DOCSLIB.ORG

The Wave Equation the Method of Characteristics Inclusion Of

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
The Wave Equation the Method of Characteristics Inclusion Of The Wave Equation ♦ The method of characteristics ♦ Inclusion of boundary conditions ♦ Traveling waves and stationary waves TRANSVERSE OSCILLATIONS OF AN ELASTIC STRING T uniform linear density ρ y 2 θ 2 small displacements y θ T 1 1 • T1 cos θ1 = T2 cos θ2 x for small θ, cos θ ≃ 1 ⇒ T1 = T2 = T ∂2y • ρ δx = T sin θ − T sin θ ∂t2 2 1 ∂y ∂2y ∂y ∂y sin θ ≃ tan θ ≃ =⇒ ρ δx = T ( ) − ( ) ∂x ∂t2 ∂x 2 ∂x 1 (∂2y/∂x2)δx+... | {z } ∂2y ∂2y Thus ρ = T ∂t2 ∂x2 ∂2y 1 ∂2y T i.e., = , c2 ≡ wave equation ∂x2 c2 ∂t2 ρ 2nd-order linear PDEs A(x, y)uxx +2B(x, y)uxy +C(x, y)uyy +D(x, y)ux +E(x, y)uy +F (x, y)u = R(x, y) 2 • B − AC < 0 elliptic. Ex.: Laplace eqn. uxx + uyy = 0 2 • B − AC = 0 parabolic. Ex.: diffusion eqn. ut − αuxx = 0 2 2 • B − AC > 0 hyperbolic. Ex.: wave eqn. utt − c uxx = 0 ♦ General solutions of PDEs depend on arbitrary functions (analogous to solutions of ODEs depending on arbitrary constants) −→ boundary conditions to determine such functions • Cauchy boundary conditions: assign function u and normal derivative ∂u/∂n on given curve γ in xy plane (relevant to hyperbolic PDEs) !2 y 1 !2 y D’Alembert’s solution 2 = 2 2 !x c !t Change variables u = x ! ct v = x + ct ! !u ! !v ! ! ! !x = !x !u + !x !v = !u + !v ! !u ! !v ! ! ! " c c !t = !t !u + !t !v = !u + !v 2 2 2 !2 y 1 !2 y ! ! % 1 ! ! % ! ! % = " + y = (c + c y = ( y !x2 c2 !t 2 $ u v$ c2 $ u v$ $ u v$ " ! ! % " ! ! % " ! ! % "2 y ! = 0 f, g arbitrary functions "u"v General solution y() u,v = f() u + g() v i.e. y() x,t = f() x ! ct + g() x + ct • The curves in the xt plane x − ct = const. x + ct = const. are called the characteristics of the wave equation. t x−ct=const. x x+ct=const. CHARACTERISTICS Consider the PDE Aytt + 2Bytx + Cyxx + Dyt + Eyx + F y = R A B Q = matrix of 2nd − order coefficients B C • Characteristics of the above PDE are defined as the curves χ(t, x) = const. such that their normal n is rotated by 90◦ by Q or is annihilated by Q, i.e., n · Q n = 0. • Since n ∝ ∇χ, for characteristics ∇χ · Q ∇χ = 0. 1 −1/c2 0 ⊲ Example. Wave eqn. : − ytt + yxx = 0 ⇒ Q = c2 0 1 The curves χ∓(t, x)= x ∓ ct =const. are the characteristics of the wave equation because ∓c −1/c2 0 ∓c ±1/c ∇χ∓ = ⇒ ∇χ · Q ∇χ =( ∓c 1) =( ∓c 1) = 0 1 0 1 1 1 • The condition ∇χ · Q ∇χ = 0 implies, using ∇χ =(χt,χx), that A B χt ( χt χx ) = 0 , B C χx 2 2 i.e., Aχt + 2Bχtχx + Cχx = 0 . ′ Expressing the derivatives in terms of x (t)= −χt/χx, A[x′(t)]2 − 2Bx′(t)+ C = 0 . ♠ Hyperbolic eqns. (B2 − AC > 0) have 2 families of characteristics ♠ Parabolic eqns. (B2 − AC = 0) have 1 (Q ∇χ = 0) ♠ Elliptic eqns. (B2 − AC < 0) have none Uses of characteristics • Characteristics χ∓(t, x)= const. can be used to solve hyperbolic equations by means of the transformation of variables u = χ−(t, x) v = χ+(t, x) ⊲ Example: D’Alembert solution of the wave equation • Characteristics serve to analyze whether boundary value problems for PDEs are well posed. ⊲ Example: Cauchy conditions on curve γ well-defined provided γ is not a characteristic [Cauchy-Kovalevska theorem] Cauchy boundary conditions and characteristics Consider the PDE Aytt + 2Bytx + Cyxx = H(yt, yx, y, t, x) • Cauchy conditions: Suppose y and the normal derivative yn are assigned on the curve γ specified by G(t, x) = 0 ⊲ The normal and tangential directions to γ are n ∝∇G =(Gt,Gx), τ ∝ (−Gx,Gt). ⊲ Given y on γ, we can compute tangential derivative yτ . From yn and yτ we can get yt and yx. • Can we determine higher derivatives as well? ∂ ytt yt = τ · ∇ yt ∝ ( −Gx Gt ) = −Gxytt + Gtyxt ∂τ yxt ∂ ytx yx = τ · ∇ yx ∝ ( −Gx Gt ) = −Gxytx + Gtyxx ∂τ yxx ⇒ 3 linear equations in ytt, ytx, yxx, with unique solution if det =6 0 A 2B C det −Gx Gt 0 =6 0 − 0 Gx Gt 2 2 ⇒ AGt + 2BGtGx + CGx =6 0 That is, Cauchy conditions on curve γ are well-defined provided γ is not a characteristic .#$%$'%().(+(,$,.,/+ !2 !2 y 1 y ! , = " + + 2 = 2 2 y() x t f() x ct g() x ct !x c !t f and g are determined by the initial conditions: Suppose at time t = 0, the wave has an initial displacement U(x) and an initial velocity V (x) y() x,0 = f() x + g() x = U() x x y() x,0 1 = "cf#() x + cg#() x = V() x $ f() x " g() x = " V() x' dx ' t c % b x 1 1 f() x = U() x ! V() x' dx ' 2 2c " b x 1 1 g() x = U() x + V() x' dx ' 2 2c ! b " x+ct x!ct $ x+ct 1 1 ( 1 1 y() x,t = "U() x ! ct + U() x + ct $ + V() x dx ! V() x dx = "U() x ! ct + U() x + ct $ + V() x dx 2 # % 2c " " ( 2 # % 2c " # b b %( x!ct !69$ &4($5(0,$(-(.&/$.(10&-23/&0$4(5-/&1(7(-0$.(/(&5(4$8.,7$.(508$$ V(x) = 0 ' x+ct % 1 1 1 " $ y x,t = "U x ct + U x + ct ( + V x dx , y() x,t = U() x ! ct + U() x + ct () # ()! ()) () 2 % 2 2c $ , ( x!ct ' U() x y x,t = f x ! ct + g x + ct () () () What is the form of f (x), g(x)? y() x,t = f() x ! ct + g() x + ct If time dependence is cos ()!t the full ()x,t dependence is given by y(x,t)=Acos ()kx ! !t + Bcos ()kx " !t ! ""$%&"%'( ' • Speed of wave c = k 2" 1 y(x,t) ! = Frequency f = = • ! 2# t " 2" 2 ( , ) ! • Wavelength ! = y x t = k k x k is "wavenumber" We can write the equation of a travelling wave in a number of analogous forms: Velocity Wavelength Period Angular frequency Asin () kx"" t " / k 2 / k 2 /" " Asin k() x" vt v 2 / k 2 / vk vk &,()x t $/ % $ % 2 /% Asin,- 2 $" * 0./$ % . v $ $ / v 2 v / $ Asin0&, 2 () x" vt / $ . N.B. Can include phase most easily by putting = &, y()() x, t Re0 A exp 0&, i kx"" t . where A is complex. N.B.2 Sometimes more convenient to switch x and t, i.e. y()() x, t= A sin " t" kx This is still a travelling wave moving to the right. For non-sinusoidal wave moving to right with speed v, can always write as f() x! vt . y() t )#$%%#'.)-#,,-) y = Asin()kx ! "t + Asin()kx + "t 2 sin cos " = A kx t y() t + !t 1-)#$%%#'.)-#,,1) .%',)/,%,'#00.) sin ! 2# sin 2# y = A ()kx t + 1 + A ()kx + t + 2 2 sin # # cos # ! # = A ()kx + 1 + 2 ()t + 1 2 !$"$%('( *&",$-. -"0- † !" !.E = 0 !"10%%%3()%).",-.()401%%)(2"3%5) !" !.B = 0 !" !" B !" E # = t !" 0 !" 1 E !" B = !" !" !" c2 t 2 " " E = # E + () .E !" !" !" B " # B 1 2 E !" # = ! = ! t t c2 t 2 !" !" 1 "2 E !2 E = c2 "t 2 !" !" EM plane wave E = E()z !" !" † !" !2 E 1 !2 E !E = = 0 " 0 ! 2 2 ! 2 ! Ez = z c t z 1".(6%1(%)0"6%) !$" %&' ($)* sin Ex = A ()kx ! t sin Ey = B ()kx ! t + # E x E y.
Load more

Recommended publications
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • Phys 234 Vector Calculus and Maxwell's Equations Prof. Alex
    Prof. Alex Small Phys 234 Vector Calculus and Maxwell's Equationsarsmall@csupomona.edu 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).
    [Show full text]
  • 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
    [Show full text]
  • 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.
    [Show full text]
  • 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 ).
    [Show full text]
  • 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> .
    [Show full text]
  • 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 dherrin@engr.uky.edu 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.
    [Show full text]
  • Analysis and Computation of the Elastic Wave Equation with Random Coefficients
    Computers and Mathematics with Applications 70 (2015) 2454–2473 Contents lists available at ScienceDirect Computers and Mathematics with Applications journal homepage: www.elsevier.com/locate/camwa Analysis and computation of the elastic wave equation with random coefficients Mohammad Motamed a,∗, Fabio Nobile b, Raúl Tempone c a Department of Mathematics and Statistics, The University of New Mexico, Albuquerque, NM 87131, USA b MATHICSE-CSQI, École Polytechnique Fédérale de Lausanne, Lausanne, Switzerland c King Abdullah University of Science and Technology, Jeddah, Saudi Arabia article info a b s t r a c t Article history: We consider the stochastic initial–boundary value problem for the elastic wave equation Received 23 August 2014 with random coefficients and deterministic data. We propose a stochastic collocation Received in revised form 6 September 2015 method for computing statistical moments of the solution or statistics of some given Accepted 12 September 2015 quantities of interest. We study the convergence rate of the error in the stochastic Available online 9 October 2015 collocation method. In particular, we show that, the rate of convergence depends on the regularity of the solution or the quantity of interest in the stochastic space, which is in Keywords: turn related to the regularity of the deterministic data in the physical space and the type Uncertainty quantification Stochastic partial differential equations of the quantity of interest. We demonstrate that a fast rate of convergence is possible in Elastic wave equation two cases: for the elastic wave solutions with high regular data; and for some high regular Collocation method quantities of interest even in the presence of low regular data.
    [Show full text]
  • Simple Derivation of Electromagnetic Waves from Maxwell's Equations
    Simple Derivation of Electromagnetic Waves from Maxwell’s Equations By Lynda Williams, Santa Rosa Junior College Physics Department Assume that the electric and magnetic fields are constrained to the y and z directions, respectfully, and that they are both functions of only x and t. This will result in a linearly polarized plane wave travelling in the x direction at the speed of light c. Ext( , ) Extj ( , )ˆ and Bxt( , ) Bxtk ( , ) ˆ We will derive the wave equation from Maxwell’s Equations in free space where I and Q are both zero. EB 0 0 BE EB με tt00 iˆˆ j kˆ E Start with Faraday’s Law. Take the curl of the E field: E( x , t )ˆ j = kˆ x y z x 0E ( x , t ) 0 EB Equating magnitudes in Faraday’s Law: (1) xt This means that the spatial variation of the electric field gives rise to a time-varying magnetic field, and visa-versa. In a similar fashion we derive a second equation from Ampere-Maxwell’s Law: iˆˆ j kˆ BBE Take the curl of B: B( x , t ) kˆ = ˆ j (2) x y z x x00 t 0 0B ( x , t ) This means that the spatial variation of the magnetic field gives rise to a time-varying electric field, and visa-versa. Take the partial derivative of equation (1) with respect to x and combining the results from (2): 22EBBEE 22 = 0 0 0 0 x x t t x t t t 22EE (3) xt2200 22BB In a similar manner, we can derive a second equation for the magnetic field: (4) xt2200 Both equations (3) and (4) have the form of the general wave equation for a wave (,)xt traveling in 221 the x direction with speed v: .
    [Show full text]
  • On Fourier Transforms and Delta Functions
    CHAPTER 3 On Fourier Transforms and Delta Functions The Fourier transform of a function (for example, a function of time or space) provides a way to analyse the function in terms of its sinusoidal components of different wavelengths. The function itself is a sum of such components. The Dirac delta function is a highly localized function which is zero almost everywhere. There is a sense in which different sinusoids are orthogonal. The orthogonality can be expressed in terms of Dirac delta functions. In this chapter we review the properties of Fourier transforms, the orthogonality of sinusoids, and the properties of Dirac delta functions, in a way that draws many analogies with ordinary vectors and the orthogonality of vectors that are parallel to different coordinate axes. 3.1 Basic Analogies If A is an ordinary three-dimensional spatial vector, then the component of A in each of the three coordinate directions specified by unit vectors xˆ1, xˆ2, xˆ3 is A · xˆi for i = 1, 2, or 3. It follows that the three cartesian components (A1, A2, A3) of A are given by Ai = A · xˆi , for i = 1, 2, or 3. (3.1) We can write out the vector A as the sum of its components in each coordinate direction as follows: 3 A = Ai xˆi . (3.2) i=1 Of course the three coordinate directions are orthogonal, a property that is summarized by the equation xˆi · xˆ j = δij. (3.3) Fourier series are essentially a device to express the same basic ideas (3.1), (3.2) and (3.3), applied to a particular inner product space.
    [Show full text]