1 General solution to wave equation - MIT

1 General solution TO WAVE EQUATION1I-campus projectSchool-wide Program on Fluid MechanicsModules on waves TWOONE-DIMENSIONAL PROPAGATIONS ince the equation 2 t2=c2 2 governs so many physical phenomena in nature and technology, its properties are basicto the understanding of wave propagation. This chapter is devoted to its analysis whenthe extent of the medium is infinite and the motion is one dimensional. To be bespecific, physical discussions are made for shallow-water waves in the sea. The resultsare however readily tranferable or modified for sound, waves in blood vessels and othertypes of General solution to wave equationRecall that for waves in an artery or over shallow water of constant depth, the governingequation is of the classical form 2 t2=c2 2 x2( )It is easy to verify by direct substitution that the most General solution of the onedimensional wave equation ( ) is (x, t)=F(x ct)+G(x+ct)( )whereFandgare arbitrary functions of their arguments.

2 In thex, t(space,time) planeF(x ct) is constant along the straight linex ct= constant. Thus to the observer(x, t)whomovesatthesteadyspeedcalong the positivwex-axis, the functionFisstationary. Thus to an observer moving from left to right at the speedc, the signaldescribed initially byF(x)att= 0 remains unchanged in form astincreases, ,Fisa wave propagating to the right at the speedc. SimilarlyGpropagates to the left at BRANCHING OF ARTERIES2speedc. The linesx ct=constant andx+ct= contant are called thecharacteristiccurves (lines) along which signals propagate.

3 Note that another way of writing ( ) is (x, t)= F(t x/c)+ G(t+x/c)( )Let us illustrates an application of this simple Branching of arteriesReferences: Y C Fung :Biomechanics, Lighthill : waves in Fluids, Cambridge that ( ) governs both the pressure and the velocity in the blood 2p t2=c2 2p x2( ) 2u t2=c2 2u x2( )Thetwounknownsarerelatedbythemomentumeq uation u t= p x( )The General solutions are :p=p+(x ct)+p (x+ct)( )u=u+(x ct)+u (x+ct)( )Since p x=p0++p0 ,and u t= cu0++ cu0 where primes indicated ordinary differentiation with repect to the argument.

4 equation ( ) can be satisfied ifp+= cu+,p = cu ( ) BRANCHING OF ARTERIES3 Denote the discharge byQ=uAthenQ =u A= Zp ( )whereZ= cA( )is the property of the tube and is call we examine the effects of branching; Refering tofigure 1, the parent tube,characterized by wave speedcand impedanceZ, branches into two characterized byc1andc2andZ1andZ2. An incident wave approaching the junction will cause reflectionp=pi(t x/c)+pr(t+x/c),x>0( )and transmitted waves in the branches arep1(t x/c1)andp2(t x/c2)inx>0. Atthe junctionx= 0, continuity of pressure andfluxes requirespi(t)+pr(t)=p1(t)=p2(t)( )andpi prZ=p1Z1+p2Z2( )Define the reflection coefficientRto be the amplitude ratio of reflected wave to incidentwave, thenR=pr(t)pi(t)=1Z 1Z1+1Z2 1Z+ 1Z1+1Z2 ( )Similarly the tranmission coefficients areT=p1(t)pi(t)=p2(t)pi(t)=2Z1Z+ 1Z1+1Z2 ( )Note that both coefficients are constants depending only on the impedances.

5 Hence thetransmitted waves propagate in the direction of increasingxand are similar in formto the incident waves except smaller by the factorT. On the incidence side waves theincident and reflected waves propagate in opposite waves DUE TO INITIAL DISTURBANCES4 Figure 1: Branching of artieries3 Shallowwaterwavesinaninfinite sea due to ini-tial disturbancesRecall for one-dimensional long waves in a shallow sea of depthh(x), the linearlizedconservation laws of mass and momentum are t+ (uh) x=0( )and u t= g x( )where (x, t) is the vertical displacement of the free surface andu(x, t) the horizontalvelocity.

6 The atmospheric pressure over the entire free surface is uniform and cross-differentiation, is seen to be governed by 2 t2=g x h x!( )In the limit of constant depth (h=constant), the above equation reduces to the classicalwave eqaution 2 t2=c2 2 x2,wherec=qgh.( )Consider now a sea of infinite extent, <x< . Let the initial surface displace-ment and velocity be prescribed along the entire surface (x,0) =F(x)( ) waves DUE TO INITIAL DISTURBANCES5 t(x, t)=G(x),( )whereF(x)andG(x) are non-zero only in thefinite domain , and / tare zero for anyfinite t.

7 In ( ) the highest time derivative is of the secondorder and initial data are prescribed for and / t. Initial conditions that specify allderivatives of all orders less than the highest in the differential equation are called theCauchy initial conditions. These conditions are best displayed in the space-time diagramas shown in Figure )tu =g(x)u=f(xu =c utxFigure 2: Summary of the initial-boundary-value problemThe present initial-boundary-value problem has a famous solution due to d Alembert,which can be derived from ( ), , = ( )+ ( )= (x+ct)+ (x ct),( )where and are so far arbitrary functions of the characteristic variables =x ctand =x+ the initial conditions we get (x,0) = (x)+ (x)=f(x) t(x,0) =c 0(x) c 0(x)=g(x).

8 ( )The last equation may be integrated with respect tox =1cZxx0g(x0)dx0+K,( ) waves DUE TO INITIAL DISTURBANCES6dependencex,tccccx(0 Domain ofinfluenceRange of1111)x-ctx+ctxtFigure 3: Domain of dependence and range of influencewhereKis an arbitrary constant. Now and can be solved from ( ) and ( asfunctions ofx, (x)=12[f(x)+K] 12cZxx0g(x0)dx0 (x)=12[f(x) K]+12cZxx0g(x0)dx0,whereKandx0are some constants. Replacing the arguments of byx+ctand of byx ctand substituting the results inu,weget (x, t)=12f(x ct) 12cZx ctx0gdx0+12f(x+ct)+12cZx+ctx0gdx0=12[f(x ct)+f(x+ct)] +12cZx+ctx ctg(x0)dx0,( )which is d Alembert s solution to the homogeneous wave equation subject to generalCauchy initial see the physical meaning, let us draw in the space-time diagram a triangle formedby two characteristic lines passing through the observer atx, t, as shown in Figure 0 begins atx ctand ends atx+ solution ( ))

9 Depends on the initial displacement at just the two cornersx waves DUE TO INITIAL DISTURBANCES7Ou,txFigure 4: waves due to initial displacementandx+ct, and on the initial velocity only along the segment fromx cttox+ outside the triangle matters. Therefore, to the observer atx, t,thedomainof dependenceis the base of the characteristic triangle formed by two characteristicspassing throughx, t. On the other hand, the data at any pointxon the initial linet=0must influence all observers in the wedge formed by two characteristics drawn fromx,0into the region oft>0; this characteristic wedge is called therange of us illustrate the physical effects of initial displacement and velocity (i): Initial displacement only:f(x)6=0andg(x) = 0.

10 The solution is (x, t)=12f(x ct)+12f(x+ct)and is shown for a simplef(x) in Figure 4 at successive time steps. Clearly, the initialdisturbance is split into two equal waves propagating in opposite directions at the speedc. The outgoing waves preserve the initial profile, although their amplitudes are reducedby (ii): Initial velocity only:f(x)=0,andg(x)6= 0. Consider the simple examplewhereg(x)=g0when|x|<b,and=0when|x|> waves DUE TO INITIAL DISTURBANCES8Ou,tABCDIEFGHxFigure 5: waves due to initial velocityReferring to Figure 5, we divide thex tdiagram into six regions by the characteristicswithBandClying on thexaxis atx= band +b, respectively.