Project 10.5C Circular Membrane Vibrations

Transcription

1 Projet 1.5C Cirular Membrane Vibrations In problems involving regions that enjoy irular symmetry about the origin in the plane (or the vertial z-axis in spae), the use of polar (or ylindrial) oordinates is advantageous. In Setion 9.7 of the text we disussed the expression of the - dimensional Laplaian u 1 u 1 u u = + + (1) r r r r θ in terms of the familiar plane polar oordinates (, r θ ) for whih x = r os θ and y= rsin θ. If ur (, θ,) t denotes the vertial displaement at time t of the point (, r θ ) of a vibrating irular membrane of radius, then the -dimensional wave equation takes the polar oordinate form u u 1 u 1 u a u a = = t + + r r r r θ. () where a = T / ρ in terms of the membrane's tension T and density ρ (per unit area). If the membrane is released from rest with given initial position funtion f (, r θ ) at time t = and thereafter its boundary is held fixed, then the membrane's displaement funtion ur (, θ,) t satisfies both () and the boundary onditions u (, θ,) t =, (fixed boundary) (3) ur (, θ,) = f(, rθ), (given initial displaement) (4) ut ( r, θ,) =. (zero initial veloity) (5) Fill in the details in the solution that is outlined as follows. Show first that the substitution in the wave equation () yields the separation of variables ur (, θ,) t = Rr () Θ ( θ) Tt () (6) 1 T R + r R Θ = + = α at R rθ (onstant). (7) Then T + α a T =, () T = (8) Appliation 1.5C 39

2 implies that (to within a onstant multiple) Tt () = osαat. (9) Next, the right equality in (43) yields the equation rr + rr Θ α r R Θ + + = (1) from whih it follows that Θ = Θ β (onstant). (11) In order that a solution Θ ( θ ) of Θ+ β Θ= have the neessary π-periodiity, the parameter β must be an integer, so we have the θ-solutions for n =, 1,, 3,.... os nθ Θ n( θ ) = (1) sin nθ Substitution of Θ / Θ= n in (1) now yields the parametri Bessel equation + + ( ) = (13) rr rr α r n R of order n, with bounded solution R( r) = Jn( αr). Sine the zero boundary ondition (3) yields Jn( α ) =, ase 1 in the table of Fig of the text yields the r- eigenfuntions γ mnr Rmn() r = Jn (m = 1,, 3,... ; n =, 1,,...) (14) where γ mn denotes the mth positive solution of the equation Jn( x ) =. Finally, substitution of α = γ / in (9) yields the t-funtion mn mn T mn γ mnat () t = os. (15) Combining (1), (14), and (15) we see that our boundary value problem for the irular membrane released from rest has the formal series solution γmnr γmnat ur (, θ, t) = Jn ( amnos nθ + bmnsin nθ) os m= 1 n=. (16) 31 Chapter 1

3 Thus the vibrating irular membrane's typial natural mode of osillation with zero initial veloity is of the form γmnr γmnat umn( r, θ, t) = Jn os nθ os (17) or the analogous form with sin nθ instead of os nθ. In this mode the membrane vibrates with m 1 fixed nodal irles (in addition to its boundary irle r = ) with radii rjn = γ jn/ γmn for j = 1,,..., m 1. It also has n fixed nodal radii spaed at angles of π/n starting with θ = π/(n). Figure in the text shows some typial onfigurations of these nodal irles and radii, whih divide the irle into annular setors that move alternately up and down as the membrane vibrates. We proeed here with a Mathematia-based investigation of irular membrane vibrations. Analogous Maple- and MATLAB-based investigations an downloaded from the DE omputing projets link at the web site Figure in the text shows the graphs of y= Jn( x) for n = 1,, 3, 4. We see there that n + 3 is (at least for small n) a rough but reasonable initial estimate of the first positive zero γ 1n of the equation Jn( x ) =. This observation motivates the Mathematia ommands inits = Table[ FindRoot[ BesselJ[n,x] ==, {x, n+3} ], {n, 1,5} ]; g1 = x /. inits { , , , , }} that aurately approximate these initial zeros of J1( x), J( x), J3( x), J4( x), and J5( x ). Now reall that the gap between suessive zeros of Jn( x ) = is approximately π, so it follows that γ γ1 + ( 1) π. Consequently the ommands mn n m zeros = Table[ FindRoot[ BesselJ[n,x] ==, {x, g1[[n]] + (m-1)pi} ], {m,1,5}, {n,1,5} ]; g = x /. zeros; g // TableForm Appliation 1.5C 311

4 yield a table displaying the mth zero For instane, γ of J ( x ) = in the mth row and nth olumn. mn n g[[,3]] so we see that the nd zero of J ( x ) = is 3 γ Now that numerial values of the zeros of Bessel funtions are available, we an employ the ommands x = r Cos[t]; y = r Sin[t]; m = ; n = 1; ParametriPlot3D[ {x,y,besselj[n, g[[m,n]] r] Cos[n t]}, {r,,1}, {t,, Pi} ] to display the initial position orresponding to an eigenfuntion defined as in (17). following ommands graph these initial positions with m = 1 and n = 1,, 3, 4. The shape = Table[ ParametriPlot3D[ {x,y, BesselJ[n, g[[1,n]] r] Cos[n t]}, {r,,1}, {t,, Pi}, Shading -> False, PlotPoints -> {1,48}, Tiks -> None, ViewPoint->{1.3, -.4, 1.}], {n,1,4} ]; 31 Chapter 1

5 Show[GraphisArray[{{shape[[1]], shape[[]]}, {shape[[3]], shape[[4]]}}]]; We suggest that you explore vibrating irular membrane possibilities by graphing onvenient linear ombinations of eigenfuntions defined as in (17). For instane, the figure below shows the initial position of the osillation ( ) ( ) ur (, θ, t) = J γ r os θ os γ t+ J γ r os θ os γ t (18) defined for for a irular membrane with a = 1 and radius = 1. Appliation 1.5C 313

6 The following Mathematia ommands generate a sequene of k p = 8 frames that an be animated to show a movie illustrating the irular membrane osillation defined in (18). k = 4; (* yles *) p = ; (* frames/yle *) g1 = g[[,1]]; g3 = g[[3,]]; w1 = g1; w = g3; (* frequenies *) x = r Cos[theta]; y = r Sin[theta]; z = BesselJ[1, g1 r] Cos[ theta] Cos[w1 t] + BesselJ[, g3 r] Cos[ theta] Cos[w t]; Do[ ParametriPlot3D[ Evaluate[{x,y,z}], {r,,1}, {theta,,pi}, PlotRange -> Automati, BoxRatios -> {1,1,.6}, ViewPoint -> {1,-,1}, PlotPoints -> {15,6}, Tiks -> None, Shading -> False], {t,, k Pi/w1, Pi/(p w1)} ]; 314 Chapter 1