lect2 - 2.1. GENERAL SOLUTION TO WAVE EQUATION 1...

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1 1 2.1. GENERAL SOLUTION TO WAVE EQUATION 1.138J/2.062J/18.376J, PROPAGATION Fall, 2006 MIT Notes by C. C. Mei CHAPTER TWO ONE DIMENSIONAL WAVES IN SIMPLE SYSTEMS General solution to wave equation It is easy to verify by direct substitution that the most general solution of the one dimensional equation: 2 φ ∂t 2 = c 2 2 φ ∂x 2 (1.1) can be solved by φ ( x, t )= f ( x ct )+ g ( x + ct ) (1.2) where f ( ξ )and g ( ξ ) are arbitrary functions of ξ .In the x,t (space,time) plane f ( x ct ) is constant along the straight line x ct = constant. Thus to the observer ( )who moves at the steady speed c along the positivwe x -axis., the function f is stationary. Thus to an observer moving from left to right at the speed c , the signal described initially by f ( x )at t = 0 remains unchanged in form as t increases, i.e., f propagates to the right at the speed c . Similarly g propagates to the left at the speed c . The lines x ct =constant and x + ct = contant are called the characteristic curves (lines) along which signals propagate. Note that another way of writing (1.2) is φ ( F ( t x/c G ( t + x/c ) (1.3) Let us illustrates an application. Branching of arteries References: Y C Fung : Biomechanics, Circulation. Springer1997 M.J. Lighthill : Waves in Fluids , Cambridge 1978. 2
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2 2.2 BRANCHING OF ARTERIES Recall the governing equations for pressure and velocity 2 p 2 2 p = c (2.1) ∂t 2 ∂x 2 2 u 2 2 u = c (2.2) 2 2 The two are related by the momentum equation ∂u ∂p ρ = (2.3) The general solutions are : p = p + ( x ct )+ p ( x + ct ) (2.4) u = u + ( x ct u ( x + ct ) (2.5) Since ± ± = p + + p , and ρ = ρcu ± + ρcu ± + where primes indicated ordinary diFerentiation with repect to the argument. Equation (2.3) can be satis±ed if p + = ρcu + ,p = ρcu (2.6) Denote the discharge by Q = uA then ZQ ² = u ² A = ± p ² (2.7) where p ² ρc Z = ± = (2.8) Q ² A is the ratio of pressure to flux rate and is call the impedance . It is the property of the tube. Now we examine the eFects of branching; Referening to ±gure x, the parent tubes branches into two charaterized by wave speeds C 1 and C 2 and impedaces Z 1 and Z 2 . An incident approaching the junction will cause reflection in the same tube p = p i ( t x/c p r ( t + x/c ) (2.9)
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± ² 2.2. WAVES DUE TO INITIAL DISTURBANCES 4 and transmitted waves in the branches are p 1 ( t x/c 1 )and p 2 ( t x/c 2 ). At the junction x = 0 we expect the continuity of pressure and fluxes, hence p i ( t )+ p r ( t )= p 1 ( t p 2 ( t ) (2.10) p i p r p 1 p 2 = + (2.11) Z Z 1 Z 2 DeFne the reflection coefficient R to be the amplitude ratio of reflected wave to incident wave, then 1 1 + 1 p r ( t ) Z Z 1 Z 2 R = = ± ² (2.12) p i ( t ) 1 + 1 + 1 Z Z 1 Z 2 Similarly the tranmission coefficients are p 1 ( t ) p 2 ( t ) Z 2 T = = = ± ² (2.13) p i ( t ) p i ( t ) 1 + 1 + 1 Z Z 1 Z 2 Note that both coefficients are constants depending only on the impedances. Hence the transmitted waves are similar in form to the incident waves except smaller by the factor T . The total on the incidence side is however very di±erent.
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This note was uploaded on 12/04/2011 for the course ESD 18.327 taught by Professor Gilbertstrang during the Spring '03 term at MIT.

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lect2 - 2.1. GENERAL SOLUTION TO WAVE EQUATION 1...

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