Application 10.5B Rectangular Membrane Vibrations

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1 Applicatio.5B Rectagular Membrae Vibratios Here we ivestigate the vibratios of a flexible membrae whose equilibrium positio is the rectagle x a, y b. Suppose it is released from rest with give iitial displacemet, ad thereafter its four edges are held fixed. The (uder the usual assumptios) its displacemet fuctio uxyt (,, ) satisfies the boudary value problem u u u c = t + x y ( c T / ρ ) = () u(, y, t) = u( a, y, t) = u( x,, t) = u( x, b, t) = () uxy (,,) = f( xy, ), u ( x, y,) =. () Accordig to Problem i Sectio.5 of the text, the solutio is give by t πy uxyt (,, ) c si si cosγ ct a b = (4) m= = where the coefficiets are defied by a b 4 πy c = f ( x, y)si si dy dx. (5) ab a b The th term i (4) correspods to the membrae's th atural mode of oscillatio with displacemet fuctio πy u( x, y, t) = si si cosγ ct (6) a b with circular frequecy ω = γ c where The th iital positio fuctio u γ m a b = + π. (7) πy ( x, y) = si si (8) a b Chapter

2 is the rectagular membrae's th eigefuctio. Ivestigatio For simplicity, take a = b = c = ad plot some eigefuctios with small values of m ad i (8). The plot liear combiatios of several eigefuctios to see some of the more iterestig possible iitial shapes of a vibratig membrae. For example, the figure below shows the graph of the iitial positio fuctio uxy (, ) = u( xy, ) u ( xy, ) = sixsi y sixsiy geerated by the Mathematica commads u = Si[x] Si[y] - Si[x] Si[y]; PlotD[ Evaluate[u], {x,,pi}, {y,,pi}, PlotPoits -> {,}, Shadig -> False, ViewPoit -> {-.5,,.5} ] - The Maple commads u := si(x)*si(y) - *si(*x)*si(*y): plotd(u, x=..pi, y=..pi); ad the MATLAB commads x = : pi/ : pi; y = x; [x,y] = meshgrid(x,y); u = si(x).*si(y) - *si(*x).*si(*y); surf(x,y,u) produce similar results. Applicatio.5B

3 Maple, Mathematica, ad MATLAB all have the capability to aimate a sequece of sapshots of a vibratig membrae so as to show a "movie" illustratig its motio. For istace the Maple commads u := (x,y) -> si(x)*si(*y) + si(*x)*si(y): w := sqrt(5): # circular frequecy p := *Pi/w: # period of oscillatio with(plots): aimated( u(x,y)*cos(w*t), x=..pi, y=..pi, t=..p, frames=, style = patch ); produce a -frame movie showig oe complete oscillatio of the membrae with iitial positio fuctio ad circular frequecy ω = 5 uxy (, ) = si xsi y+ si xsi y (9). The Mathematica commads u = Si[x] Si[y] + Si[x] Si[y]; w = Sqrt[5]; (* circular frequecy *) P = Pi/w; (* period of oscillatio *) frame = Table[ PlotD[ Evaluate[u Cos[w t]], {x,,pi}, {y,,pi}, PlotRage -> {-.5,.5}, BoxRatios -> {,,}, ViewPoit->{-.5,.8,.75} ], {t,,p/, P/ } ]; produce a movie of a half-oscillatio which (with the Aimate Graphics selectio) ca be played back-ad-forth to show successive oscillatios cotiuously. The commad Show[ GraphicsArray[ {{frame[[]], frame[[ ]]}, {frame[[5]], frame[[ 7]]}, {frame[[9]], frame[[]]}} ]] displays the array of successive sapshots show o the ext page. Correspodig MATLAB commads are icluded i the m-file for this project that ca be dowloaded from the DE projects page at the web site Experimet i this way with liear combiatios of two, three, or more membrae eigefuctios of the form u ( x, y, t) = si mx si y cosω t () 4 Chapter

4 Applicatio.5B 5 where m ω = +. Vary the coefficiets so as to produce a visually attractive movie of sufficiet complexity to be iterestig. The Plucked Square Membrae Suppose the square membrae x, y π is plucked at its ceter poit ad set i motio from rest with the iitial positio fuctio (,,) (, ) mi(,,, ) uxy f xy xy x y π π = = () whose graph over the square x, y π looks like a square tet or pyramid with height π/ at its ceter. Thus the "tet fuctio" (, ) f xy is the -dimesioal aalogue of the familiar -dimesioal triagle fuctio that describes the iitial positio of a plucked strig. It ca be defied "piecewise" as idicated i the figure o the ext page. This diagram idicates how to subdivide the domai of defiitio of the fuctio f i the itegral i (5) with a = b = π i order to calculate the coefficiets { } c i (4)

5 y (π, π) f(x,y) = π - y f(x,y) = x f(x,y) = π - x f(x,y) = y Evidetly we ca write x f( x, y) x if x< y < π x, < x< π /, y if y < x< π y, < y < π /, = π y if π y < x < y, π / < y < π, π x if π x< y < x, π / < x< π. We proceed here with a Maple-based ivestigatio of the motio of the square membrae if it starts from rest with the iitial positio fuctio defied i (). Aalogous Mathematica- ad MATLAB-based ivestigatios ca dowloaded from the DE computig projects web site metioed previously. The coefficiet itegral i (5) is the sum of four double itegrals correspodig to the four triagles i the figure above. To evaluate these itegrals symbolically, we eter the Maple commads I := it( it(x*si(m*x)*si(*y), y=x..pi-x), x=..pi/): I := it( it(y*si(m*x)*si(*y), x=y..pi-y), y=..pi/): I := it( it((pi-y)*si(m*x)*si(*y), x=pi-y..y), y=pi/..pi): I4 := it( it((pi-x)*si(m*x)*si(*y), y=pi-x..x), x=pi/..pi): 6 Chapter

6 The the sum i (5) is give by c := simplify((4/pi^)*(i+i+i+i4)); si( π) + cos( πm)si( π) + si( πm) m si( πm)cos( π) m c : = 4 mπ ( m + ) We see (from the deomiator here) that Maple is assumig m ad ot equal, i which case it is obvious that c = because of the sie factors. To calculate the o-zero "diagoal coefficiets" i the Fourier series (4), we repeat the computatio above with m = from the begiig. The result is c () = = π 4 for odd, π for eve. () Thus the Fourier series of the tet fuctio f ( xy, ) defied i () is f( x, y) 4 six siy =. () π odd It follows that the solutio of our origial vibratig membrae problem with iitial positio fuctio f (x, y) is give by uxyt (,, ) 4 six siy cost =. (4) π odd We ivite you check out the computatio outlied here usig either Maple, Mathematica, or MATLAB. Try uequal values of a ad b to see whether you still get a "diagoal" series as i (4). Is it ow clear because (4) cotais o terms with m that the fuctio uxyt (,, ) is periodic (i t) with period P = π? This fact, that the tet fuctio i (4) thus yields a "musical" vibratio of a square membrae, was first poited out to us by Joh Polkig of Rice Uiversity. To ivestigate this vibratio visually, we proceed to defie a partial sum of the series i (4). Applicatio.5B 7

7 c := -> 4/(Pi*^): # with odd N := : # so N- = 5 P := Pi*sqrt(): # period of oscillatio u := (x,y,t) -> sum(c(*k-)*si((*k-)*x)* si((*k-)*y)*cos((*k-)*p*t/pi), k=..n): The followig commads plot ow sapshots of the resultig vibratio with t = ad with t = P. plotd(u(x,y,), x=..pi, y=..pi); plotd(u(x,y,p/8),x=..pi, y=..pi); Thus a vibratig plucked membrae exhibits a "flat spot" that is remiiscet of the flat spot we see i vibratios of a plucked strig. Fially, the commads with(plots): aimated( u(x,y,t), x=..pi, y=..pi, t=..p, frames=7, style = patch ); costruct a7-frame movie that you ca play (usig the Aimate meu) either cotiuously or oe-frame-at-a-time to ivestigate the periodic oscillatio. For istace, you will fid that the 8th frame (after oe-half oscillatio) shows a pyramid "poited" dowward istead of upward. 8 Chapter

φ φ sin φ θ sin sin u = φθu

φ φ sin φ θ sin sin u = φθu Project 10.5D Spherical Harmoic Waves I problems ivolvig regios that ejoy spherical symmetry about the origi i space, it is appropriate to use spherical coordiates. The 3-dimesioal Laplacia for a fuctio