.) Page 152 of the French textbook gives an example of the infrared absorption spectrum of sodium chloride (common salt). The case of sodium chloride is very simple. The absorption shown would remove a “colour” from the spectrum. If a number of substances (with various infrared absorption spectra) are put together, many colours could be removed, and the remaining transmitted light would have the colour that has not been removed (this assumes that it was originally white, that is, composed of all colours).
The fact that sodium chloride is colourless in visible light indicates that it has no strong absorption at the wavelengths in the visible spectrum. The solar spectrum (below) shows light at many wavelengths (the wavelengths appear below the spectrum in units of 10 -10 m). Newton showed that white light is made up of all of the colours blended together. The subtle absorption lines at specific wavelengths correspond to resonant absorption as described on pages 106-107 of the French book. Please do not attempt to see these lines by looking at the Sun. They are difficult to see, they require specialized apparatus, and you should never look directly at the Sun. Study Guide Unit 4: Forced Vibrations and Resonance Supplemental Material.
You may test the frequency response of an RLC circuit using this interactive web page:. You can find some interesting applets related to oscillations and waves at. Annenberg Media’s Learner.org hosts a recommended general discussion of (scroll down to Program 17), available for viewing in Canada or the U.S. You will need to enable pop-ups from to view this video.
It is not available for download. It includes footage of both the dramatic Tacoma Narrows Bridge Collapse and the shattering of a wine glass at specific vocal pitches. Study Guide Unit 5: Normal Modes Topic.
Coupled Oscillators Learning Objectives After completing this unit, you should be able to. analyze an undamped coupled harmonic oscillator. define normal modes and describe how they may be combined. find the normal modes of coupled pendulums. determine the motion of coupled pendulums from their initial conditions.
identify how beats can originate through coupled modes. apply the formula for determining normal modes (in general). describe the normal modes of three coupled bodies. Study Guide Unit 5: Normal Modes Exercises. Download Sample Problem Set #3, Problem Set #3 solutions, and the instructions for Take-Home Experiment #2 from MIT from the links below. Do the take-home experiment, which is Problem 3.1. Compare your results to those given in the solutions.
(PDF) (PDF). Solve Problems 3.2, 3.3, and 3.4 from Sample Problem Set #3. Problem 3.2 and Problem 3.3 correspond to problems 5-10 and 5-11, respectively, in the French textbook. Compare your solutions to those provided.
Do problems 5-2, 5-4, 5-5, 5-6, 5-7, and 5-9 in the French textbook. Answers are provided near the back of the book. Study Guide Unit 6: Driven Coupled Systems Topics.
Driven Coupled Oscillators. Cramer's Rule.
Triple Pendulum. Steady State and Transient Solutions Learning Objectives After completing this unit, you should be able to. derive the (small angle) motion of a double pendulum. use Cramer’s Rule to solve an algebraic system. modify the equation of motion of a double pendulum to incorporate driving.
find the solutions for a steady state driven double pendulum. describe how to find a general solution for a coupled system. outline the relationship between driving, transients, steady state, and normal modes. describe the normal modes of the triple pendulum. Study Guide Unit 7: Wave Equation Moving from a large number of coupled oscillators to a continuous string of them is in many ways a natural transition. In a continuous system, rather than having individual masses joined by springs, both the mass and the restoring force are distributed (in many cases uniformly).
The continuous system is an approximation in our atomic world, but it is a good one since atoms are very small. At the individual atomic component level of a string, we must also consider other effects such as quantum mechanics. The modes and waves we will study in this unit have lower frequencies (by many orders of magnitude) than those involved in atomic vibrations. Topics. Many Coupled Oscillators.
Wave Equation. Transverse Travelling Pulses and Waves Learning Objectives After completing this unit, you should be able to. discuss transverse modes, longitudinal modes, and the boundary conditions doe a linear system. analyze the motion of a particle on a string of many particles. use normal modes to solve the system of differential equations for particles on a string.
describe how a combination of normal mode motions can make up a specified motion of a set of particles. outline the transition of a large number of coupled oscillators to a continuous string, including the change in normal mode frequencies. set up the equation of motion for a small piece of a string and obtain a wave equation. demonstrate that any single-valued function f(x±ct) solves the wave equation. describe the effect of boundary conditions on waves (reflections). Study Guide Unit 8: Waves Having examined the wave equation, we proceed to consider periodic waves.
Periodic waves are an important type both because they appear frequently and because many motions can be described as a superposition of periodic motions. The textbook and lectures consider periodic waves in inverse directions.
The textbook proceeds from standing waves to travelling waves, while the lecture proceeds from travelling waves to standing waves. Longitudinal waves are examined via the example of sound, which is easy to demonstrate. The final topic in Unit 8 is wave energy, which is examined quantitatively. Wave energy requires an extension of the mathematics to include integral calculus, which is described in an intuitive way in the course materials. Topics. Travelling Waves. Boundary Conditions.
Standing Waves. Sound (Longitudinal Waves). Energy in Waves Learning Objectives After completing this unit, you should be able to. define a wavelength and show why the wave repeats on the given scale.
apply the kx-ωt form of the wave solution. superpose waves and describe how to form a standing wave. determine the normal mode frequencies for continuous media with fixed ends. define nodes, antinodes, fundamental, harmonics, eigenvalue, and eigenstate (eigenfunction). describe the boundary conditions for (over)pressure waves (sound) in an enclosure. explain how to determine the speed of sound in a standing wave enclosure.
discuss why waves transport energy without necessarily transporting matter. calculate the power in a travelling wave. identify that travelling waves carry momentum.
outline the consequences of the wave equation having different forms in different coordinate systems. apply boundary conditions to obtain the ratio of reflection and the transmission coefficients at a boundary.
identify that particles have a wavelength, and define the de Broglie wavelength. identify that wave effects similar to those explored in the course also apply to particles on some scale. identify that free particles can have any energy, momentum, and wavelength. This material is found only in the course materials, not in the textbook or MIT video lecture. See, included in the Unit 8 Reading Assignment section. Study Guide Unit 8: Waves Reading Assignment (PDF) French, A.
Vibrations and Waves. New York: W.W. Norton & Company, 1971.
Chapter 7: Progressive Waves (pp. 201-216, up to “Wave Pulses”).
Chapter 6: Normal Modes of Continuous Systems. Fourier Analysis (pp. 167-170, up to “Longitudinal Vibrations of a Rod”). Chapter 7: Progressive Waves (pp. 228-230, up to “Dispersion. Chapter 8: Boundary Effects and Interference (pp. 253-259, up to “Impedences.”).
Chapter 8: Boundary Effects and Interference (p. 264, bottom, and p. 265, up to “Waves in Two Dimensions”). Waves: Quantum Mechanical Waves. (PDF). Study Guide Unit 8: Waves Exercises.
Work out the potential energy in a travelling wave (see French). Download Sample Problem Set #4 and solutions from MIT from the links below. Do Problems 4.1 and 4.2 from Sample Problem Set #4. These correspond to Problems 7-12 and 7-13, respectively, in the French textbook. When you have completed your solutions, compare them to the solutions provided.
(PDF) (PDF). Do problems 7-1, 7-2, 7-4, 7-8, and 7-9 in the French text. Answers for 7-2, 7-4, and 7-8 are found near the end of the book. Do problem 6-10 in the French text. Answers are found near the end of the book. Study Guide Unit 8: Waves Supplemental Material. (see Lecture 24)—This short video shows the equivalence of mechanical and electromagnetic standing waves, and the effects of boundary conditions.
—Java applet. Annenberg Media’s Learner.org hosts a recommended general discussion of (scroll down to Program 18), including water waves that are not otherwise treated in this course, available for viewing in Canada or the U.S. You will need to enable pop-ups from to view this video. It is not available for download.
Study Guide Unit 9: Bounded Waves Topics. Musical Instruments. Sound Cavities. Normal Modes Learning Objectives After you have completed this unit, you should be able to. discuss how normal modes are controlled by boundary conditions.
describe the basic structure of a stringed instrument. explain the role of tension, mass per unit length, length, boundary conditions, and/or speed of sound, as appropriate, in producing musical notes. write the wave equation in higher dimensions than 1 (in Cartesian coordinates) and solve these equations, incorporating boundary conditions. explain the principle behind Chladni patterns. identify that confining a particle leads to discrete energy levels for the particle and quantum states similar to normal modes.
determine the allowable wavelengths and energies for a particle in a box. This material is found in the course notes only, not in the textbook or video lecture. Study Guide Unit 9: Bounded Waves Other Notes. Note that 48:10 to 52:40 of the video lecture shows students trying out instruments. The Spring 2004 MIT exam has a 3-d sound box solution (online). You may find this interesting. Lewin recommends in his lecture, do not memorize the equation for the speed of sound, but recall the 340 m/s usual figure.
Please do not attempt any “inhalation” experiments based on what you see in the video lecture!. If you are interested in musical instruments, you may wish to view the supplemental video “The Sounds of Music” (URL available under Supplemental Material).
This is a public lecture, with limited mathematics. Study Guide Unit 9: Bounded Waves Supplemental Material. Supplementary lecture: “” by Walter Lewin.
—This PowerPoint presentation gives a pretty good explanation about organ pipes. —You may find this review lecture useful, but be aware that not all of this material is relevant for PHYS 302. Annenberg Media’s Learner.org hosts a recommended general discussion of (scroll down to Program 50), available for viewing in Canada or the U.S. This video contains material relevant to the discussions of quantum mechanical waves in Units 8 to 15. You will need to enable pop-ups from to view this video. It is not available for download. Study Guide Unit 10: Fourier Techniques In this unit, there are slight differences in notation between the textbook and the video lecture.
It is suggested that you view the video lecture first. Topics. Fourier Analysis. Time Evolution of Pulses on Strings Learning Objectives After completing this unit, you should be able to. write the general expression for a Fourier series.
determine the terms of the Fourier series of a simple function. outline how even and odd functions contribute to Fourier integrals. sketch the build-up of a function from the parts of its Fourier series. identify how the normal modes, each at its own frequency, contribute to development of a travelling wave.
describe why the location along the string where a musical instrument string is plucked affects the sound. construct a power density spectrum from the Fourier components. describe the relation between length of a wave packet and its frequency spread. relate frequency spread to radio bandwidth and to the Heisenberg Uncertainty Principle of quantum mechanics.
This material is found in the course notes only, not in the textbook or video lecture. Study Guide Unit 10: Fourier Techniques Exercises. Find the indefinite integral of cos 2x. Use this result to show why the definite integral of this function, over one wavelength, is ½. Download Sample Problem Set #5 and solutions from MIT from the links below. (PDF) (PDF).
If you have access to a piano, perform the investigations given in Problems 5.1 and 5.6. If you can access the book referenced in Problem 5.2 (Benade, A.
Horns, Strings, and Harmony. NY: Dover Publications, 1992), examine Problem 5.2.
The pages referenced are not available in “preview” versions online. Do Problems 5.3, 5.4, and 5.5.
Note that 5.3 is problem 6-12 and 5.5 is problem 6-14 from the French text. Review your findings and solutions with the information provided. Do problems 6-14 and 6-15 in the French text. Answers are provided near the back of the book. Study Guide Unit 11: Dispersive Waves Topics. Dispersion. Phase Velocity.
Group Velocity Learning Objectives After completing this unit, you should be able to. define dispersive medium and non-dispersive medium and give examples of each. combine two waves of approximately the same ω and k in a dispersive medium, to show the phase and group velocities.
describe the propagation of a wave in a dispersive medium. define and draw the slopes corresponding to the phase and group velocities on an ω-k diagram. identify why sharp features in a pulse cannot be supported in a dispersive medium. identify speed of light and calculate the wavelengths and frequencies of electromagnetic waves. identify that water and glass are dispersive media for electromagnetic waves. identify (on a graph) the dispersion relation of microwaves and the cut-off.
find the dispersion relation of de Broglie waves. This material is found in the course notes only, not in the textbook or video lecture. Study Guide Unit 11: Dispersive Waves Exercises.
Download Sample Problem Set #6 and solutions from MIT from the links below. (PDF) (PDF). Do Problems 6.1 and 6.2, including the hands-on work if possible. Lighting could be critical for seeing the waves, but please do not put lamps near the bathtub, and please do not touch anything connected to your household electrical service with wet hands. Do Problem 6.3 of Sample Problem Set #5.
This corresponds to Problem 7-20 of the French textbook. Compare your answers to solutions provided. Problems 6-4 and 6-5 relate to material on energy of waves covered in an earlier unit, but they are worth a look. The remainder of the problems relate to material not included in this course.
Do Problems 7-17, 7-18, 7-19, and 7-21 in the French textbook. Answers for 7-17, 7-18, and 7-21 are found near the back of the book. Study Guide Unit 11: Dispersive Waves Other Notes Ignore any reference that the MIT materials make to the Bekefi and Barrett book. PHYS 302 does not address Maxwell’s equations, although PHYS 302 discusses electromagnetic waves from a wave point of view (the description of electromagnetic waves comes from Maxwell’s equations).
At this point, we can appreciate some real-life physics with some sophistication. The diagram (Fig. 1117) in the article shows the computed transverse velocity of a C piano string after being hit at the point H. Time increases by 62.5 μs on each successive line up from the bottom. The initial pulse from the third to sixth lines (likely the times at which the hammer contacted the strings) spreads both to the right and left.
The left-travelling pulse reflects inversely after which two pulses travel to the right. Other pulses follow, likely because the support point (agraffe) at the left in not totally rigid. When these pulses reach the far right, they reflect and head back towards the left. By the top line, the whole string is moving, although an identifiable pulse would still exist. In addition to the travelling of pulses in both directions (reflection with reversal of the wave at a rigid boundary—velocity in this case), you can see spreading of the pulse due to dispersion (you must look carefully). We have studied all of these effects. Although this situation is far more complex than the terms of harmonic waves in an idealized case (as first discussed), you should now understand most of the physics of this real-life situation.
You may read an entire article about simulations of piano strings in the following resource: Chaigne, A, & Askenfelt, A. Am., 95(2), 1112-1118. Figure 4, above, is reproduced with permission from the journal cited here. Study Guide Unit 13: Spatial Combination of Waves II Topics. Diffraction.
Gratings. Pin Holes. Angular Resolution Learning Objectives After completing this unit you should be able to. make a diagram showing the addition of many rotating vectors. determine the amplitude arising from N combined vectors of amplitude A 0, phased δ apart. describe a diffraction grating. determine the phase difference between slits based on geometry.
draw vector diagrams for the minima and maxima of a given diffraction pattern. describe the width and height of diffraction maxima in terms of conservation of energy. describe single slit diffraction using equivalent half slits. state Huygen’s principle and draw a sketch based on it. calculate amplitude in single slit diffraction. combine the results of ideal diffraction gratings and single slits to describe real gratings. describe a number of physical systems that give diffraction.
describe quantum mechanical diffraction. Study Guide Unit 13: Spatial Combination of Waves II Exercises. At the, try Diffraction and Interference II, which you can access via the left-hand menu on the RCLs page. Note: If the site comes up in German, click on the small British flag (Union Jack) below the home tab on the home page to change the language to English. Download, a very useful, free graphing program. You are encouraged to investigate gnuplot, through which you can display data files from the (you may need to edit the files to remove certain textual material). Gnuplot is also useful for plotting functions.
The following commands will produce the graph of a two slit “grating.” These commands may be saved to a file and loaded, or copied into gnuplot. Sep=10000 lam=500 wid=2000 n=2 set ylabel “I/I0 relative intensity” set xlabel “angle theta (radians)” bet(x)=pi/lam.wid.sin(x) del(x)=2.pi/lam.sep.sin(x) in(x)=(sin(n.del(x)/2)/sin(del(x)/2)).2 df(x)=(sin(bet(x))/bet(x)).2 set sample 1000 plot x=-0.5:0.5 in(x).df(x) The names given are fairly intuitive:. sep is the separation of slits. lam is the wavelength.
wid is the width of the slits. n is the number of slits. bet(x) is the beta function that occurs in diffraction equations—called df(x). del(x) is the delta function that occurs in interference equations—called in(x). set sample 1000—an important command, since the complicated function for finite slit width diffraction/interference requires high screen resolution to be seen properly.
plot—plots the product of in(x) and df(x), the overall result of interference with finite width slits. You should try changing the various parameters to see how the grating function looks in many situations. The plot will be produced no matter what parameters you give, but for purposes of learning, it is recommended that you apply physically possible values. For example, a separation less than the width of the slits is meaningless. In Sample Problem Set #10, downloaded in Unit 12, do Problem 10.5. Check your solution against the one provided.
(PDF) (PDF). In Sample Problem Set #11, downloaded in Unit 12, do Problem 11.1 if you have a grating available. You will need to download the procedure below. (PDF) (PDF). In Sample Problem Set #11, downloaded in Unit 12, do Problem 11.2.
Check your solution against the one provided. Do problems 8-15 and 8-16 in the French textbook. Answers can be found near the end of the book. Study Guide Unit 14: Doppler Effect This unit discusses wave and vibration applications in astronomy. Much of it explains simple facts of astronomy, as the wave nature of light is the basis of much of what we know about the universe.
Topics. Doppler Effect. Sound. EM Radiation. Binary Stars. Neutron Stars and Black Holes. The Expanding Universe Learning Objectives After completing this unit, you should be able to.
identify the speed of sound through air. identify the Doppler formula for sound, and explain how sonic booms originate.
explain how the period, speed, and radius for circular motion can be determined using the Doppler shift. identify the Doppler formula for electromagnetic radiation for slow radial motion. describe how the absorption lines in stars indicate their radial speed, and outline the importance of this in astronomy. explain how the absorption lines in double stars (whether both are seen or not) show their orbital motion. identify that the masses of stars and the size of a binary orbit may be determined from the Doppler effect. describe accretion disks (qualitatively), and explain why X-ray emission would be expected from them. identify the maximum mass of a white dwarf and that of a neutron star, and explain how scientists infer that black holes.
use Hubble’s Law to find the distance (span) of a galaxy (given its radial velocity) and to determine the age of the universe. explain why the expansion of the universe makes it look as if Earth is at the centre (although it isn’t). Study Guide Unit 14: Doppler Effect Other Notes. This unit discusses some aspects of electromagnetic radiation. While the (nonrelativistic) Doppler effect acts similarly for any periodic phenomenon, electromagnetic radiation does not need a medium for transmission, so it can cross (essentially empty) cosmic space. Sound cannot do so.
So, while the Doppler effect is similar for both sound and electromagnetic radiation, only electromagnetic radiation can bring us information about the Universe. With sound, an object moving away from is different from you moving away from the object. The reason why may not be immediately obvious. Bear in mind that motion relative to the medium carrying the sound is important.
With light, the relativistic Doppler effect is involved, and the speed relative to any medium is always given as the speed of light. The general form of the Doppler formula given reduces to. This is a good form to remember, because although intuitively one would think that a change in wavelength is proportional to the velocity, in fact, a relative change in wavelength is equal to the velocity relative to that of light. It is slightly misleading to claim that universal redshifts are actually Doppler shifts, since the scale of space has changed over time. However, the math works identically.
Note that astronomers Tom Bolton and Wendy Freedman, mentioned in Walter Lewin’s lecture, are Canadians. The determination of the Hubble constant was one of the main goals of the Hubble Space Telescope. Study Guide Unit 15: Resonant Cavities Topics. Wave Guides. Resonance Cavities of EM-Radiation and Sound Learning Objectives After completing this unit, you should be able to. apply boundary conditions to obtain a combination standing and travelling wave that corresponds to a guided wave.
discuss the dispersion relation for a waveguide. obtain the phase and group velocities of guided waves from the dispersion relation. draw an ω-kz diagram showing the dispersion relation of guided waves. discuss the cut-off frequencies of guided waves.
discuss the allowable frequencies for waves in a rectangular enclosure and relate them to membrane vibrations. identify why pressure antinodes exist for sound waves at the walls of a cavity. explain that “sin” solutions arise with nodes, and “cos” solutions arise with antinodes at the walls of a cavity. explain how a resonant cavity can select frequencies from a broadband source or transient driver. identify that the wave equation can be solved in cylindrical and spherical coordinate systems. describe boundary conditions suited to cylindrical and spherical coordinate systems.
describe the atomic energy levels arising from wave effects in atoms and determine the energy of EM-radiation arising from transitions between two levels. This material is found in the course notes only, not in the textbook or video lecture. Study Guide Unit 15: Resonant Cavities Other Notes.
There can be no electric field parallel to nor interior to the surface of a conductor, because such an electric field would cause the charges to move in the conductor until the electric field was cancelled out. The boundary condition used in deriving the equations for waves in a waveguide originates from this. In the lecture (32:00:00), the boundary condition is removed by changing the polarization of the waves. The gap used is a polarizer, since it will pass only EM waves with polarization perpendicular to the gap. For small gap width, however, it will not pass EM waves with polarization parallel to the gap.
We did not include the lecture on boundary conditions for electromagnetic waves in this course, since PHYS 302 does not require a sophisticated understanding of electromagnetism. Some understanding of boundary conditions is needed to understand the general application of EM boundary conditions. However, to understand the boundary conditions for an electric field, all we need to know is that an electric field causes forces on charged particles, and that charged particles (that are free to move) are present in a conductor. This means that if an electric field is applied to a perfect conductor, charged particles will move around very quickly, to cancel it out. Essentially, we can say that the electric field parallel to the surface of a conductor must be zero.
We cannot, however, say this about the field perpendicular to the surface of a conductor, since in this case, charged particles are free to move only up to the surface. We do not need to explore this further in this course, since our discussion is limited to cases in which the boundary condition arises for fields parallel to the conducting surface. In the beginning example in the lecture, the electric field is required to be zero at the conducting boundary; it refers to the parallel component. The resulting boundary condition is analogous to that of sound applied earlier in the course. The argument that follows is general.
For example, Dr. Lewin refers to water waves, which reflects this generality. As pointed out about half-way through, in this lecture, c is used both as the speed of light and as an index, so be careful with c. Broadband driving: If many frequencies impinge on a resonator, the response will be at the resonant frequencies, creating a spectrum. In this way, noise can be organized into standing waves, or a signal can be extracted from noise.
This is the basis of tuning of a radio and of amplifying sound by the soundbox of a guitar, etc. In the example given in the lecture, a transient signal drives (puts energy into) the cavity and causes beats.
Introductory Physics Series is the result of a program of careful study, planning, and development that began in 1960. The Education Research Center at the Massachusetts Institute of Technology (formerly the Science Teaching Center) was established to study the process of instruction, aids thereto, and the learning process itself, with special reference to science teaching at the university level. Generous support from a number of foundations provided the means for assembling and maintaining an experienced staff to co-operate with members of the Institute's Physics Department in the examination, improvement, and development of physics curriculum materials for students planning careers in the sciences. After careful analysis of objectives and the problems involved, preliminary versions of textbooks were prepared, tested through classroom use at M.I.T. And other institutions, re-evaluated, rewritten, and tried again. Only then were the final manuscripts undertaken.