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Group 2 Bhadouria, Arjun Singh Glave, Theodore Dean Han, Zhe Chapter 5. Laplace Transform Chapter 19. Wave Equation

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Wave Equation Chapter 19

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Overview 19.1 – Introduction – Derivation – Examples 19.2 – Separation of Variables / Vibrating String – – Solution by Separation of Variables – – Travelling Wave Interpretation 19.3 – Separation of Variables/ Vibrating Membrane 19.4 – Solution of wave equation – – d’Alembert’s solution – – Solution by integral transforms

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Introduction

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Derivation u(x, t) = vertical displacement of the string from the x axis at position x and time t θ(x, t) = angle between the string and a horizontal line at position x and time t T(x, t) = tension in the string at position x and time t ρ(x) = mass density of the string at position x

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Derivation

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Derivation Vertical Component of Motion Divide by Δx and taking the limit as Δx → 0.

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Derivation

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Derivation For small vibrations: Therefore,

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Derivation Substitute into (2) into (1)

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Derivation Horizontal Component of the Motion Divide by Δx and taking the limit as Δx → 0.

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Derivation

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Solution

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Separation of Variables; Vibrating String Solution by Separation of Variables

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Scenario

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Procedure

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Step 1 – Finding Factorized Solutions

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Step 1 – Finding Factorized Solutions

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Step 1 – Finding Factorized Solutions

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Step 1 – Finding Factorized Solutions

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Step 1 – Finding Factorized Solutions

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Step 2 – Imposition of Boundaries

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Step 2 – Imposition of Boundaries

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Step 2 – Imposition of Boundaries

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Step 2 – Imposition of Boundaries

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Step 3 – Imposition of the Initial Condition

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Step 3 – Imposition of the Initial Condition The previous expression must also satisfy the initial conditions (4) and (5): (4 ’ ) (5 ’ )

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Step 3 – Imposition of the Initial Condition For any (reasonably smooth) function, h(x) defined on the interval 0

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Step 3 – Imposition of the Initial Condition

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Step 3 – Imposition of the Initial Condition Therefore, (8) Where,

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Step 3 – Imposition of the Initial Condition

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Step 3 – Imposition of the Initial Condition The first 3 modes at fixed t’s.

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Step 3 – Imposition of the Initial Condition

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Example Problem:

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Example

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Separation of Variables; Vibrating String Travelling Wave Interpretation

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Travelling Wave Start with the Transport Equation: where, u(t, x) – function c – non-zero constant (wave speed) x – spatial variable Initial Conditions

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Travelling Wave Let x represents the position of an object in a fixed coordinate frame. The characteristic equation: Represents the object’s position relative to an observer who is uniformly moving with velocity c. Next, replace the stationary space-time coordinates (t, x) by the moving coordinates (t, ξ).

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Travelling Wave Re-express the Transport Equation: Express the derivatives of u in terms of those of v:

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Travelling Wave Using this coordinate system allows the conversion of a wave moving with velocity c to a stationary wave. That is,

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Travelling Wave For simplicity, we assume that v(t, ξ) has an appropriate domain of definition, such that, Therefore, the transport equation must be a function of the characteristic variable only.

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The Travelling Wave Interpretation

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Travelling Wave Revisiting the transport equation, Also recall that:

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Travelling Wave At t = 0, the wave has the initial profile When c > 0, the wave translates to the right. When c < 0, the wave translates to the left. While c = 0 corresponds to a stationary wave form that remains fixed at its original location.

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Travelling Wave As it only depends on the characteristic variable, every solution to the transport equation is constant on the characteristic lines of slope c, that is: where k is an arbitrary constant. At any given time t, the value of the solution at position x only depends on its original value on the characteristic line passing through (t, x).

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Travelling Wave

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19.3 Separation of Variables Vibrating Membranes Let us consider the motion of a stretched membrane This is the two dimensional analog of the vibrating string problem To solve this problem we have to make some assumptions

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Physical Assumptions 1.The mass of the membrane per unit area is constant. The membrane is perfectly flexible and offers no resistance to bending 2.The membrane is stretched and then fixed along its entire boundary in the xy plane. The tension per unit length T is the same at all points and does not change 3.The deflection u(x,y,t) of the membrane during the motion is small compared to the size of the membrane

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Vibrating Membrane Ref: Advanced Engineering Mathematics, 8 th Edition, Erwin Kreyszig

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Derivation of differential equation We consider the forces acting on the membrane Tension T is force per unit length For a small portion ∆x, ∆y forces are approximately T∆x and T∆y Neglecting horizontal motion we have vertical components on right and left side as T ∆y sin β and -T ∆y sin α Hence resultant is T∆y(sin β – sin α) As angles are small sin can be replaced with tangents F res = T∆y(tan β – tan α)

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F res = TΔy[u x (x+ Δx,y 1 )-u x (x,y 2 )] Similarly F res on other two sides is given by F res = TΔx[u y (x 1, y+ Δy)-u y (x 2,y)] Using Newtons Second Law we get Which gives us the wave equation: …..(1)

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Vibrating Membrane: Use of double Fourier series The two-dimensional wave equation satisfies the boundary condition (2) u = 0 for all t ≥ 0 (on the boundary of membrane) And the two initial conditions (3) u(x,y,0) = f(x,y) (given initial displacement f(x,y) And (4)

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Separation of Variables Let u(x,y,t) = F(x,y)G(t)…..(5) Using this in the wave equation we have Separating variables we get

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This gives two equations: for the time function G(t) we have …..(6) And for the Amplitude function F(x,y) we have …..(7) which is known as the Helmholtz equation Separation of Helmholtz equation: F(x,y) = H(x)Q(y)…..(8) Substituting this into (7) gives

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Separating variables Giving two ODE’s (9) And (10) where

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Satisfying boundary conditions The general solution of (9) and (10) are H(x) = Acos(kx)+Bsin(kx) and Q(y) = Ccos(py)+Dsin(py) Using boundary condition we get H(0) = H(a) = Q(0) = Q(b) = 0 which in turn gives A = 0; k = mπ/a; C = 0; p = nπ/b m,n Ε integer

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We thus obtain the solution H m (x) = sin (mπx/a) and Q n (y) = sin(nπy/b) Hence the functions (11)F mn (x) = H m (x)Q n (y) = sin(mπx/a)sin (nπy/b) Turning to time function As p 2 = ν 2 -k 2 and λ=cν we have λ = c(k 2 +p 2 ) 1/2 Hence λ mn = cπ(m 2 /a 2 +n 2 /b 2 ) 1/2 …..(12) Therefore …(13)

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Solution of the Entire Problem: Double Fourier Series …..(14) Using (3)

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Using Fourier analysis we get the generalized Euler formula And using (4) we obtain

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Example Vibrations of a rectangular membrane Find the vibrations of a rectangular membrane of sides a = 4 ft and b = 2 ft if the Tension T is 12.5 lb/ft, the density is 2.5 slugs/ft 2, the initial velocity is zero and the initial displacement is

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Solution

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Which gives Ref: Advanced Engineering Mathematics, 8 th Edition, Erwin Kreyszig

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19.4 Vibrating String Solutions d’Alembert’s Solution Solution for the wave equation can be obtained by transforming (1) by introducing independent variables

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u becomes a function of v and z. The derivatives in (1) can be expressed as derivatives with respect to v and z. We transform the other derivative in (1) similarly to get

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Inserting these two results in (1) we get which gives This is called the d’Alembert’s solution of the wave equation (1)

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D’Alembert’s solution satisfying initial conditions Dividing (8) by c and integrating we get

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Solving (9) with (7) gives Replacing x by x+ct for φ and x by x-ct for ψ we get the solution

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Solution by integral transforms Laplace Transform Semi Infinite string Find the displacement w(x,t) of an elastic string subject to: (i)The string is initially at rest on the x axis (ii)For time t>0 the left end of the string is moved by (iii)

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Solution Wave equation: With f as given and using initial conditions Taking the Laplace transform with respect to t

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We thus obtain Which gives

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Using initial condition This implies A(s) = 0 because c>0 so e sx/c increases as x increases. So we have W(0,s) = B(s)=F(s) So W(x,s)=F(s)e -sx/c Using inverse Laplace we get

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Travelling wave solution Ref: Advanced Engineering Mathematics, 8 th Edition, Erwin Kreyszig

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References H. Brezis. Functional Analysis, Sobolev Spaces and Partial Differential Equations. 1st Edition., 2011, XIV, 600 p. 9 illus R. Baber. The Language of Mathematics: Utilizing Math in Practice. Appendix F Poromechanics III - Biot Centennial ( ) tion.pdf

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Advanced engineering mathematics, 2 nd edition, M. D. Greenberg Advanced engineering mathematics, 8 th edition, E. Kreyszig Partial differential equations in Mechanics, 1 st edition, A.P.S. Selvadurai Partial differential equations, Graduate studies in mathematics, Volume 19, L. C. Evans Advanced engineering mathematics, 2 nd edition, A.C. Bajpai, L.R. Mustoe, D. Walker

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