EP225 Note No. 5 Mechanical Waves
|
|
- Georgiana Stafford
- 6 years ago
- Views:
Transcription
1 EP5 Note No. 5 Mechanical Wave 5. Introduction Cacade connection of many ma-pring unit conitute a medium for mechanical wave which require that medium tore both kinetic energy aociated with inertia (ma) and potentil energy aociated with elaticity (pring). In thi Chapter, baic propertie of mechanical wave are eplained. 5. Wave in Ma-Spring Tranmiion Line A ma-pring tranmiion line coniting of many cacaded ma-pring unit can model variou media of mechanical wave. We aume each unit ha a ma m and a pring with an equilibrium length and a pring contant k (N m ) a hown above. (Soon we will ee that k K (elatic modulu in Newton) i more convenient. Lower cae k for the wavenumber will be retored.) When a wave i ecited in the tranmiion line, the mae are diplaced from their original equilibrium poition. We conider ma-pring unit located at and and denote the repective ma diplacement by (; t) and ( ; t): The length of the pring to the right change by ( + ; t) (; t) and eert a retoring force on the ma originally at ; Similarly, the pring to the left eert a force The net force to act on the ma i F F + F F + k [( + ; t) (; t)] () F k [(; t) ( ; t)] () k [( + ; t) + ( ; t) (; t)] (3) Therefore, the equation of motion for the ma can be written down (; t) k [( + ; t) + ( ; t) (; t)] (4)
2 If i mall compared with the wavelength, the dicrete tranmiion line can be conidered continuou. In thi cae, the diplacement ( ; t) can be Taylor epanded a and the equation of motion become ( ; t) + (; t) k (; t) Thi i in the form of wave di erential equation and we can readily identify the wave velocity, r k () m k m The quantity K k (N) i called the elatic modulu of the pring. It i a material contant and more convenient than the pring contant k which depend on the length of pring. Likewie, the linear ma denity l m (kg/m) i a normalized quantity appropriate for analyzing local motion of the tranmiion line. Uing K and l ; we rewrite the wave velocity in the form K (7) l The wave motion we are conidering i aociated with ma motion along the tranmiion line, that i, in the direction of wave propagation. Such wave are called longitudinal. Sound wave in gae, liquid, and olid are typical longitudinal wave. Wave in a tring are tranvere in the ene that tring diplacement i perpendicular to the direction of wave propagation. However, velocitie of all mechanical wave are in the ame form, Elatic Modulu (8) Ma Denity For eample, the wave velocity in a tring i T (N) l (kg/m) (9) where T i the tenion. The velocity of ound wave in a olid rod take a imilar form, c Y (N/m ) v (kg/m 3 ) (0)
3 where Y i the Young modulu and v i the volume ma denity. In a ga, the ound velocity i given by P c () v where P i the ga preure (in N/m ) and ( 75 for diatomic gae) i the adiabaticity contant (or the ratio of peci c heat). Eample Find the velocity of each of the following wave. (a) Longitudinal wave in a pring 50 cm long having a total ma of 40 g and pring contant of 30 N/m. (b) Tranvere wave in a tring with a linear ma denity of 0 g/m under a tenion of N. (c) Tranvere wave in a membrane with a urface tenion of 0 N/m and urface ma denity of 50 g m : (d) Sound wave in water. The bulk modulu of water i M : 0 9 N m : Solution (a) The elatic modulu i K 30 N/m 0:5 m 5 N and the linear ma denity i l 0:08 kg m. Then, r K 5 l 0:08 m 7.3 m (b) (c) (d) T N l 0:0 kg/m 0 m T 0 N/m 0:5 kg/m 8: m c M : 0 9 N/m v 0 3 kg/m 3 : m Eample. Show that Eq. (; t) k [( + ; t) + ( ; t) (; t)] can be ati ed by a inuoidal wave (; t) A in(k! 4k m in k!t) provided 3
4 Solution The econd order time derivative of A in(k!t) A in(k!t)! A in(k!t): ( + ; t) (; t) A in k co k + k!t we nd ( ; t) (; t) A in k co k k!t ( + ; t) (; t) + ( ; t) A in k co k + k!t 4A in k in(k!t) Subtitution into the original equation yield In the limit of! 0; and we recover The diperion relation i applicable to arbitrary :! 4 k m in k in k! k! k () m k! 4 k m in k! r k m in k co k k!t i plotted below for the cae of poitive group velocitie d!dk > 0: The frequency i normalized by! 0 p k m and the wavenumber by. 8 < in if 3 < < f() in if < < : in if < < 3 f 4
5 f Energy Carried by Mechanical Wave A in mechanical ocillation, kinetic energy and potential energy are aociated with mechanical wave. One major di erence from the cae of ocillation i that in wave, kinetic energy and potential energy are in phae. When and where the kinetic energy i maimum, o i the potential energy. Let u conider the ma-pring tranmiion line. The velocity of the ma located at can be found by di erentiating the diplacement (; t) with repect to time, v(; Therefore, the kinetic energy of the ma i The potential energy tored in the pring i m P.E. k [( + ; t) (; t)] k () Since (; t) decribe wave motion, it mut be a function of X Note 4. Then c w t a dicued in d mc w (5) and P.E. k d k() (6) 5
6 Recalling c w k () m; we indeed ee that the kinetic energy and potential energy are identical to each other everywhere at anytime for propagating wave. (For tanding wave which are eentially ocillation, thi tatement doe not hold. In tanding wave, the potential and kinetic energie are mutually ecluive a in ocillation.) Thi concluion hold regardle of wave hape, for we have not aumed any particular wave form in the analyi. The total energy in the ma-pring unit i d mc w d mc w and the total energy denity i l c w d (J m ) (7) For a inuoidal wave decribed by (k here i the wavenumber in rad/m) Eq. (6) give (; t) 0 in(k!t);! c w k l c wk 0 co (k l! 0 co (k!t) A naphot of thi function at, ay, t 0; i hown below. The patial average of the function co (k) i /. Therefore, the average energy denity aociated with the wave i!t) l! 0 (J m ) (8) Since all of the energy clump travel at the wave velocity c w ; we nally obtain the rate of energy tranfer (power) RMS power lc w! 0 (J Watt) (9) For wave in volume media uch a ound wave, the linear ma denity in Eq. (8) can be replaced with the volume ma denity v (kg m 3 ) which yield the rate of energy tranfer per unit area, vc w! 0 (J m Watt m ) (0) Thi quantity i called the intenity of (ound) wave. Eample 3. A tring having a linear ma denity of 0 g/m i under a tenion of 40 N. A inuoidal tranvere wave i ecited in the tring with an amplitude 0 5 mm and frequency 80 Hz. Find the energy tranfer rate due to the wave. 6
7 Solution The wave velocity i Uing Eq. (8), we nd the energy tranfer rate T 40 N l 0:0 kg/m 44:7 m. lc w! 0 0:0 44:7 ( 80) (0:005) J :8 Watt Eample 4. Typical intenity of ound wave in converation i 0 6 W m : Calculate the amplitude of diplacement wave of air molecule 0 : Aume c (ound peed) 340 m (which correpond to 0 C temperature a we will tudy later), one atmopheric preure and (wave frequency) 00 Hz. Solution The air ma denity at 0 C, one atmopheric preure can be found by recalling that the ma of one mole of air i 9 g (approimately 80% nitrogen N (8 g) and 0% oygen O (3 g)). At 0 C, one atmopheric preure, 9 g of air occupie a volume of.4 l: Therefore, at 0 C, one atmopheric preure, the ma denity of air i Uing Eq. (9), we nd 0 v 0:09 kg :4 0 3 m : kg m 3 : I v c! 0 6 W/m : kg/m m/ec 00 ec 5: m Thi i a very mall diplacement and air molecule hardly move. However, there are o many air molecule that participate in wave motion collectively for audible ound intenity. 5.4 Momentum Carried by Wave Whenever energy i tranferred, momentum i tranferred a well and we epect mechanical wave hould carry momentum with them. In fact, we have already derived an epreion for the momentum tranfer rate in Eq. (8), l! 0 (J m ) which we interpreted a the average energy denity along the pring (or tring). Let u take a look at the dimenion, J m N m m N ec ec 7
8 which indeed ha the dimenion of momentum (Nec) tranfer per unit time. Thi obervation ugget that there i a imple relationhip between the energy tranfer rate and momentum tranfer rate, momentum tranfer rate energy tranfer rate : () wave velocity In fact thi relationhip hold true for any wave, mechanical and electromagnetic wave. Since momentum tranfer i done at the wave velocity, the average momentum denity aociated with a wave can be found by further dividing Eq. (0) by the wave velocity, momentum denity energy tranfer rate c w energy denity c w () Eample 5. The above derivation of wave momentum i baed on dimenional analyi. Derive the relation in Eq. (0, ) from the rt principle, i.e., equation of motion applied to longitudinal wave in a pring. Solution Let the linear ma denity of the pring be l and pring elatic modulu be K: Conider a egment of length located at : The ma of the egment in the abence of wave i l With wave perturbation, the egment i elongated by ( + ) and the ma l i now ditributed over a ditance : Therefore, with perturbation, the ma denity i perturbed a well, and given by 0 l ' l where The momentum denity i ma denity time @t For a inuoidal wave, the average of the rt imply vanihe. However, the average of the econd term @t lk! 0 average 8 l (! 0 ) c
9 where the diperion relation! c w k i recalled. Thi prove the relation in Eq. () between the momentum denity and energy denity. Eample 6: Show that the average momentum tranfer rate due to a harmonic tranvere wave 0 in(k!t) i a tring (tenion T; ma denity l ) i given by l(! 0 ) Solution It may ound odd that tranvere wave in a tring tranfer momentum along the tring becaue the diplacement vector (; t) i perpendicular to the tring. A in the cae of longitudinal wave, mall econd order e ect mut be retained in the analyi. The diplacement of a egment of the tring i not vertically traight but i of arc hape a hown in the gure. The velocity ha a parallel component given by v k If a inuoidal diplacement (; t) 0 @ i aumed, v k 0!k co (k!t) k! (! 0) co (k!t) The average momentum denity i therefore given by and the average momentum tranfer rate by k! l(! 0 ) c l (! 0 ) w l(! 0 ) ; (N) 9
EELE 3332 Electromagnetic II Chapter 10
EELE 333 Electromagnetic II Chapter 10 Electromagnetic Wave Propagation Ilamic Univerity of Gaza Electrical Engineering Department Dr. Talal Skaik 01 1 Electromagnetic wave propagation A changing magnetic
More informationOnline supplementary information
Electronic Supplementary Material (ESI) for Soft Matter. Thi journal i The Royal Society of Chemitry 15 Online upplementary information Governing Equation For the vicou flow, we aume that the liquid thickne
More information1. Basic introduction to electromagnetic field. wave properties and particulate properties.
Lecture Baic Radiometric Quantitie. The Beer-Bouguer-Lambert law. Concept of extinction cattering plu aborption and emiion. Schwarzchild equation. Objective:. Baic introduction to electromagnetic field:
More informationLecture 3 Basic radiometric quantities.
Lecture 3 Baic radiometric quantitie. The Beer-Bouguer-Lambert law. Concept of extinction cattering plu aborption and emiion. Schwarzchild equation.. Baic introduction to electromagnetic field: Definition,
More informationA) At each point along the pipe, the volume of fluid passing by is given by dv dt = Av, thus, the two velocities are: v n. + ρgy 1
1) The horizontal pipe hon in Fig. 1 ha a diameter of 4.8 cm at the ider portion and 3.7 cm at the contriction. Water i floing in the pipe and the dicharge from the pipe i 6.50 x -3 m 3 /. A) Find the
More informations much time does it take for the dog to run a distance of 10.0m
ATTENTION: All Diviion I tudent, START HERE. All Diviion II tudent kip the firt 0 quetion, begin on #.. Of the following, which quantity i a vector? Energy (B) Ma Average peed (D) Temperature (E) Linear
More informationLecture 23 Date:
Lecture 3 Date: 4.4.16 Plane Wave in Free Space and Good Conductor Power and Poynting Vector Wave Propagation in Loy Dielectric Wave propagating in z-direction and having only x-component i given by: E
More informationFluid-structure coupling analysis and simulation of viscosity effect. on Coriolis mass flowmeter
APCOM & ISCM 11-14 th December, 2013, Singapore luid-tructure coupling analyi and imulation of vicoity effect on Corioli ma flowmeter *Luo Rongmo, and Wu Jian National Metrology Centre, A*STAR, 1 Science
More information@(; t) p(;,b t) +; t), (; t)) (( whih lat line follow from denition partial derivative. in relation quoted in leture. Th derive wave equation for ound
24 Spring 99 Problem Set 5 Optional Problem Phy February 23, 999 Handout Derivation Wave Equation for Sound. one-dimenional wave equation for ound. Make ame ort Derive implifying aumption made in deriving
More informationPhysics 2. Angular Momentum. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB
Phyic Angular Momentum For Campu earning Angular Momentum Thi i the rotational equivalent of linear momentum. t quantifie the momentum of a rotating object, or ytem of object. To get the angular momentum,
More informationSOLUTIONS
SOLUTIONS Topic-2 RAOULT S LAW, ALICATIONS AND NUMERICALS VERY SHORT ANSWER QUESTIONS 1. Define vapour preure? An: When a liquid i in equilibrium with it own vapour the preure exerted by the vapour on
More informationELECTROMAGNETIC WAVES AND PHOTONS
CHAPTER ELECTROMAGNETIC WAVES AND PHOTONS Problem.1 Find the magnitude and direction of the induced electric field of Example.1 at r = 5.00 cm if the magnetic field change at a contant rate from 0.500
More informationAssessment Schedule 2017 Scholarship Physics (93103)
Scholarhip Phyic (93103) 201 page 1 of 5 Aement Schedule 201 Scholarhip Phyic (93103) Evidence Statement Q Evidence 1-4 mark 5-6 mark -8 mark ONE (a)(i) Due to the motion of the ource, there are compreion
More informationBlackbody radiation. Main radiation laws. Sun as an energy source. Solar spectrum and solar constant.
Lecture 3. lackbody radiation. Main radiation law. Sun a an energy ource. Solar pectrum and olar contant. Objective:. Concept of a blackbody, thermodynamical equilibrium, and local thermodynamical equilibrium..
More informationFermi Distribution Function. n(e) T = 0 T > 0 E F
LECTURE 3 Maxwell{Boltzmann, Fermi, and Boe Statitic Suppoe we have a ga of N identical point particle in a box ofvolume V. When we ay \ga", we mean that the particle are not interacting with one another.
More information= 16.7 m. Using constant acceleration kinematics then yields a = v v E The expression for the resistance of a resistor is given as R = ρl 4 )
016 PhyicBowl Solution # An # An # An # An # An 1 C 11 C 1 B 31 E 41 D A 1 B E 3 D 4 B 3 D 13 A 3 C 33 B 43 C 4 D 14 E 4 B 34 C 44 E 5 B 15 B 5 A 35 A 45 D 6 D 16 C 6 C 36 B 46 A 7 E 17 A 7 D 37 E 47 C
More informationtwo equations that govern the motion of the fluid through some medium, like a pipe. These two equations are the
Fluid and Fluid Mechanic Fluid in motion Dynamic Equation of Continuity After having worked on fluid at ret we turn to a moving fluid To decribe a moving fluid we develop two equation that govern the motion
More informationMassless fermions living in a non-abelian QCD vortex based on arxiv: [hep-ph]
Male fermion living in a non-abelian QCD vortex baed on arxiv:1001.3730 [hep-ph] Collaborator : S.Yaui KEK and M. Nitta Keio U. K. Itakura KEK Theory Center, IPNS, KEK New frontier in QCD @ Kyoto March
More informationPHYSICSBOWL March 29 April 14, 2017
PHYSICSBOWL 2017 March 29 April 14, 2017 40 QUESTIONS 45 MINUTES The ponor of the 2017 PhyicBowl, including the American Aociation of Phyic Teacher, are providing ome of the prize to recognize outtanding
More informationUSPAS Course on Recirculated and Energy Recovered Linear Accelerators
USPAS Coure on Recirculated and Energy Recovered Linear Accelerator G. A. Krafft and L. Merminga Jefferon Lab I. Bazarov Cornell Lecture 6 7 March 005 Lecture Outline. Invariant Ellipe Generated by a Unimodular
More informationPhysics 741 Graduate Quantum Mechanics 1 Solutions to Final Exam, Fall 2014
Phyic 7 Graduate Quantum Mechanic Solution to inal Eam all 0 Each quetion i worth 5 point with point for each part marked eparately Some poibly ueful formula appear at the end of the tet In four dimenion
More information2015 PhysicsBowl Solutions Ans Ans Ans Ans Ans B 2. C METHOD #1: METHOD #2: 3. A 4.
05 PhyicBowl Solution # An # An # An # An # An B B B 3 D 4 A C D A 3 D 4 C 3 A 3 C 3 A 33 C 43 B 4 B 4 D 4 C 34 A 44 E 5 E 5 E 5 E 35 E 45 B 6 D 6 A 6 A 36 B 46 E 7 A 7 D 7 D 37 A 47 C 8 E 8 C 8 B 38 D
More informationFinal Comprehensive Exam Physical Mechanics Friday December 15, Total 100 Points Time to complete the test: 120 minutes
Final Comprehenive Exam Phyical Mechanic Friday December 15, 000 Total 100 Point Time to complete the tet: 10 minute Pleae Read the Quetion Carefully and Be Sure to Anwer All Part! In cae that you have
More informationLinear Motion, Speed & Velocity
Add Important Linear Motion, Speed & Velocity Page: 136 Linear Motion, Speed & Velocity NGSS Standard: N/A MA Curriculum Framework (006): 1.1, 1. AP Phyic 1 Learning Objective: 3.A.1.1, 3.A.1.3 Knowledge/Undertanding
More informationTHE EXPERIMENTAL PERFORMANCE OF A NONLINEAR DYNAMIC VIBRATION ABSORBER
Proceeding of IMAC XXXI Conference & Expoition on Structural Dynamic February -4 Garden Grove CA USA THE EXPERIMENTAL PERFORMANCE OF A NONLINEAR DYNAMIC VIBRATION ABSORBER Yung-Sheng Hu Neil S Ferguon
More informationThe Influence of the Load Condition upon the Radial Distribution of Electromagnetic Vibration and Noise in a Three-Phase Squirrel-Cage Induction Motor
The Influence of the Load Condition upon the Radial Ditribution of Electromagnetic Vibration and Noie in a Three-Phae Squirrel-Cage Induction Motor Yuta Sato 1, Iao Hirotuka 1, Kazuo Tuboi 1, Maanori Nakamura
More informationChapter 2 Sampling and Quantization. In order to investigate sampling and quantization, the difference between analog
Chapter Sampling and Quantization.1 Analog and Digital Signal In order to invetigate ampling and quantization, the difference between analog and digital ignal mut be undertood. Analog ignal conit of continuou
More informationFinite Element Truss Problem
6. rue Uing FEA Finite Element ru Problem We tarted thi erie of lecture looking at tru problem. We limited the dicuion to tatically determinate tructure and olved for the force in element and reaction
More informationSynchrotorn Motion. A review:
A review: H e cm c Synchrotorn Motion ( p ea ( x / ( p x ea x ( p z ea / z The phae pace coordinate are (x,,z with independent coordinate t. In one revolution, the time advance T, called the orbital period.
More informationOn the Isentropic Forchheimer s Sound Waves Propagation in a Cylindrical Tube Filled with a Porous Media
5th WSEAS Int. Conf. on FLUID MECHANICS (FLUIDS') Acapulco, Mexico, January 5-7, On the Ientropic Forchheimer Sound Wave Propagation in a Cylindrical Tube Filled with a Porou Media H. M. Dwairi Civil Engineering
More informationUNITS FOR THERMOMECHANICS
UNITS FOR THERMOMECHANICS 1. Conitent Unit. Every calculation require a conitent et of unit. Hitorically, one et of unit wa ued for mechanic and an apparently unrelated et of unit wa ued for heat. For
More informationPhysics 6A. Angular Momentum. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB
Phyic 6A Angular Momentum For Campu earning Angular Momentum Thi i the rotational equivalent of linear momentum. t quantifie the momentum of a rotating object, or ytem of object. f we imply tranlate the
More informationMODELLING OF FRICTIONAL SOIL DAMPING IN FINITE ELEMENT ANALYSIS
MODELLING OF FRICTIONAL SOIL DAMPING IN FINITE ELEMENT ANALYSIS S. VAN BAARS Department of Science, Technology and Communication, Univerity of Luxembourg, Luxembourg ABSTRACT: In oil dynamic, the oil i
More informationEuler-Bernoulli Beams
Euler-Bernoulli Beam The Euler-Bernoulli beam theory wa etablihed around 750 with contribution from Leonard Euler and Daniel Bernoulli. Bernoulli provided an expreion for the train energy in beam bending,
More informationBernoulli s equation may be developed as a special form of the momentum or energy equation.
BERNOULLI S EQUATION Bernoulli equation may be developed a a pecial form of the momentum or energy equation. Here, we will develop it a pecial cae of momentum equation. Conider a teady incompreible flow
More informationPressure distribution in a fluid:
18/01/2016 LECTURE 5 Preure ditribution in a fluid: There are many intance where the fluid i in tationary condition. That i the movement of liquid (or ga) i not involved. Yet, we have to olve ome engineering
More informationModeling of Transport and Reaction in a Catalytic Bed Using a Catalyst Particle Model.
Excerpt from the Proceeding of the COMSOL Conference 2010 Boton Modeling of Tranport and Reaction in a Catalytic Bed Uing a Catalyt Particle Model. F. Allain *,1, A.G. Dixon 1 1 Worceter Polytechnic Intitute
More informationc = ω k = 1 v = ω k = 1 ²µ
3. Electromagtic Wave 3.5. Plama wave A plama i an ionized ga coniting of charged particle (e.g., electron and ion). Variou wave can be excited eaily in a plama. Wave phenomena have been an important ubject
More informationMAE 101A. Homework 3 Solutions 2/5/2018
MAE 101A Homework 3 Solution /5/018 Munon 3.6: What preure gradient along the treamline, /d, i required to accelerate water upward in a vertical pipe at a rate of 30 ft/? What i the anwer if the flow i
More informationElastic Collisions Definition Examples Work and Energy Definition of work Examples. Physics 201: Lecture 10, Pg 1
Phyic 131: Lecture Today Agenda Elatic Colliion Definition i i Example Work and Energy Definition of work Example Phyic 201: Lecture 10, Pg 1 Elatic Colliion During an inelatic colliion of two object,
More informationIII.9. THE HYSTERESIS CYCLE OF FERROELECTRIC SUBSTANCES
III.9. THE HYSTERESIS CYCLE OF FERROELECTRIC SBSTANCES. Work purpoe The analyi of the behaviour of a ferroelectric ubtance placed in an eternal electric field; the dependence of the electrical polariation
More information84 ZHANG Jing-Shang Vol. 39 of which would emit 5 He rather than 3 He. 5 He i untable and eparated into n + pontaneouly, which can alo be treated a if
Commun. Theor. Phy. (Beijing, China) 39 (003) pp. 83{88 c International Academic Publiher Vol. 39, No. 1, January 15, 003 Theoretical Analyi of Neutron Double-Dierential Cro Section of n+ 11 B at 14. MeV
More informationEmittance limitations due to collective effects for the TOTEM beams
LHC Project ote 45 June 0, 004 Elia.Metral@cern.ch Andre.Verdier@cern.ch Emittance limitation due to collective effect for the TOTEM beam E. Métral and A. Verdier, AB-ABP, CER Keyword: TOTEM, collective
More informationSimulation and Analysis of Linear Permanent Magnet Vernier Motors for Direct Drive Systems
Available online at www.ijpe-online.com vol. 3, no. 8, December 07, pp. 304-3 DOI: 0.3940/ijpe.7.08.p.3043 Simulation and Analyi of Linear Permanent Magnet Vernier Motor for Direct Drive Sytem Mingjie
More informationMechanics. Free rotational oscillations. LD Physics Leaflets P Measuring with a hand-held stop-clock. Oscillations Torsion pendulum
Mechanic Ocillation Torion pendulum LD Phyic Leaflet P.5.. Free rotational ocillation Meauring with a hand-held top-clock Object of the experiment g Meauring the amplitude of rotational ocillation a function
More informationCake ltration analysis the eect of the relationship between the pore liquid pressure and the cake compressive stress
Chemical Engineering Science 56 (21) 5361 5369 www.elevier.com/locate/ce Cake ltration analyi the eect of the relationhip between the pore liquid preure and the cake compreive tre C. Tien, S. K. Teoh,
More informationA novel protocol for linearization of the Poisson-Boltzmann equation
Ann. Univ. Sofia, Fac. Chem. Pharm. 16 (14) 59-64 [arxiv 141.118] A novel protocol for linearization of the Poion-Boltzmann equation Roumen Tekov Department of Phyical Chemitry, Univerity of Sofia, 1164
More informationBUBBLES RISING IN AN INCLINED TWO-DIMENSIONAL TUBE AND JETS FALLING ALONG A WALL
J. Autral. Math. Soc. Ser. B 4(999), 332 349 BUBBLES RISING IN AN INCLINED TWO-DIMENSIONAL TUBE AND JETS FALLING ALONG A WALL J. LEE and J.-M. VANDEN-BROECK 2 (Received 22 April 995; revied 23 April 996)
More informationSMALL-SIGNAL STABILITY ASSESSMENT OF THE EUROPEAN POWER SYSTEM BASED ON ADVANCED NEURAL NETWORK METHOD
SMALL-SIGNAL STABILITY ASSESSMENT OF THE EUROPEAN POWER SYSTEM BASED ON ADVANCED NEURAL NETWORK METHOD S.P. Teeuwen, I. Erlich U. Bachmann Univerity of Duiburg, Germany Department of Electrical Power Sytem
More informationTRAVELING WAVES. Chapter Simple Wave Motion. Waves in which the disturbance is parallel to the direction of propagation are called the
Chapte 15 RAVELING WAVES 15.1 Simple Wave Motion Wave in which the ditubance i pependicula to the diection of popagation ae called the tanvee wave. Wave in which the ditubance i paallel to the diection
More informationWake Field. Impedances: gz j
Collective beam Intability in Accelerator A circulating charged particle beam reemble an electric circuit, where the impedance play an important role in determining the circulating current. Liewie, the
More informationDesign By Emulation (Indirect Method)
Deign By Emulation (Indirect Method he baic trategy here i, that Given a continuou tranfer function, it i required to find the bet dicrete equivalent uch that the ignal produced by paing an input ignal
More informationFLEXOELECTRIC SIGNALS ON RINGS IN TRANSVERSE MOTIONS
Proceeding of the ASME 011 International Deign Engineering Technical Conference & Computer and Information in Engineering Conference IDETC/CIE 011 Augut 8-31, 011, Wahington, DC, USA DETC011-4819 FLEXOELECTRIC
More informationMolecular Dynamics Simulations of Nonequilibrium Effects Associated with Thermally Activated Exothermic Reactions
Original Paper orma, 5, 9 7, Molecular Dynamic Simulation of Nonequilibrium Effect ociated with Thermally ctivated Exothermic Reaction Jerzy GORECKI and Joanna Natalia GORECK Intitute of Phyical Chemitry,
More informationLateral vibration of footbridges under crowd-loading: Continuous crowd modeling approach
ateral vibration of footbridge under crowd-loading: Continuou crowd modeling approach Joanna Bodgi, a, Silvano Erlicher,b and Pierre Argoul,c Intitut NAVIER, ENPC, 6 et 8 av. B. Pacal, Cité Decarte, Champ
More informationDetermination of the local contrast of interference fringe patterns using continuous wavelet transform
Determination of the local contrat of interference fringe pattern uing continuou wavelet tranform Jong Kwang Hyok, Kim Chol Su Intitute of Optic, Department of Phyic, Kim Il Sung Univerity, Pyongyang,
More informationAP Physics Charge Wrap up
AP Phyic Charge Wrap up Quite a few complicated euation for you to play with in thi unit. Here them babie i: F 1 4 0 1 r Thi i good old Coulomb law. You ue it to calculate the force exerted 1 by two charge
More informationPHYS 110B - HW #6 Spring 2004, Solutions by David Pace Any referenced equations are from Griffiths Problem statements are paraphrased
PHYS B - HW #6 Spring 4, Solution by David Pace Any referenced equation are from Griffith Problem tatement are paraphraed. Problem. from Griffith Show that the following, A µo ɛ o A V + A ρ ɛ o Eq..4 A
More informationBasics of a Quartz Crystal Microbalance
Baic of a Quartz Crytal Microbalance Introduction Thi document provide an introduction to the quartz crytal microbalance (QCM) which i an intrument that allow a uer to monitor mall ma change on an electrode.
More information1. Intensity of Periodic Sound Waves 2. The Doppler Effect
1. Intenity o Periodic Sound Wae. The Doppler Eect 1-4-018 1 Objectie: The tudent will be able to Deine the intenity o the ound wae. Deine the Doppler Eect. Undertand ome application on ound 1-4-018 3.3
More informationDimension Effect on Dynamic Stress Equilibrium in SHPB Tests
International Journal of Material Phyic. ISSN 97-39X Volume 5, Numer 1 (1), pp. 15- International Reearch Pulication Houe http://www.irphoue.com Dimenion Effect on Dynamic Stre Equilirium in SHPB Tet Department
More informationTHE THERMOELASTIC SQUARE
HE HERMOELASIC SQUARE A mnemonic for remembering thermodynamic identitie he tate of a material i the collection of variable uch a tre, train, temperature, entropy. A variable i a tate variable if it integral
More informationA FUNCTIONAL BAYESIAN METHOD FOR THE SOLUTION OF INVERSE PROBLEMS WITH SPATIO-TEMPORAL PARAMETERS AUTHORS: CORRESPONDENCE: ABSTRACT
A FUNCTIONAL BAYESIAN METHOD FOR THE SOLUTION OF INVERSE PROBLEMS WITH SPATIO-TEMPORAL PARAMETERS AUTHORS: Zenon Medina-Cetina International Centre for Geohazard / Norwegian Geotechnical Intitute Roger
More informationDimensional Analysis A Tool for Guiding Mathematical Calculations
Dimenional Analyi A Tool for Guiding Mathematical Calculation Dougla A. Kerr Iue 1 February 6, 2010 ABSTRACT AND INTRODUCTION In converting quantitie from one unit to another, we may know the applicable
More informationQuestion 1 Equivalent Circuits
MAE 40 inear ircuit Fall 2007 Final Intruction ) Thi exam i open book You may ue whatever written material you chooe, including your cla note and textbook You may ue a hand calculator with no communication
More informationDomain Optimization Analysis in Linear Elastic Problems * (Approach Using Traction Method)
Domain Optimization Analyi in Linear Elatic Problem * (Approach Uing Traction Method) Hideyuki AZEGAMI * and Zhi Chang WU *2 We preent a numerical analyi and reult uing the traction method for optimizing
More informationIsentropic Sound Waves Propagation in a Tube Filled with a Porous Media
INTERNATIONAL JOURNAL OF ECHANICS Ientropic Sound Wave Propagation in a Tube Filled with a Porou edia H.. Duwairi Abtract A rigid frame, cylindrical capillary theory of ound propagation in porou media
More informationBACKSCATTER FROM A SPHERICAL INCLUSION WITH COMPLIANT INTERPHASE CHARACTERISTICS. M. Kitahara Tokai University Shimizu, Shizuoka 424, Japan
BACKSCATTER FROM A SPHERICAL INCLUSION WITH COMPLIANT INTERPHASE CHARACTERISTICS M. Kitahara Tokai Univerity Shimizu, Shizuoka 424, Japan K. Nakagawa Total Sytem Intitute Shinjuku, Tokyo 162, Japan J.
More informationStability. ME 344/144L Prof. R.G. Longoria Dynamic Systems and Controls/Lab. Department of Mechanical Engineering The University of Texas at Austin
Stability The tability of a ytem refer to it ability or tendency to eek a condition of tatic equilibrium after it ha been diturbed. If given a mall perturbation from the equilibrium, it i table if it return.
More informationLecture 10 Filtering: Applied Concepts
Lecture Filtering: Applied Concept In the previou two lecture, you have learned about finite-impule-repone (FIR) and infinite-impule-repone (IIR) filter. In thee lecture, we introduced the concept of filtering
More informationTime [seconds]
.003 Fall 1999 Solution of Homework Aignment 4 1. Due to the application of a 1.0 Newton tep-force, the ytem ocillate at it damped natural frequency! d about the new equilibrium poition y k =. From the
More informationChapter 5 Consistency, Zero Stability, and the Dahlquist Equivalence Theorem
Chapter 5 Conitency, Zero Stability, and the Dahlquit Equivalence Theorem In Chapter 2 we dicued convergence of numerical method and gave an experimental method for finding the rate of convergence (aka,
More informationElectromechanical Dynamics for Micro Film
Senor & Tranducer, Vol. 76, Iue 8, Augut 04, pp. 99-06 Senor & Tranducer 04 by IFSA Publihing, S. L. http://www.enorportal.com Electromechanical Dynamic for Micro Film Lizhong Xu, Xiaorui Fu Mechanical
More informationFUNDAMENTALS OF POWER SYSTEMS
1 FUNDAMENTALS OF POWER SYSTEMS 1 Chapter FUNDAMENTALS OF POWER SYSTEMS INTRODUCTION The three baic element of electrical engineering are reitor, inductor and capacitor. The reitor conume ohmic or diipative
More informationUnit I Review Worksheet Key
Unit I Review Workheet Key 1. Which of the following tatement about vector and calar are TRUE? Anwer: CD a. Fale - Thi would never be the cae. Vector imply are direction-conciou, path-independent quantitie
More informationSound waves. Content. Chapter 21. objectives. objectives. When we use Sound Waves. What are sound waves? How they work.
Chapter 21. Sound wae Content 21.1 Propagation o ound wae 21.2 Source o ound 21.3 Intenity o ound 21.4 Beat 21.5 Doppler eect 1 2 objectie a) explain the propagation o ound wae in air in term o preure
More informationTo appear in International Journal of Numerical Methods in Fluids in Stability analysis of numerical interface conditions in uid-structure therm
To appear in International Journal of Numerical Method in Fluid in 997. Stability analyi of numerical interface condition in uid-tructure thermal analyi M. B. Gile Oxford Univerity Computing Laboratory
More informationSECTION x2 x > 0, t > 0, (8.19a)
SECTION 8.5 433 8.5 Application of aplace Tranform to Partial Differential Equation In Section 8.2 and 8.3 we illutrated the effective ue of aplace tranform in olving ordinary differential equation. The
More informationCHAPTER 3 LITERATURE REVIEW ON LIQUEFACTION ANALYSIS OF GROUND REINFORCEMENT SYSTEM
CHAPTER 3 LITERATURE REVIEW ON LIQUEFACTION ANALYSIS OF GROUND REINFORCEMENT SYSTEM 3.1 The Simplified Procedure for Liquefaction Evaluation The Simplified Procedure wa firt propoed by Seed and Idri (1971).
More informationPHYS 110B - HW #2 Spring 2004, Solutions by David Pace Any referenced equations are from Griffiths Problem statements are paraphrased
PHYS 11B - HW # Spring 4, Solution by David Pace Any referenced equation are from Griffith Problem tatement are paraphraed [1.] Problem 7. from Griffith A capacitor capacitance, C i charged to potential
More informationFinite Element Analysis of a Fiber Bragg Grating Accelerometer for Performance Optimization
Finite Element Analyi of a Fiber Bragg Grating Accelerometer for Performance Optimization N. Baumallick*, P. Biwa, K. Dagupta and S. Bandyopadhyay Fiber Optic Laboratory, Central Gla and Ceramic Reearch
More informationPhysics 41 Homework Set 3 Chapter 17 Serway 7 th Edition
Pyic 41 Homework Set 3 Capter 17 Serway 7 t Edition Q: 1, 4, 5, 6, 9, 1, 14, 15 Quetion *Q17.1 Anwer. Te typically iger denity would by itelf make te peed of ound lower in a olid compared to a ga. Q17.4
More informationPotential energy of a spring
PHYS 7: Modern Mechanic Spring 0 Homework: It i expected that a tudent work on a a homework #x hortly after lecture #x, ince HWx i on material of LECx. While the due date for HW are typically et to about
More informationBogoliubov Transformation in Classical Mechanics
Bogoliubov Tranformation in Claical Mechanic Canonical Tranformation Suppoe we have a et of complex canonical variable, {a j }, and would like to conider another et of variable, {b }, b b ({a j }). How
More informationCritical behavior of slider-block model. (Short title: Critical ) S G Abaimov
Critical behavior of lider-bloc model (Short title: Critical ) S G Abaimov E-mail: gabaimov@gmail.com. Abtract. Thi paper applie the theory of continuou phae tranition of tatitical mechanic to a lider-bloc
More informationDynamic response of a double Euler Bernoulli beam due to a moving constant load
Journal of Sound and Vibration 297 (26) 477 49 JOURNAL OF SOUND AND VIBRATION www.elevier.com/locate/jvi Dynamic repone of a double Euler Bernoulli beam due to a moving contant load M. Abu-Hilal Department
More informationOnline Appendix for Managerial Attention and Worker Performance by Marina Halac and Andrea Prat
Online Appendix for Managerial Attention and Worker Performance by Marina Halac and Andrea Prat Thi Online Appendix contain the proof of our reult for the undicounted limit dicued in Section 2 of the paper,
More informationEffects of vector attenuation on AVO of offshore reflections
GEOPHYSICS, VOL. 64, NO. 3 MAY-JUNE 1999); P. 815 819, 9 FIGS., 1 TABLE. Effect of vector attenuation on AVO of offhore reflection J. M. Carcione ABSTRACT Wave tranmitted at the ocean bottom have the characteritic
More informationSolution to Theoretical Question 1. A Swing with a Falling Weight. (A1) (b) Relative to O, Q moves on a circle of radius R with angular velocity θ, so
Solution to Theoretical uetion art Swing with a Falling Weight (a Since the length of the tring Hence we have i contant, it rate of change ut be zero 0 ( (b elative to, ove on a circle of radiu with angular
More informationMidterm Review - Part 1
Honor Phyic Fall, 2016 Midterm Review - Part 1 Name: Mr. Leonard Intruction: Complete the following workheet. SHOW ALL OF YOUR WORK. 1. Determine whether each tatement i True or Fale. If the tatement i
More informationLecture Notes II. As the reactor is well-mixed, the outlet stream concentration and temperature are identical with those in the tank.
Lecture Note II Example 6 Continuou Stirred-Tank Reactor (CSTR) Chemical reactor together with ma tranfer procee contitute an important part of chemical technologie. From a control point of view, reactor
More informationMath 273 Solutions to Review Problems for Exam 1
Math 7 Solution to Review Problem for Exam True or Fale? Circle ONE anwer for each Hint: For effective tudy, explain why if true and give a counterexample if fale (a) T or F : If a b and b c, then a c
More informationFirst Principles Derivation of Differential Equations of Equilibrium of Anisotropic Rectangular Thin Plates on Elastic Foundations
Journal of Geotechnical and Tranportation Engineering Volume 4 Iue 1 Firt Principle Derivation of Differential Equation of Equilibrium of Aniotropic Rectangular Thin Plate... Ike Received 5/12/2018 Accepted
More informationA Group Theoretic Approach to Generalized Harmonic Vibrations in a One Dimensional Lattice
Virginia Commonwealth Univerity VCU Scholar Compa Mathematic and Applied Mathematic Publication Dept. of Mathematic and Applied Mathematic 984 A Group Theoretic Approach to Generalized Harmonic Vibration
More informationAt the end of this lesson, the students should be able to understand:
Intructional Objective: At the end of thi leon, the tudent hould be able to undertand: Baic failure mechanim of riveted joint. Concept of deign of a riveted joint. 1. Strength of riveted joint: Strength
More informationPhysics 2212 G Quiz #2 Solutions Spring 2018
Phyic 2212 G Quiz #2 Solution Spring 2018 I. (16 point) A hollow inulating phere ha uniform volume charge denity ρ, inner radiu R, and outer radiu 3R. Find the magnitude of the electric field at a ditance
More informationFRTN10 Exercise 3. Specifications and Disturbance Models
FRTN0 Exercie 3. Specification and Diturbance Model 3. A feedback ytem i hown in Figure 3., in which a firt-order proce if controlled by an I controller. d v r u 2 z C() P() y n Figure 3. Sytem in Problem
More informationNumerical algorithm for the analysis of linear and nonlinear microstructure fibres
Numerical algorithm for the anali of linear and nonlinear microtructure fibre Mariuz Zdanowicz *, Marian Marciniak, Marek Jaworki, Igor A. Goncharenko National Intitute of Telecommunication, Department
More informationFALL TERM EXAM, PHYS 1211, INTRODUCTORY PHYSICS I Saturday, 14 December 2013, 1PM to 4 PM, AT 1003
FALL TERM EXAM, PHYS 111, INTRODUCTORY PHYSICS I Saturday, 14 December 013, 1PM to 4 PM, AT 1003 NAME: STUDENT ID: INSTRUCTION 1. Thi exam booklet ha 14 page. Make ure none are miing. There i an equation
More informationMacromechanical Analysis of a Lamina
3, P. Joyce Macromechanical Analyi of a Lamina Generalized Hooke Law ij Cijklε ij C ijkl i a 9 9 matri! 3, P. Joyce Hooke Law Aume linear elatic behavior mall deformation ε Uniaial loading 3, P. Joyce
More information