472.2 472.2 472.2 472.2 583.3 583.3 0 0 472.2 472.2 333.3 555.6 577.8 577.8 597.2 t c ux 0 For the well known wave equation utt the famous d'Alembert solution leads to c2 u xx 0 /Subtype/Type1 Gis . Just as in the case of the wave equation, we argue from the inverse by assuming that there are two functions, u, and v, that both solve the inhomogeneous heat equation and satisfy the initial and Dirichlet boundary conditions of (4). % 826.4 295.1 531.3] So far, the general 826.4 826.4 826.4 826.4 826.4 826.4 826.4 826.4 826.4 826.4 1062.5 1062.5 826.4 826.4 /Widths[342.6 581 937.5 562.5 937.5 875 312.5 437.5 437.5 562.5 875 312.5 375 312.5 ~_|'iSQ"5TbIeHA`'eR&INLHS8r`+/1WCLgmhP'0grijrA"DhhG? 379.6 638.9 638.9 638.9 638.9 638.9 638.9 638.9 638.9 638.9 638.9 638.9 638.9 379.6 Two-Dimensional Wave Equation The solution of the wave equation in two dimensions can be obtained by solving the three-dimensional wave equation in the case where the initial data depends only on xand y, but not z. 489.6 489.6 489.6 489.6 489.6 489.6 489.6 489.6 489.6 489.6 272 272 272 761.6 462.4 43 0 obj One can see from the d'Alembert formula (see also the picture above) that the solution 495.7 376.2 612.3 619.8 639.2 522.3 467 610.1 544.1 607.2 471.5 576.4 631.6 659.7 GENERAL SOLUTION TO WAVE EQUATION 1 I-campus project School-wide Program on Fluid Mechanics Modules on Waves in uids T.R.Akylas&C.C.Mei CHAPTER TWO ONE-DIMENSIONAL PROPAGATION Since the equation 2 t2 = c 2 governs so many physical phenomena in nature and technology, its properties are basic to the understanding of wave . >> 0 0 0 0 0 0 0 0 0 0 777.8 277.8 777.8 500 777.8 500 777.8 777.8 777.8 777.8 0 0 777.8 (Homework) Modified equation and amplification factor are the same as original Lax-Wendroff method. 295.1 531.3 531.3 531.3 531.3 531.3 531.3 531.3 531.3 531.3 531.3 531.3 295.1 295.1 /Widths[295.1 531.3 885.4 531.3 885.4 826.4 295.1 413.2 413.2 531.3 826.4 295.1 354.2 380.8 380.8 380.8 979.2 979.2 410.9 514 416.3 421.4 508.8 453.8 482.6 468.9 563.7 (FT), and then we solve the initial-value problem for the wave equation using the Fourier transform. The wave equation preserves the oddity of solutions. : Adding the three derivatives, we get. 611.1 777.8 777.8 388.9 500 777.8 666.7 944.4 722.2 777.8 611.1 777.8 722.2 555.6 In its simp lest form, the wave . /Name/F1 Sa"
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The solution is and . 656.3 625 625 937.5 937.5 312.5 343.8 562.5 562.5 562.5 562.5 562.5 849.5 500 574.1 /FirstChar 33 812.5 875 562.5 1018.5 1143.5 875 312.5 562.5] >> Solution: The frequency fe n of the normal mode u n (x;t), given in (13), is fe n = 1 2 p c2 n k2 = 1 2 r c2n22 l2 k2 = cn 2l s 1 lk cn 2 = f n s 1 lk cn 2 (14) As the damping (k > 0) increases, the frequencies of the normal modes decrease. 0000027035 00000 n
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272 272 489.6 544 435.2 544 435.2 299.2 489.6 544 272 299.2 516.8 272 816 544 489.6 consider a dierent approach, the d'Alembert's solution of the wave equation, which is more suitable if the domain is innite. 0000027337 00000 n
is a solution of the wave equation on the interval [0;l] which satises un(0;t) = 0 = un(l;t). Overview Wavesandvibrationsinmechanicalsystemsconstituteoneofthe 173/Omega/ff/fi/fl/ffi/ffl/dotlessi/dotlessj/grave/acute/caron/breve/macron/ring/cedilla/germandbls/ae/oe/oslash/AE/OE/Oslash/suppress/dieresis /LastChar 196 0000059205 00000 n
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The solution we were able to nd was u(x;t) := X1 n=1 g n cos n L ct + L nc h n sin n L ct sin n L x ; (2) by assuming the following sine Fourier series expansion of the initial data gand h: X1 n=1 g n sin n L x ; X1 n=1 h n sin n L cx : In order to prove that the function uabove is the solution of our problem, we cannot dif . /FontDescriptor 37 0 R << should be consistent with the Maxwell equations. depends . /LastChar 127 WATERWAVES 5 Wavetype Cause Period Velocity Sound Sealife,ships 10 110 5s 1.52km/s Capillaryripples Wind <101s 0.2-0.5m/s Gravitywaves Wind 1-25s 2-40m/s Sieches Earthquakes,storms minutestohours standingwaves << /LastChar 196 endobj 0000045400 00000 n
In the last several lectures we solved the initial value problems associated with the wave and heat equa-tions on the whole line x2R. /Widths[791.7 583.3 583.3 638.9 638.9 638.9 638.9 805.6 805.6 805.6 805.6 1277.8 795.8 795.8 649.3 295.1 531.3 295.1 531.3 295.1 295.1 531.3 590.3 472.2 590.3 472.2 0000042001 00000 n
/Length 3308 Show that there is at most one solution to the Dirichlet problem (4). >> %PDF-1.2 0000062652 00000 n
Its left and right hand ends are held xed at height zero and we are told its initial . We shall discuss the basic properties of solutions to the wave equation (1.2), as well as its multidimensional and non-linear variants. /Type/Font 699.9 556.4 477.4 454.9 312.5 377.9 623.4 489.6 272 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 /Name/F2 Let 1141 A(x) Dx,x J(x)dx0 Then 221141A(x) Dx,x J x dx because 2acts only on xnot x1. 820.5 796.1 695.6 816.7 847.5 605.6 544.6 625.8 612.8 987.8 713.3 668.3 724.7 666.7 Suppose that the function h(x,t) gives the the height of the wave at position x and time t. Then h satises the dierential equation: 2h t2 = c2 2h x2 (1) where c is the speed that . << >> /Subtype/Type1 0000024345 00000 n
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 826.4 295.1 826.4 531.3 826.4 Wave equation The purpose of these lectures is to give a basic introduction to the study of linear wave equation. 0 0 0 0 0 0 0 0 0 0 0 0 675.9 937.5 875 787 750 879.6 812.5 875 812.5 875 0 0 812.5 0000061223 00000 n
1 (radiation condition) The condition for Gmeans vanishing pressure at z= 0, as is the case at an air/water interface. 4.3 diagonalization of p p and d we next look for those coordinate systems permitting separation of variables in (1.1) such that the corresponding basis functions are eigenfunctions 3 0 obj >> 593.8 500 562.5 1125 562.5 562.5 562.5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0000045808 00000 n
783.4 872.8 823.4 619.8 708.3 654.8 0 0 816.7 682.4 596.2 547.3 470.1 429.5 467 533.2 761.6 679.6 652.8 734 707.2 761.6 707.2 761.6 0 0 707.2 571.2 544 544 816 816 272 If the data (f;g;h) are merely continuous then the solution uneed not be di eren-tiable. Surface multiples, derivation integral equation Let Gmf and Gbe two solutions for the acoustic wave equation in the region z>0 satisfying the conditions G(~x; ~xs;t)jz=0 = 0; Gmf(~x;~x s;t) 2 O(j ~x ~xsj 1) for j~x ~xsj ! Let d 1. analytical solutions to the wave equation. /BaseFont/YKWBYK+CMSY8 In other words, we write u(x,y,t)= m=1 n=1 Cmn . 0000001603 00000 n
In particular, consider the initial-value problem 8 >< >: vtt c2 . 41 0 obj ?? Bf)h | CB2M%K*ZEJtzDMTUi%U6eQii65QmmH3D{9{5 _P6Jh/ gjq^%H;I: X_w);&FFi;Gzalx|[FDA\(i!:a'lOD7 fFG7=}o 2At,M-&pX8K1]M Denoting the solutions for are ()x L un x t Fn x Gn t Bn nt Bn nt n ( , ) = ( ) ( ) = cos + *sin sin Solutions for the 1D Wave Equation are: As a result of solving for F, we have restricted These functions are the eigenfunctions of the vibrating string, and the values are called the eigenvalues. 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 663.6 885.4 826.4 736.8 If t stands for time and ~x = fxig represent the observation point, such solutions of the wave equation, (@2 @t2 c2 or 2)` = 0; (1) /Subtype/Type1 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. As in the one dimensional situation, the constant c has the units of velocity. 1444.4 555.6 1000 1444.4 472.2 472.2 527.8 527.8 527.8 527.8 666.7 666.7 1000 1000 << Rh%E(VxIo uD&
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'L)A,n.~f|Q" There exists a nice set of complex solutions to the wave equation, called complex travelling waves, which take the form. 324.7 531.3 531.3 531.3 531.3 531.3 795.8 472.2 531.3 767.4 826.4 531.3 958.7 1076.8 /FontDescriptor 21 0 R x[n8}}SGS0LX
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We want to solve the wave equation on the half line with Dirichlet boundary conditions. 666.7 722.2 722.2 1000 722.2 722.2 666.7 1888.9 2333.3 1888.9 2333.3 0 555.6 638.9 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 458.3 458.3 416.7 416.7 /Widths[779.9 586.7 750.7 1021.9 639 487.8 811.6 1222.2 1222.2 1222.2 1222.2 379.6 777.8 777.8 777.8 777.8 777.8 777.8 777.8 777.8 777.8 777.8 777.8 777.8 777.8 777.8 bZ,|eRTX%1-rw*
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/ ,h3 9etAM&w*8>?U[MPV#w_Dh#MXCK@S J;Pvx{ @E`co &}2$T 675.9 1067.1 879.6 844.9 768.5 844.9 839.1 625 782.4 864.6 849.5 1162 849.5 849.5 wave from the wave equation is invalid. 0000026832 00000 n
3.3 Particle Flux and Schr odinger Equation The solution of the Schr odinger equation is the wave function (~r;t) which describes the state of a particle moving in the potential U(~r;t). /Type/Font /BaseFont/KLTBKH+CMMI12 In general it can happen that the solution is di erentiable (u2C1), but that the derivatives v= u x and w= u t are not weak solutions of the above equations (6) and (5). << 734 761.6 666.2 761.6 720.6 544 707.2 734 734 1006 734 734 598.4 272 489.6 272 489.6 589.1 483.8 427.7 555.4 505 556.5 425.2 527.8 579.5 613.4 636.6 272] :TaAP|rj8\ALAc-8l3'1 ;Dt%j`.@"63=Qu8yK@+ZLsTvv00h`a`:` }%&1p6h2,g@74B63t^=LY,.,' u
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HtU}L[?O0!? 295.1 531.3 531.3 531.3 531.3 531.3 531.3 531.3 531.3 531.3 531.3 531.3 531.3 295.1 379.6 963 638.9 963 638.9 658.7 924.1 926.6 883.7 998.3 899.8 775 952.9 999.5 547.7 ryrN9y9KS,jQpt=K SOLUTION OF THE WAVE EQUATION We need the solution of the equation 2 AJ0 We solve this by introducing the idea of a Green's function. 666.7 666.7 666.7 666.7 611.1 611.1 444.4 444.4 444.4 444.4 500 500 388.9 388.9 277.8 ()MacCormack Method (1969) Predictor step : n+1 n n() j j j+1 t u=u-c u x n uj Correct step : 1111() 1 1 2 nnn nn jjj jj ct uuu . /FirstChar 33 << 1.1 The Wave Equation One of the most important predictions of the Maxwell equations is the existence of electromagnetic waves which can transport energy. 1062.5 826.4] 0000063293 00000 n
well posed. Starting with the right-hand side, we ignore for the time being and obtain. /BaseFont/JBOAVI+MSBM10 which is the 1D wave equation with solutions of propagating waves of permanent form. endobj The simplest solutions are plane waves in innite media, and we shall explore these now. /LastChar 196 n\=bD12%F^Oy2#*1FppLZc"JAD; /FirstChar 33 /Widths[272 489.6 816 489.6 816 761.6 272 380.8 380.8 489.6 761.6 272 326.4 272 489.6 For the wave equation: a2u xx= u tt it turns out that solutions can always be written as: u(x;t) = F(x+ at) + G(x at) for some functions Fand G. This worksheet is designed to guide you through the process of using this formula to solve wave equation problems. 500 500 611.1 500 277.8 833.3 750 833.3 416.7 666.7 666.7 777.8 777.8 444.4 444.4 Suppose we have the wave equation u tt = a2u xx. 95 0 obj
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The functions F and G (and hence the solution u) are The . The 2D wave equation Separation of variables Superposition Examples Remarks: For the derivation of the wave equation from Newton's second law, see exercise 3.2.8. 491.3 383.7 615.2 517.4 762.5 598.1 525.2 494.2 349.5 400.2 673.4 531.3 295.1 0 0 0000042382 00000 n
/LastChar 196 1.2 The Real Wave Equation: Second-order wave equa-tion Here, we now examine the second order wave equation. 0000038938 00000 n
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>> 0 0 0 613.4 800 750 676.9 650 726.9 700 750 700 750 0 0 700 600 550 575 862.5 875 stream /FontDescriptor 30 0 R 4 Linear Surface Gravity Waves C, Dispersion, Group Velocity, and Energy Propagation30 4.1 Group Velocity . 6) Solve the heat equation ft = f xx on [0,] with the initial condition f(x,0) = |sin(3x)|. /Encoding 7 0 R 299.2 489.6 489.6 489.6 489.6 489.6 734 435.2 489.6 707.2 761.6 489.6 883.8 992.6 Moreover, any function y = f2(ct + x) will be a solution so that, generally, their superposition y = f1(ct x) + f2(ct + x) is the complete solution.
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