Lecture 6: Waves in Strings and Air
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Glossary Physics (I-Introduction)
1 Glossary Physics (I-introduction) - Efficiency: The percent of the work put into a machine that is converted into useful work output; = work done / energy used [-]. = eta In machines: The work output of any machine cannot exceed the work input (<=100%); in an ideal machine, where no energy is transformed into heat: work(input) = work(output), =100%. Energy: The property of a system that enables it to do work. Conservation o. E.: Energy cannot be created or destroyed; it may be transformed from one form into another, but the total amount of energy never changes. Equilibrium: The state of an object when not acted upon by a net force or net torque; an object in equilibrium may be at rest or moving at uniform velocity - not accelerating. Mechanical E.: The state of an object or system of objects for which any impressed forces cancels to zero and no acceleration occurs. Dynamic E.: Object is moving without experiencing acceleration. Static E.: Object is at rest.F Force: The influence that can cause an object to be accelerated or retarded; is always in the direction of the net force, hence a vector quantity; the four elementary forces are: Electromagnetic F.: Is an attraction or repulsion G, gravit. const.6.672E-11[Nm2/kg2] between electric charges: d, distance [m] 2 2 2 2 F = 1/(40) (q1q2/d ) [(CC/m )(Nm /C )] = [N] m,M, mass [kg] Gravitational F.: Is a mutual attraction between all masses: q, charge [As] [C] 2 2 2 2 F = GmM/d [Nm /kg kg 1/m ] = [N] 0, dielectric constant Strong F.: (nuclear force) Acts within the nuclei of atoms: 8.854E-12 [C2/Nm2] [F/m] 2 2 2 2 2 F = 1/(40) (e /d ) [(CC/m )(Nm /C )] = [N] , 3.14 [-] Weak F.: Manifests itself in special reactions among elementary e, 1.60210 E-19 [As] [C] particles, such as the reaction that occur in radioactive decay. -
Linear Elastodynamics and Waves
Linear Elastodynamics and Waves M. Destradea, G. Saccomandib aSchool of Electrical, Electronic, and Mechanical Engineering, University College Dublin, Belfield, Dublin 4, Ireland; bDipartimento di Ingegneria Industriale, Universit`adegli Studi di Perugia, 06125 Perugia, Italy. Contents 1 Introduction 3 2 Bulk waves 6 2.1 Homogeneous waves in isotropic solids . 8 2.2 Homogeneous waves in anisotropic solids . 10 2.3 Slowness surface and wavefronts . 12 2.4 Inhomogeneous waves in isotropic solids . 13 3 Surface waves 15 3.1 Shear horizontal homogeneous surface waves . 15 3.2 Rayleigh waves . 17 3.3 Love waves . 22 3.4 Other surface waves . 25 3.5 Surface waves in anisotropic media . 25 4 Interface waves 27 4.1 Stoneley waves . 27 4.2 Slip waves . 29 4.3 Scholte waves . 32 4.4 Interface waves in anisotropic solids . 32 5 Concluding remarks 33 1 Abstract We provide a simple introduction to wave propagation in the frame- work of linear elastodynamics. We discuss bulk waves in isotropic and anisotropic linear elastic materials and we survey several families of surface and interface waves. We conclude by suggesting a list of books for a more detailed study of the topic. Keywords: linear elastodynamics, anisotropy, plane homogeneous waves, bulk waves, interface waves, surface waves. 1 Introduction In elastostatics we study the equilibria of elastic solids; when its equilib- rium is disturbed, a solid is set into motion, which constitutes the subject of elastodynamics. Early efforts in the study of elastodynamics were mainly aimed at modeling seismic wave propagation. With the advent of electron- ics, many applications have been found in the industrial world. -
1 Fundamental Solutions to the Wave Equation 2 the Pulsating Sphere
1 Fundamental Solutions to the Wave Equation Physical insight in the sound generation mechanism can be gained by considering simple analytical solutions to the wave equation. One example is to consider acoustic radiation with spherical symmetry about a point ~y = fyig, which without loss of generality can be taken as the origin of coordinates. If t stands for time and ~x = fxig represent the observation point, such solutions of the wave equation, @2 ( − c2r2)φ = 0; (1) @t2 o will depend only on the r = j~x − ~yj. It is readily shown that in this case (1) can be cast in the form of a one-dimensional wave equation @2 @2 ( − c2 )(rφ) = 0: (2) @t2 o @r2 The general solution to (2) can be written as f(t − r ) g(t + r ) φ = co + co : (3) r r The functions f and g are arbitrary functions of the single variables τ = t± r , respectively. ± co They determine the pattern or the phase variation of the wave, while the factor 1=r affects only the wave magnitude and represents the spreading of the wave energy over larger surface as it propagates away from the source. The function f(t − r ) represents an outwardly co going wave propagating with the speed c . The function g(t + r ) represents an inwardly o co propagating wave propagating with the speed co. 2 The Pulsating Sphere Consider a sphere centered at the origin and having a small pulsating motion so that the equation of its surface is r = a(t) = a0 + a1(t); (4) where ja1(t)j << a0. -
Phys 234 Vector Calculus and Maxwell's Equations Prof. Alex
Prof. Alex Small Phys 234 Vector Calculus and Maxwell's [email protected] This document starts with a summary of useful facts from vector calculus, and then uses them to derive Maxwell's equations. First, definitions of vector operators. 1. Gradient Operator: The gradient operator is something that acts on a function f and produces a vector whose components are equal to derivatives of the function. It is defined by: @f @f @f rf = ^x + ^y + ^z (1) @x @y @z 2. Divergence: We can apply the gradient operator to a vector field to get a scalar function, by taking the dot product of the gradient operator and the vector function. @V @V @V r ⋅ V~ (x; y; z) = x + y + z (2) @x @y @z 3. Curl: We can take the cross product of the gradient operator with a vector field, and get another vector that involves derivatives of the vector field. @V @V @V @V @V @V r × V~ (x; y; z) = ^x z − y + ^y x − z + ^z y − x (3) @y @z @z @x @x @y 4. We can also define a second derivative of a vector function, called the Laplacian. The Laplacian can be applied to any function of x, y, and z, whether the function is a vector or a scalar. The Laplacian is defined as: @2f @2f @2f r2f(x; y; z) = r ⋅ rf = + + (4) @x2 @y2 @z2 Notes that the Laplacian of a scalar function is equal to the divergence of the gradient. Second, some useful facts about vector fields: 1. If a vector field V~ (x; y; z) has zero curl, then it can be written as the gradient of a scalar function f(x; y; z). -
No. 2990907 ACOUSTIC FILTER
July 4, 1961 W. S. EVERETT 2,990,907 ACOUSTIC FILTER Filed June 11, 1959 3 Sheets-Sheet 1 74 40 0.---.>2 .58 to 70 "1l!I.~a--34 22. /8 20 , 30 28 _~X 70 24 ~~~~~~~f;:;26 \:l Z 58 ~2B.6. 32 =- 76 y-r~.z. 77 J~ 00 7- rf ~8 /' oX. / 54 / 0 I ~' 40 7~- 58 70 8'1 July 4, 1961 W. S. EVERETT 2,990,907 ACOUSTIC FILTER Filed June 11, 1959 3 Sheets-Sheet 2 120 104~~--;18~ INVE IV TOR. 140 ILH£L~. ErR.£TT :By ~~ ATTCRN£'t, July 4, 1961 W. S. EVERETT 2,990,907 ACOUSTIC FILTER Filed June 11, 1959 3 Sheets-Sheet 3 o o No o (\,J o , .... 0 In° '\ 0;)2 I""'" I\. t- ID ....... ~ '\ \Cl ~ Q , I\. \ Z "- ", 0 I\. ~ tJ ~ ~ ~ I\. ru U) ~ " 1\ IL1 -r-J 1'\ ll. c:r 1\ r\ V'J '\ -OLiJ 1110,.J C()~v t- ~ lD 1..1 U) Z -t >- v II) Z UJ 1/ ::> (\j C!J IIJ \ J Q: I \ IJ.. I '" 1'0... I ........ II . I--.. I--~ ~ 1....- .... ~ 1/ IJ INVENTOR WILHELI'1 S. EVERETT B)/ ~t1~ ~ ATTORNEY, 2,990,907 United States Patent Office Patented July 4, 1961 1 2 acoustic spectrum without introducing any substantial 2,990,907 back pressure into the system. ACOUSTIC FILTER Wilhelm S. Everett, 1349, E. Main St., Santa P.aula, Calif. I have devised an acoustic mute which absorbs se Filed June 11, 1959, Ser. No. 819,779 lected frequencies of the acoustic speotrum, which may 6 Claims. -
Fundamentals of Acoustics Introductory Course on Multiphysics Modelling
Introduction Acoustic wave equation Sound levels Absorption of sound waves Fundamentals of Acoustics Introductory Course on Multiphysics Modelling TOMASZ G. ZIELINSKI´ bluebox.ippt.pan.pl/˜tzielins/ Institute of Fundamental Technological Research of the Polish Academy of Sciences Warsaw • Poland 3 Sound levels Sound intensity and power Decibel scales Sound pressure level 2 Acoustic wave equation Equal-loudness contours Assumptions Equation of state Continuity equation Equilibrium equation 4 Absorption of sound waves Linear wave equation Mechanisms of the The speed of sound acoustic energy dissipation Inhomogeneous wave A phenomenological equation approach to absorption Acoustic impedance The classical absorption Boundary conditions coefficient Introduction Acoustic wave equation Sound levels Absorption of sound waves Outline 1 Introduction Sound waves Acoustic variables 3 Sound levels Sound intensity and power Decibel scales Sound pressure level Equal-loudness contours 4 Absorption of sound waves Mechanisms of the acoustic energy dissipation A phenomenological approach to absorption The classical absorption coefficient Introduction Acoustic wave equation Sound levels Absorption of sound waves Outline 1 Introduction Sound waves Acoustic variables 2 Acoustic wave equation Assumptions Equation of state Continuity equation Equilibrium equation Linear wave equation The speed of sound Inhomogeneous wave equation Acoustic impedance Boundary conditions 4 Absorption of sound waves Mechanisms of the acoustic energy dissipation A phenomenological approach -
Ultrasonic Concentration of Microorganisms
University of Kentucky UKnowledge Theses and Dissertations--Biosystems and Agricultural Engineering Biosystems and Agricultural Engineering 2012 Ultrasonic Concentration of Microorganisms Samuel J. Mullins University of Kentcuky, [email protected] Right click to open a feedback form in a new tab to let us know how this document benefits ou.y Recommended Citation Mullins, Samuel J., "Ultrasonic Concentration of Microorganisms" (2012). Theses and Dissertations-- Biosystems and Agricultural Engineering. 7. https://uknowledge.uky.edu/bae_etds/7 This Master's Thesis is brought to you for free and open access by the Biosystems and Agricultural Engineering at UKnowledge. It has been accepted for inclusion in Theses and Dissertations--Biosystems and Agricultural Engineering by an authorized administrator of UKnowledge. For more information, please contact [email protected]. STUDENT AGREEMENT: I represent that my thesis or dissertation and abstract are my original work. Proper attribution has been given to all outside sources. I understand that I am solely responsible for obtaining any needed copyright permissions. I have obtained and attached hereto needed written permission statements(s) from the owner(s) of each third-party copyrighted matter to be included in my work, allowing electronic distribution (if such use is not permitted by the fair use doctrine). I hereby grant to The University of Kentucky and its agents the non-exclusive license to archive and make accessible my work in whole or in part in all forms of media, now or hereafter known. I agree that the document mentioned above may be made available immediately for worldwide access unless a preapproved embargo applies. I retain all other ownership rights to the copyright of my work. -
Elastic Wave Equation
Introduction to elastic wave equation Salam Alnabulsi University of Calgary Department of Mathematics and Statistics October 15,2012 Outline • Motivation • Elastic wave equation – Equation of motion, Definitions and The linear Stress- Strain relationship • The Seismic Wave Equation in Isotropic Media • Seismic wave equation in homogeneous media • Acoustic wave equation • Short Summary Motivation • Elastic wave equation has been widely used to describe wave propagation in an elastic medium, such as seismic waves in Earth and ultrasonic waves in human body. • Seismic waves are waves of energy that travel through the earth, and are a result of an earthquake, explosion, or a volcano. Elastic wave equation • The standard form for seismic elastic wave equation in homogeneous media is : ρu ( 2).u u ρ : is the density u : is the displacement , : Lame parameters Equation of Motion • We will depend on Newton’s second law F=ma m : mass ρdx1dx2dx3 2u a : acceleration t 2 • The total force from stress field: body F Fi Fi body Fi fidx1dx2dx3 ij F dx dx dx dx dx dx i x j 1 2 3 j ij 1 2 3 Equation of Motion • Combining these information together we get the Momentum equation (Equation of Motion) 2u 2 j f t ij i • where is the density, u is the displacement, and is the stress tensor. Definitions • Stress : A measure of the internal forces acting within a deformable body. (The force acting on a solid to deform it) The stress at any point in an object, assumed to behave as a continuum, is completely defined by nine component stresses: three orthogonal normal stresses and six orthogonal shear stresses. -
The Wave Equation (30)
GG 454 March 31, 2002 1 THE WAVE EQUATION (30) I Main Topics A Assumptions and boundary conditions used in 2-D small wave theory B The Laplace equation and fluid potential C Solution of the wave equation D Energy in a wavelength E Shoaling of waves II Assumptions and boundary conditions used in 2-D small wave theory Small amplitude surface wave y L H = 2A = wave height η x Still water u level ε v ζ Depth above Water depth = +d bottom = d+y Particle Particle orbit velocity y = -d Modified from Sorensen, 1978 A No geometry changes parallel to wave crest (2-D assumption) B Wave amplitude is small relative to wave length and water depth; it will follow that particle velocities are small relative to wave speed C Water is homogeneous, incompressible, and surface tension is nil. D The bottom is not moving, is impermeable, and is horizontal E Pressure along air-sea interface is constant F The water surface has the form of a cosine wave Hx t ηπ=−cos2 2 L T Stephen Martel 30-1 University of Hawaii GG 454 March 31, 2002 2 III The Laplace equation and fluid potential A Conservation of mass Consider a box the shape of a cube, with fluid flowing in and out or it. ∆z v+∆v (out) u+∆u (out) ∆ u y (in) ∆x v (in) The mass flow rate in the left side of the box is: ∆m ∆ρV ∆V ∆∆xyz∆ ∆x 1a 1 = = ρρ= =∆ ρyz∆ =∆∆()( ρyzu )() ∆t ∆t ∆t ∆t ∆t The mass flow rate out the right side is ∆m 1b 2 =∆∆+()(ρ yzu )(∆ u ). -
Generation of Sub-Wavelength Acoustic Stationary Waves in Microfluidic Platforms
GENERATION OF SUB-WAVELENGTH ACOUSTIC STATIONARY WAVES IN MICROFLUIDIC PLATFORMS: THEORY AND APPLICATIONS TO THE CONTROL OF MICRO-NANOPARTICLES AND BIOLOGICAL ENTITIES. A Dissertation Presented to the Faculty of the Graduate School of Cornell University in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy by Muhammet Kursad Araz February 2010 c 2010 Muhammet Kursad Araz ALL RIGHTS RESERVED GENERATION OF SUB-WAVELENGTH ACOUSTIC STATIONARY WAVES IN MICROFLUIDIC PLATFORMS: THEORY AND APPLICATIONS TO THE CONTROL OF MICRO-NANOPARTICLES AND BIOLOGICAL ENTITIES. Muhammet Kursad Araz, Ph.D. Cornell University 2010 Presented in this dissertation are the theoretical modeling and experimental re- sults of a novel method for manipulating micro and nanoparticles in an acoustically actuated microfluidic glass capillary. Here, the PZT (Lead Zirconate Titanate)- glass capillary actuator mechanism performs bioanalytical methods such as collec- tion, separation and mixing at microscale, at low voltage drives, enabling produc- tion of battery operated inexpensive portable microfluidic systems. Analytical and finite element modeling of the vibrational modes of the fluid filled thick walled cylindrical capillary has been also studied. Torsional, longitudi- nal and flexural modes and their dispersion relationships are presented. Through the excitation of the various vibrational modes of the silica capil- lary, sub-wavelength acoustic pressure modes in the microfluidic cavity are formed. More than 20 such sub-harmonic modes are generated harmonically in the 20kHz- 2MHz regime whereas naturally occurring radial modes have a cut off frequency around 9 MHz. The amplitude of these stationary acoustic pressure fields are high enough to generate nonlinear acoustic forces and streaming effects for micro and nanoparticle manipulation. -
The Acoustic Wave Equation and Simple Solutions
Chapter 5 THE ACOUSTIC WAVE EQUATION AND SIMPLE SOLUTIONS 5.1INTRODUCTION Acoustic waves constitute one kind of pressure fluctuation that can exist in a compressible fluido In addition to the audible pressure fields of modera te intensity, the most familiar, there are also ultrasonic and infrasonic waves whose frequencies lie beyond the limits of hearing, high-intensity waves (such as those near jet engines and missiles) that may produce a sensation of pain rather than sound, nonlinear waves of still higher intensities, and shock waves generated by explosions and supersonic aircraft. lnviscid fluids exhibit fewer constraints to deformations than do solids. The restoring forces responsible for propagating a wave are the pressure changes that oc cur when the fluid is compressed or expanded. Individual elements of the fluid move back and forth in the direction of the forces, producing adjacent regions of com pression and rarefaction similar to those produced by longitudinal waves in a bar. The following terminology and symbols will be used: r = equilibrium position of a fluid element r = xx + yy + zz (5.1.1) (x, y, and z are the unit vectors in the x, y, and z directions, respectively) g = particle displacement of a fluid element from its equilibrium position (5.1.2) ü = particle velocity of a fluid element (5.1.3) p = instantaneous density at (x, y, z) po = equilibrium density at (x, y, z) s = condensation at (x, y, z) 113 114 CHAPTER 5 THE ACOUSTIC WAVE EQUATION ANO SIMPLE SOLUTIONS s = (p - pO)/ pO (5.1.4) p - PO = POS = acoustic density at (x, y, Z) i1f = instantaneous pressure at (x, y, Z) i1fO = equilibrium pressure at (x, y, Z) P = acoustic pressure at (x, y, Z) (5.1.5) c = thermodynamic speed Of sound of the fluid <I> = velocity potential of the wave ü = V<I> . -
Chapter 4 – the Wave Equation and Its Solution in Gases and Liquids
Chapter 4 – The Wave Equation and its Solution in Gases and Liquids Slides to accompany lectures in Vibro-Acoustic Design in Mechanical Systems © 2012 by D. W. Herrin Department of Mechanical Engineering University of Kentucky Lexington, KY 40506-0503 Tel: 859-218-0609 [email protected] Simplifying Assumptions • The medium is homogenous and isotropic • The medium is linearly elastic • Viscous losses are negligible • Heat transfer in the medium can be ignored • Gravitational effects can be ignored • Acoustic disturbances are small Dept. of Mech. Engineering 2 University of Kentucky ME 510 Vibro-Acoustic Design Sound Pressure and Particle Velocity pt (r,t) = p0 + p(r,t) Total Pressure Undisturbed Pressure Disturbed Pressure Particle Velocity ˆ ˆ ˆ u(r,t) = uxi + uy j + uzk Dept. of Mech. Engineering 3 University of Kentucky ME 510 Vibro-Acoustic Design Density and Temperature ρt (r,t) = ρ0 + ρ (r,t) Total Density Undisturbed Density Disturbed Density Absolute Temperature T (r,t) Dept. of Mech. Engineering 4 University of Kentucky ME 510 Vibro-Acoustic Design Equation of Continuity A fluid particle is a volume element large enough to contain millions of molecules but small enough that the acoustic disturbances can be thought of as varying linearly across an element. Δy # ∂ρ &# ∂u & ρ + t Δx u + x Δx ρ u % t (% x ( x x Δz $ ∂x '$ ∂x ' Δx Dept. of Mech. Engineering 5 University of Kentucky ME 510 Vibro-Acoustic Design Equation of Continuity Δy # ∂ρt &# ∂ux & %ρt + Δx(%ux + Δx( ρ u $ x '$ x ' x x Δz ∂ ∂ Δx ∂ $ ∂ρt '$ ∂ux ' (ρtΔxΔyΔz) = ρtuxΔyΔz −&ρt + Δx)&ux + Δx)ΔyΔz ∂t % ∂x (% ∂x ( Dept.