Is it possible to tune a drum?

Accepted Manuscript Is it possible to tune a drum?

Pedro R.S. Antunes

PII: DOI: Reference:

S0021-9991(17)30159-6 http://dx.doi.org/10.1016/j.jcp.2017.02.056 YJCPH 7188

To appear in:

Journal of Computational Physics

Received date: Revised date: Accepted date:

10 August 2016 1 February 2017 23 February 2017

Please cite this article in press as: P.R.S. Antunes, Is it possible to tune a drum?, J. Comput. Phys. (2017), http://dx.doi.org/10.1016/j.jcp.2017.02.056

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IS IT POSSIBLE TO TUNE A DRUM? PEDRO R.S. ANTUNES

Abstract. It is well known that the sound produced by string instruments has a well defined pitch. Essentially, this is due to the fact that all the resonance frequencies of the string have integer ratio with the smallest eigenfrequency. However, it is enough to use Ashbaugh-Benguria bound for the ratio of the smallest two eigenfrequencies to conclude that it is impossible to build a drum with a uniform density membrane satisfying harmonic relations on the eigenfrequencies. On the other hand, it is known since the antiquity, that a drum can produce an almost harmonic sound by using different densities, for example adding a plaster to the membrane. This idea is applied in the construction of some Indian drums like the tabla or the mridangam. In this work we propose a density and shape optimization problem of finding a composite membrane that satisfy approximate harmonic relations of some eigenfrequencies. The problem is solved by a domain decomposition technique applied to the Method of Fundamental Solutions and Hadamard shape derivatives for the optimization of inner and outer boundaries. This method allows to present new configurations of membranes, for example a two-density membrane for which the first 21 eigenfrequencies have approximate five harmonic relations or a three-density membrane for which the first 45 eigenfrequencies have eight harmonic relations, both involving some multiple eigenfrequencies.

1. Introduction Let Ω ⊂ R be a bounded domain, and consider the Dirichlet eigenvalue problem,  −Δu = ρλu in Ω (1) u=0 on ∂Ω, 2

defined in the Sobolev space H01 (Ω). This is a model for the vibration of a drum where the shape of the membrane is defined by the geometry of Ω. ρ is the density of the membrane and we will assume that ρ = ρ1 χΩ\D + ρ2 χD and ρi > 0, i=1,2. We will denote the eigenvalues by 0 < λ1 (Ω, D, ρ) ≤ λ2 (Ω, D, ρ) ≤ ... where each λi (Ω, D, ρ) is counted with its multiplicity and the corresponding orthonormal real  eigenfunctions by ui , i = 1, 2, .... We will also use the notation κi (Ω, D, ρ) = λi (Ω, D, ρ) to denote an eigenfrequency. In this work we will focus on the acoustic properties of a drum that can be modeled by problem (1). In particular, in this simplified model we are neglecting the body of the drum. Date: February 28, 2017. Key words and phrases. harmonic drum, Dirichlet Laplacian, eigenvalues, shape optimization. P.A. was partially supported by FCT, Portugal, through the program ”Investigador FCT” with reference IF/00177/2013 and the scientific project PTDC/MAT- CAL/4334/2014. 1

2

Pedro R. S. Antunes

The one-dimensional problem with ρ ≡ 1 is a model for the vibration of a uniform string. In that case, it is well known that the eigenfrequencies are given by (2)

κi (Ω, 1) = iκ1 (Ω, 1), i = 2, 3, 4, ...

In particular, all the eigenfrequencies have integer ratio with the smallest eigenfrequency which is called fundamental. This property implies that the sound produced by string instruments like the violin or the piano have a well defined pitch (e.g. [FR]). Indeed, this also justifies that the same happens for the sound produced by a broad class of musical instruments. For example, the clarinet, the flute or the trumpet and most wind instruments are essentially one-dimensional resonators which implies that the eigenfrequencies satisfy the harmonic relation (2). If we consider the two-dimensional problem (again with ρ ≡ 1) we see that it is impossible to have the harmonic property (2) . This follows immediately from Ashbaugh-Benguria bound for the ratio of the first two eigenvalues ([AB]), which can be written in terms of the eigenfrequencies as j1,1 κ2 (Ω, 1) ≤ κ1 (Ω, 1), j0,1 where j0,1 and j1,1 are respectively the smallest positive roots of Bessel functions J0 and J1 . We have j1,1 ≈ 1.593 < 2 j0,1 which implies that it is impossible to build a harmonic drum with uniform density ρ ≡ 1. The optimization of the ratios κn (Ω, 1)/κ1 (Ω, 1) was considered in [O, A] and these numerical studies suggest that these ratios are clearly below the corresponding integer number, i.e. κn (Ω, 1)/κ1 (Ω, 1) < n. However, it is known since the antiquity that a drum can produce an almost harmonic sound by using different densities, for example adding a plaster to the membrane. This reduce the ’harshness of sound’ ([B2], chapter 33, verse 25-26) and is applied in the construction of some Indian drums like the tabla or the mridangam (Figure 1). The tabla consists of a pair of drums, the dayan with a concentric loading and the bayan with eccentric loading, while the mridangam is a double headed drum which is played with both hands. This type of drums are pictured in the paintings of the walls of Ajanta caves, which clearly indicates that it is a very ancient invention. The technique of using different densities allows to modify the

Figure 1. The Tabla (left plot) and the mridangam (right plot). eigenfrequencies to have an almost harmonic relation. This effect was studied experimentally by the Nobel Prize-winning physicist Raman and coworkers (eg. [RK, R])

Harmonic membranes

3

by calculating the eigenfrequencies of the mridangam. They observed that the first eigenfrequencies had approximate five harmonic relations, involving some multiple eigenfrequencies. The higher eigenfrequencies were not harmonic but were almost suppressed. This almost harmonic relation was also obtained in several computational works (eg. [RS, GGL]), where the drum head was modeled as a composite circular membrane with a concentric circular inner region. In this case, we can use separation of variables and (1) is reduced to a one dimensional eigenvalue problem that can be solved analytically. Thus, the problem considered in those references is an optimization problem of minimizing a function of two variables, whose values are obtained by Bessel functions. One of the conclusions that arise from those studies is that the configuration of the dayan that was certainly obtained by a trial and error procedure provides very good approximations for a few harmonic relations. In this work we will consider a shape and density optimization of inner and outer sub-regions of the membrane aiming to suggest non trivial configurations that can provide some approximate harmonic relations. This problem will be solved by a domain decomposition technique applied to the Method of Fundamental Solutions and Hadamard shape derivatives for the optimization of both sub-regions. 2. Numerical solution of the eigenvalue problem In this work we will search for configurations of membranes that provide some harmonic relations at the beginning of the spectrum. For now we will assume that we have just one sub-region D, as illustrated in Figure 2, but the case of several sub-regions could be considered in a similar manner.

Figure 2. A two-density membrane. We will denote by n the unitary vector that is normal to the boundaries ∂D or ∂Ω and is oriented towards the exterior of the domain. This means that for a ¯ ∀x ∈ ∂D. sufficiently small  > 0, x + n ∈ Ω\D, We will assume that Ω and D are star-shaped domains with boundaries parameterized by (3)

∂Ω = {rΩ (t) (cos(t), sin(t)) : t ∈ [0, 2π)} ,

(4)

∂D = {rD (t) (cos(t), sin(t)) + (x0 , y0 ) : t ∈ [0, 2π)}

4

Pedro R. S. Antunes

and consider the approximations, for some Nr ∈ N, (5)

rΩ (t) ≈ r˜Ω (t) =

Nr 

aΩ j cos(jt) +

j=0

Nr 

bΩ j sin(jt)

j=1

and (6)

rD (t) ≈ r˜D (t) =

Nr 

aD j cos(jt) +

j=0

Nr 

bD j sin(jt).

j=1

The eigenvalue problem (1) can be solved by a domain decomposition technique. We will denote by uD and uΩ\D the restrictions of an eigenfunction respectively to D and Ω\D. Thus, (1) is equivalent to the problem

(7)

⎧ −ΔuΩ\D = ρ1 λuΩ\D ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ −ΔuD = ρ2 λuD uD = uΩ\D ⎪ ⎪ ⎪ ∂uD = ∂uΩ\D ⎪ ⎪ ∂n ∂n ⎪ ⎩ uΩ\D = 0

in Ω\D in D on ∂D on ∂D on ∂Ω.

(7) Note that, since ρ1 and ρ2 are constants, the PDE’s involved in the problem √ are both the Helmholtz equation, but with different frequencies kΩ\D = λρ1 and √ kD = λρ2 . We take a fundamental solution of the Helmholtz equation in R2 , (8)

Φk (x) =

i (1) H (k x ) , 4 0 (1)

where . is the Euclidean norm and H0 consider the approximations (9)

is a Hankel function of the first kind and

uD (x) ≈ u ˜D (x) =

ND 

αi φj (x),

i=1

and NΩ\D

(10)

uΩ\D (x) ≈ u ˜Ω\D (x) =



βi ψj (x),

i=1

where φj = ΦkD (· − yjD ), Ω\D

ψj = ΦkΩ\D (· − yj

)

Ω\D ˆ D and Γ ˆ Ω\D , are some points distributed on admissible source sets Γ and yjD , yj D ˆ such that Ω ⊂ Ω, ˆ ˆ is the boundary of a domain Ω respectively. The source set Γ ˆ Ω\D = Γ ˆ1 ∪ Γ ˆ 2 , where Γ ˆ 1 and Γ ˆ 2 are the boundaries of domains Ω ˆ 1, Ω ˆ 2 , such while Γ ˆ 2 , as illustrated in Figure 3. that Ω1 ⊂ D and Ω ⊂ Ω We follow the choice of source points described in [AA1, AA2]. Given a sample of points xi , i = 1, ..., M (almost) uniformly distributed on some boundary, we calculate the source points by

(11)

yj = xj + δnj ,

Harmonic membranes

5

Figure 3. The admissible source set for a non simply connected domain. where nj is the unitary outward normal vector to the boundary at the point xj and δ is a positive parameter. ˜D can be justified by the The approximation of uD by the linear combination u following density result, that was proven in [AA2]. ˆ D is an admissible source set, then Theorem 1. If Γ

ˆD ˆ D ) = span Φκ (· − y)|Ω : y ∈ Γ S(Γ D  is dense in HκD (D) = v ∈ H 1 (D) : (Δ + λρ2 )v = 0 , with the H 1 (D) topology. With a similar argument to that was used to prove Theorem 1, we can obtain a density result for the approximation in uΩ\D , but in that case we must assume ˆ 1 , whose boundary is Γ ˆ 1 (see that κΩ\D is not an eigenfrequency of the domain Ω Figure 3). It is straightforward to calculate the gradient of a MFS linear combination. We know that for r > 0, (1)

∂H0 (r) (1) = −H1 (r) ∂r and thus, ∇φj (x) = −

  ikD x − yjD

H1(1) kD x − yjD D 4 x − yj

and ∇ψj (x) = −

ikΩ\D 4

Ω\D

  x − yj Ω\D

H (1) kΩ\D

x − yj .

Ω\D 1

x − yj

For arbitrary sets of points X = {xi , i = 1, 2, ..., NX }, Y = {yi , i = 1, 2, ..., NY } and Z = {zi , i = 1, 2, ..., NZ } will define the matrices ⎞ ⎛ Φk (x1 − y1 ) · · · Φk (x1 − yNY ) ⎟ ⎜ .. .. .. M(k, X, Y ) = ⎝ ⎠ . . . Φk (xNX − y1 )

···

Φk (x1 − yNY )

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Pedro R. S. Antunes

and

⎛

n1,1 ⎜ .. N(k, X, Y, Z) = ⎝ . nNX ,1 where ni,j

ik =− 4



··· .. . ···

⎞ n1,NY ⎟ .. ⎠, . nNX ,NY

 xi − yj (1) .zi H1 (k xi − yj ) . xi − yj

We assume that we have sets of collocation points  X ∂D = x∂D ∈ ∂D, i = 1, ..., M∂D , i  X ∂Ω = x∂Ω ∈ ∂Ω, i = 1, ..., M∂Ω , i a set of normal vectors

 N ∂D = n∂D i , i = 1, ..., M∂D ,

where n∂D is the normal vector at the point x∂D and sets of source points i i

ˆ D , i = 1, ..., ND , Y D = yiD ∈ Γ

Ω\D ˆ Ω\D , i = 1, ..., NΩ . Y Ω\D = yi ∈Γ We will also define sets of points randomly distributed in each of the sub-regions  W D = wiD ∈ D, i = 1, ..., MD ,

Ω\D ∈ Ω\D, i = 1, ..., MΩ\D . W Ω\D = wi Note that by construction, the linear combinations u ˜D and u ˜Ω\D satisfy the corresponding PDE’s in problem (7), since we are using a fundamental solution of the Helmholtz equation. Thus, for solving (7), we can focus just on the approximations of the boundary conditions on ∂Ω and continuity conditions at the interface ∂D, which can be written as a linear system, A(λ)v = 0, where

⎛

M(kD , X ∂D , Y D ) ⎝ A(λ) = N(kD , X ∂D , Y D , N ∂D ) 0

⎞ −M(kΩ\D , X ∂D , Y Ω\D ) −N(kΩ\D , X ∂D , Y Ω\D , N ∂D )⎠ M(kΩ\D , X ∂Ω , Y Ω\D )

and v = [α1 , · · · , αND , β1 , · · · , βNΩ\D ]T . We will also define the matrix   M(kD , W D , Y D ) 0 B(λ) = 0 M(kΩ\D , W Ω\D , Y Ω\D ) The numerical approximations for the eigenvalues of the two-density membrane can be calculated by plotting σ1 (λ), the smallest generalized singular value of {A(λ), B(λ)} and calculating the local minima (eg. [B1]).

Harmonic membranes

7

3. Optimization of the shape and the densities of the membrane We will search for optimal configurations of membranes, in the sense that we have almost harmonic relations in the first eigenfrequencies. We will denote by [x], the integer number that is closest to x and to avoid ambiguities we define [i + 12 ] = i, i = 1, 2, .... Then, for each domains Ω, D and density ρ we define an integer number H, which is the number of harmonic relations to be involved in the optimization. Then, we build two vectors Q = (q1 , ..., qNκ −1 ) ,

qi =

κi+1 (Ω, D, ρ) , i = 1, ..., Nκ − 1 κ1 (Ω, D, ρ)

and ˜ = (˜ Q q1 , ..., q˜Nκ −1 ) ,

q˜i = [qi ] ,

where Nκ is the number of eigenfrequencies that are considered in the optimization and is defined such that q˜Nκ −1 = H and q˜Nκ = H + 1. To measure how far we are from harmonicity de define 2 N κ −1  qj − q˜j (12) F(Ω, D, ρ, H) = q˜j j=1 and consider the optimization problem inf F(Ω, D, ρ, H).

(13)

Ω,D,ρ

Next we study the dependence of F(Ω, D, ρ, H) in terms of perturbations of Ω, D and ρ. For a given deformation field V , we consider an application Ψ(t) such that Ψ : t ∈ [0, T [→ W 1,∞ (Rd , Rd ) is differentiable at 0 with Ψ(0) = I and Ψ (0) = V . By W 1,∞ (Rd , Rd ) we denote the set of bounded Lipschitz maps from Rd into itself and I is the identity. We will denote by Ωt = Ψ(t)(Ω), Dt = Ψ(t)(D), λi (t) = λi (Ωt , ρ) and κi (t) = κi (Ωt , ρ) and will assume that λi (t) is simple  and the corresponding eigenfunction ui (t) satisfies the normalization condition Ωt ρui (t)2 dx = 1. We will also denote by u , the derivative of ui (t) at t = 0. Next, we derive the Hadamard shape derivate with respect to perturbations of the boundaries ∂Dt and ∂Ωt . It is well known that if Ω is an open domain with Lipschitzian boundary and we define  y(t, x)dx, J(t) = Ωt

for some C 1 function y, then ([S, SZ]) the Hadamard derivative is given by   ∂y (0, x)dx + y(0, x)V.ndsx . (14) J  (0) = Ω ∂t ∂Ω As a consequence we have the following Theorem 2. The functional λi (t) is differentiable at t = 0 and 2    ∂ui u2i V.ndsx − V.ndsx . (15) λi (0) = −(2 − 1 )λi (Ω, ρ) ∂n ∂D ∂Ω Proof: The eigenfunction satisfies the normalization condition   1 u2i (t)dx + 2 u2i (t)dx = 1 Ωt \Dt

Dt

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Pedro R. S. Antunes

and differentiating we obtain      1 ui u dx+ 1 u2i V.ndsx − 1 u2i V.ndsx +2 2 ui u dx+ 2 u2i V.ndsx = 0. 2  Ω\D ∂Ω ∂D D ∂D =0

Thus,



(16)

1 ui u dx + 2

2



Ω\D

2 ui u dx = − (2 − 1 )

D

On the other hand,  λi (Ωt ) =

∂D

 2

|∇ui (t)| dx +

Ωt

Dt



∇ui .∇u dx + 2

2 D



u

2 ∂D

∂ui dsx −2 ∂n 





∇ui .∇u dx + 

D

u ∂D

D

u ∂Ω

Ωt \Dt

Ω\D

Δui u dx−2 

2

|∇ui (t)| dx

∇ui .∇u dx+

Ω\D

Δui u dx + 2

−2

 2

|∇ui | V.ndsx +2 ∂D

u2i V.ndsx .



2

|∇ui (t)| dx =

and differentiating,   λi (0) = 2 ∇ui .∇u dx+ D



∂ui dsx +2 ∂n

∂ui dsx − 2 ∂n





 2

∂Ω



2

|∇ui | V.ndsx −

|∇ui | V.ndsx = ∂D

2

|∇ui | V.ndsx = 

∂Ω

u ∂Ω

∂ui dsx −2 ∂n

Δui u dx + Ω\D



Δui u dx+ Ω\D

 2

|∇ui | V.ndsx = ∂Ω

 2

|∇ui | V.ndsx ∂Ω

by using Green’s first identity. Now using the PDE of the problem, we have λi (0)







=2

λi (Ω, ρ)2 ui u dx+2 D

u ∂Ω

 ∂ui

∂n



 dsx +2

∂D

u2i V.ndsx + 2

u ∂Ω





λi (Ω, ρ)1 ui u dx+ Ω\D



−λi (Ω, ρ) (2 − 1 )



∂ui dsx + ∂n

∂Ω



 ∂Ω

∂ui ∂n

∂ui ∂n

2 V.ndsx =

2 V.ndsx

using (16). Finally we note that we can determine u |∂Ω by differentiating the homogeneous Dirichlet boundary conditions to get (e.g. [HP]) u = − Thus,

∂ui V.n, on ∂Ω. ∂n

2 2   ∂ui ∂ui V.ndsx + V.ndsx = ∂n ∂n ∂D ∂Ω ∂Ω 2    ∂ui  2 ui V.ndsx − V.ndsx .  λi (0) = −λi (Ω, ρ) (2 − 1 ) ∂n ∂D ∂Ω Since the functional F(Ω, ρ, Nκ ) was defined in terms of the eigenfrequencies (not the eigenvalues) we have λi (0) = −λi (Ω, ρ) (2 − 1 )







u2i V.ndsx −2

Corollary 1. The functional κi (t) is differentiable at t = 0 and 2    (2 − 1 ) 1 ∂ui (17) κi (0) = − κi (Ω, ρ) u2i V.ndsx − V.ndsx . 2 2κi (Ω, ρ) ∂Ω ∂n ∂D

Harmonic membranes

9

Proof: Is follows immediately from Theorem 2 using the fact that  κi (t) = λi (t) thus,

λ (0) κ (0) =   2 λi (0) Regarding the derivative of an eigenvalue with respect to perturbations of the density, we know that (eg. [H])  ∂λi (Ω, ρ + tη) |t=0 = −λi (Ω, ρ) ηu2i dx. ∂t Ω which implies that  ∂κi (Ω, ρ + tη) κi (Ω, ρ) |t=0 = − ηu2i dx. ∂t 2 Ω Now we note that we have ρ = ρ1 χΩ\D + ρ2 χD , thus, the derivatives of the eigenfrequency with respect to perturbations of the parameters ρ1 and ρ2 are given respectively by   ∂κi (Ω, ρ + tχΩ\D ) κi (Ω, ρ) κi (Ω, ρ) |t=0 = − χΩ\D u2i dx = − u2i dx ∂t 2 2 Ω Ω\D and

  κi (Ω, ρ) κi (Ω, ρ) ∂κi (Ω, ρ + tχD ) |t=0 = − χD u2i dx = − u2i dx. ∂t 2 2 Ω D The optimization problem for the harmonic membrane is solved by the minimization of F(Ω, ρ, Nκ ) with some constraints, namely ρ1 > 0, ρ2 > 0. Moreover, we assumed that D ⊂ Ω, which can be written as a constraint. We will denote by θ(x, y) the four-quadrant inverse tangent function that for each point (x, y) ∈ R2 \ {(0, 0)} associates an angle in (−π, π]. Thus, for some MT , we define ti =

2π(i − 1) , i = 1, 2, ..., MT MT

and θi = θ(˜ rD (ti ) cos(ti ) + x0 , r˜D (ti ) sin(ti ) + y0 ), i = 1, 2, ..., MT and using (3), (4), (5) and (6), for each ti we have a nonlinear constraint  (˜ rD (ti ) cos(ti ) + x0 )2 + (˜ rD (ti ) sin(ti ) + y0 )2 ≤ r˜Ω (θi ). 4. Numerical results In this section we present some numerical results obtained with our algorithm. Typically we used the following parameters ND = 150, NΩ\D = 300, M∂D = 150, M∂Ω = 200, MD = MΩ\D = 50, and δ ∈ [0.1, 0.3], except for the optimizers of example 1 which were obtained with ND = 700, NΩ\D = 800, M∂D = 500, ˆ D and Γ ˆ1 M∂Ω = 600, MD = MΩ\D = 50, and δ = 0.02 for the point-sources on Γ 2 ˆ . and δ = 0.2, for the point-sources on Γ In all the numerical experiments, we considered the optimization on both ρ1 and ρ2 . However, we present the numerical results just for the quantity R := ρ2 /ρ1 , as in previous studies on this subject (eg. [GGL]). Indeed we could consider just

10

Pedro R. S. Antunes

the optimization of R. The main objective is to have a membrane with a few harmonic relations on the first eigenfrequencies, but not to have a membrane with a prescribed fundamental frequency. Of course, once we modify the densities ρ1 and ρ2 , keeping R constant, the fundamental will change but not the qualitative acoustic properties of the membrane in terms of the harmonic relations on the eigenfrequencies. Assuming that we have a configuration that allows to have some harmonic relations on a few eigenfrequencies, we can easily prescribe the pitch of the drum, just by considering an appropriate homothety of the region. The choice of H may affect slightly the configurations of membranes that are obtained with the optimization procedure. However, it is not very important to prescribe a very large number H, since the higher overtones are damped and the corresponding eigenmodes contain little energy (eg. [R, FR]). Moreover, the choice of a large value of H would imply an increase of the computational cost, since there would be a large number of eigenvalues involved in the objective function. We considered examples obtained with H = 6 and H = 9. In the first example we considered H = 6, which means that the optimization procedure is applied to determine configurations that provide five (approximate) harmonic relations. Table 1 shows the value of the harmonicity function (12) and the approximate harmonic relations of four configurations of membranes. The first configuration that was considered corresponds to the one usually associated with the dayan. In this case Ω and D areconcentric disks. The ratio of the radii is equal to 0.4 and the densities satisfy ρ2 /ρ1 = 3.125 (eg. [GGL]). As was mentioned in the introduction, this configuration allows to have good approximations for some harmonic relations. However, we can identify some (double) eigenfrequencies for which the corresponding ratios with the fundamental are not close to an integer number, for example the ratios 4.81979, 5.43927 or 6.23221. The second configuration is the optimal two-density membrane for which Ω and D are concentric disks. This problem can be simplified using separation os variables in polar coordinates and we obtain a one dimensional eigenvalue problem whose solutions are given by Bessel functions. For obtaining this optimal configuration, we considered the minimization of the harmonicity function (12) by a direct search method. The eigenvalues were calculated solving the nonlinear equations involving Bessel functions. This problem is much simpler than the general optimization considered in this paper, because it can be formulated as a minimization of a function that depends just on  two variables, for example the ratio of the radii of /ρ1 . We obtained that the optimal ratio of the Ω and D and the quantity ρ2 radii is equal to 0.38910786 and ρ2 /ρ1 = 2.9264663. In this case we have better approximations for harmonic relations, but we still have some ratios not close to be integer numbers, namely 4.59992 and 5.19960. Finally, we show results for the numerical optimizer  obtained with our algorithm, which is plotted in Figure 4-left. In this case we have ρ2 /ρ1 ≈ 4.2268822 and we obtained good approximations for harmonic relations that are presented in the third column of Table 1. However, probably this configuration is too complex to be used in a real drum. Thus, in order to avoid a highly irregular interface, instead of the the minimization of the function F(Ω, ρ, H) defined in (12), we tried to minimize a new cost function ! 2 2 |∂Ω| − 4π |Ω| |∂D| − 4π |D| ˜ + , (18) F(Ω, D, ρ, H) = F(Ω, D, ρ, H) + η |Ω| |D|

Harmonic membranes

11

for some parameter η > 0. Note that, due to the classical isoperimetric inequality, the penalization term is always non-negative and vanish only if Ω and D are both disks. Thus, if η = 0, we have ˜ F(Ω, D, ρ, H) = F(Ω, D, ρ, H) and we obtain the previous optimizer. On the other hand, if η is very large, we expect the optimizer to be the optimal union of disks. We considered the optimization with η = 10−4 and obtained a new optimizer which is plotted in Figure4-right. The corresponding results are shown in last column of Table 1.



ρ2 /ρ1 F (Ω, ρ, 6) 1st overtone 2nd overtone

3rd overtone

4th overtone

5th overtone

Dayan 3.125 0.0148 1.93676 1.93676 2.94697 2.94697 3.05200 3.96657 3.96657 4.09639 4.09639 4.81979 4.96506 4.96506 5.14483 5.14483 5.43927 5.43927 5.93686 5.93686 6.03206 6.18827 6.18827 6.23221 6.23221

Membrane configurations Optimal concentric disks Numerical optimizer 1 2.9264663 4.2268822 0.0092 0.0020 1.94638 1.95205 1.94638 1.95276 2.97217 2.99619 2.97217 2.99778 3.04475 3.00005 4.01244 4.02045 4.01244 4.02182 4.03739 4.03679 4.03739 4.05203 4.59992 4.98024 5.00319 5.01228 5.00319 5.01777 5.03353 5.04287 5.03353 5.04553 5.19960 5.19960 5.97358 5.93681 5.98564 5.94061 5.98564 5.95031 5.99570 5.96059 5.99570 6.00555 6.02711 6.04563 6.02711

Numerical optimizer 2 3.8206642 0.0055 1.91556 1.93350 2.93763 2.95033 2.97968 3.98922 3.99379 4.02802 4.03431 5.0060 5.02323 5.03512 5.08064 5.11317

5.91110 5.94020 6.01866 6.04571 6.08130 6.11954

 Table 1. The optimal value ρ2 /ρ1 , the value of the harmonicity function (12) and the approximate harmonic relations for four configurations of membranes - the dayan, the optimal configuration obtained with concentric disks, and two numerical optimizers obtained with our numerical algorithm which are plotted in Figure 4.

In last example we considered the optimization problem with eight approximate overtones (H = 9) among two- and three-density membranes. Table 2 shows the main results that we gathered. The first column contain the results corresponding to the dayan. The second column is related with the optimal configuration among two-density membranes with concentric disks. In this case, the optimal ratio of the

12

Pedro R. S. Antunes

1

1

0.5 0.5

0 0

−0.5

−0.5 −1

−1 −1

−0.5

0

0.5

1

−1

−0.5

0

0.5

1

Figure 4. Numerical optimizers that allow to have five approximate harmonic relations.  radii is equal to 0.38872192 and ρ2 /ρ1 = 2.9372957. The third column presents the results obtained for the numerical optimizer which is plotted in Figure 7-left. In this case, each overtone is generated by degenerate eigenfrequencies with increasing multiplicity. The first overtone (the octave) is generated by a double eigefrequency, the second overtone by a triple eigenfrequency and so on. In Figure 5 we plot eigenfunctions associated with the fundamental, two eigenfunctions associated to the first overtone and three eigenfunctions generating the second overtone. Figure 6 presents the plots of nine (linearly independent) eigenfunctions generating the 8th overtone. We can observe that some of the eigenfunctions (for example the last six eigenfunctions) are mostly localized inside D, while some of them (the first three eigenfunctions) have also resonance in Ω\D.

(a)

(b)

(c) Figure 5. (a) plot of the eigenfunction associated with the fundamental; (b) plots of two eigenfunctions associated with the first overtone; (c) plots of three eigenfunctions associated to a triple eigenfrequency that generate the second overtone. We considered also optimization among three-density membranes. In this case, we assume that we have domains satisfying D ⊂ D ⊂ Ω and the density is given

Harmonic membranes

13

Figure 6. Plots of nine eigenfunctions associated to the 8th overtone. by ρ = ρ1 χΩ\D + ρ2 χD \D + ρ3 χD . The fourth column of Table 2 shows results for the optimal configuration among concentric disks. In this case, is we assume that the radius of the outer circle is equal to one, then the radii  defining the interior disks are equal to 0.34971358 and 0.49022465 and we have ρ2 /ρ1 = 3.6980882 and  ρ3 /ρ1 = 5.0535524. In last column we present results obtained for the numerical optimizerwith three-density membrane  that is plotted in Figure 7-right for which we have ρ2 /ρ1 = 8.4976065 and ρ3 /ρ1 = 12.5093430. These optimizers were obtained running the optimization process with H = 9. However, these optimal configurations allow to have some additional quotients qi close to being integer numbers. Figure 8 shows the quotients qi , i = 1, 2, ..., 100 and the integer numbers are marked with a dashed red line. 5. Conclusions and future work In this work we considered the problem of finding harmonic composite membranes. We developed a numerical method to solve the shape and density optimization of the composite membrane that allowed to present new configurations for composite membranes for which some of the first eigenfrequencies are multiple of the fundamental. In particular, we obtained two- and three-density membranes allowing to have (approximate) eight harmonic overtones. Besides these eight overtones that were involved in the optimization, these optimal membranes provide also some additional higher eigenfrequencies close to be harmonic overtones. All the coefficients that define the optimal domains are included in the appendix, so that all the results can be reproduced by other studies. We hope that these new configurations can be used in the construction of real percussion instruments in order to build drums producing sounds with definite

14

Pedro R. S. Antunes

1

1

0.5 0.5

0 0

−0.5

−0.5

−1

−1

−1

−0.5

0

0.5

1

−1

−0.5

0

0.5

1

Figure 7. Two- and three-density membrane optimizers with H = 9.

Figure 8. Quotients qi , i = 1, 2, ...100 for the two- and threedensity optimizers plotted in Figure 7. pitch. A further step towards the construction of these harmonic drums would involve an optimization of the largest number of eigenfrequencies such that the distance between the ratio with the fundamental to the closest integer number is smaller than the limits of human ear. Another question that should be taken into account is related with the point of the membrane where we strike the drum. We know that all the eigenfunctions ui , i = 2, 3, ... have nodal lines. Moreover, if we strike the drum at a point that is located on a nodal line of a certain eigenfunction, then the corresponding eigenmode is not excited. This technique is well known and used by the percussionists. They choose the location where to strike the membrane in order to have a desired sound, by selecting just some eigenmodes. For example, the configuration plotted in Figure 4-left allows to obtain five approximate harmonic relations, involving the first 21 eigenfrequencies. The highest ratio that was considered in the objective function was κ21 /κ1 . If we calculate the next ratios for this configuration we obtain κ23 κ24 κ25 κ26 κ22 = 6.4159, = 6.7082, = 6.9178, = 6.9248, = 6.9934. κ1 κ1 κ1 κ1 κ1 κ26 κ24 κ25 , and are good apThis means, that we may assume that the ratios κ1 κ1 κ1 κ22 proximations for the integer number 7 and would like to suppress the ratios κ1

Harmonic membranes

15

κ23 which are far from being integer numbers. In Figure 9 we plot the nodal κ1 lines of the eigenfunctions associated with the eigenfrequencies κ22 and κ23 , marked respectively with a thin blue line and a thick red line. A good strategy is to strike the membrane at any intersection of the nodal lines of both eigenfunctions. and

1

0.5

0

−0.5

−1

−1.5 −1.5

−1

−0.5

0

0.5

1

Figure 9. Nodal lines of the eigenfunctions associated with the eigenfrequencies κ22 (thin blue line) and κ23 (thick red line). If we strike the membrane at any of the intersections of the the nodal lines, these eigenfrequencies will be suppressed. Besides optimizing the shape and density of the membrane, we could try to optimize the point where to play the drum, so that we avoid the eigenfrequencies whose ratios with the fundamental are far from being harmonic. Of course, a more realistic model for the vibration of a drum must include the effect of the body of the drum and the effect of the surrounding air. We hope to address this study in a future work. Acknowledgements I would like to thank OldDelhiMusic.com for permission to reproduce Figure 1. References [AA1]

[AA2] [A] [AB] [B1]

C.J.S. Alves and P.R.S. Antunes, The Method of Fundamental Solutions applied to the calculation of eigenfrequencies and eigenmodes of 2D simply connected shapes, Computers, Materials & Continua 2, (4) (2005), 251–266. C.J.S. Alves and P.R.S. Antunes, The Method of Fundamental Solutions applied to some inverse eigenproblems, SIAM J. Sci. Comp. 35,(3) (2013), A1689–A1708. P.R.S. Antunes, Optimization of sums and quotients of Dirichlet-Laplacian eigenvalues, Applied Mathematics and Computation 219(9) (2013), 4239–4254. M. Ashbaugh and R. Benguria, Proof of the Payne-P´ olya-Weinberger conjecture, Bull. Amer. Math. Soc. (N.S.), 25 (1991), 19–29. T. Betcke, A GSVD formulation of a domain decomposition method for planar eigenvalue problems, IMA J. Numer. Anal., 27 (2007), 451–478.

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[B2] [FR] [GGL] [H] [HP] [O] [RS] [R] [RK] [S] [SZ]

Pedro R. S. Antunes

Bharata-Muni, The Natyasastra, Bibliotheca Indica, Calcuta, (1951), (Translated by Manmohan Ghosh). N.H. Fletcher and Rossing D.T., The physics of musical instruments, Springer, Houston, (1998). S. Gaudet, C. Gauthier, and S. L´ eger, The evolution of harmonic indian musical drums: A mathematical perspective, Journal of Sound and Vibration 291 (2006), 388–394. A. Henrot, Extremum problems for eigenvalues of elliptic operators, Frontiers in Mathematics. Birkh¨ auser Verlag, Basel, (2006). A. Henrot and M. Pierre, Variation et optimisation de formes. Une analyse g´eom´ etrique. Springer, Series Math´ ematiques et Applications, 48, (2005). B. Osting, Optimization of spectral functions of Dirichlet-Laplacian eigenvalues, J. Comp. Phys. 229 (2010), 8578–8590. B.S. Ramakrishna and M. M. Sondhi, Vibration of indian musical dums regarded as composite membranes, J. Acoust. Soc. Am. 26 (1954), 523–529. C.V. Raman, Indian musical drums, Proc. Indian Acad. Sci., Sect. A 1A (1935), 179–188. C.V. Raman and S. Kumar, Musical drums with harmonic overtones, Nature 104 (1920), 500–500. J. Simon, Differentiation with respect to the domain in boundary value problems, Numer. Funct. Anal. Optim. 2 (1980), 649-687. J. Sokolowski and J. P. Zolesio, Introduction to shape optimization: shape sensitity analysis, Springer Series in Computational Mathematics Vol 10, Springer, Berlin, (1992).

Harmonic membranes

F (Ω, ρ, 9) 1st overtone 2nd overtone

3rd overtone

4th overtone

5th overtone

6th overtone

7th overtone

8th overtone

Dayan 0.0448 1.93676 1.93676 2.94697 2.94697 3.05200 3.96657 3.96657 4.09639 4.09639 4.81979 4.96506 4.96506 5.14483 5.14483 5.43927 5.43927 5.93686 5.93686 6.03206 6.18827 6.18827 6.23221 6.23221 6.88599 6.88599 6.98854 6.98854 7.04945 7.04945 7.35689 7.35689 7.81775 7.81775 7.98008 7.98008 8.02386 8.02386 8.07476 8.49194 8.49194 8.73628 8.73628 8.95761 8.95761 9.07664 9.07664 9.11831 9.11831

17

Membrane configurations Two-density membranes Three-density membranes Opt. conc. disks Num. optimizer Opt. conc. disks Num. optimizer 0.0154 0.0128 0.0049 0.0042 1.94721 1.89995 1.93088 1.93469 1.94721 1.90130 1.93088 1.93675 2.97382 2.89332 2.93732 2.95183 2.97382 2.89387 2.93732 2.95241 3.04691 2.91467 3.00794 3.01077 4.01456 3.91305 3.96660 3.97651 4.01456 3.93018 3.96660 3.99764 4.04196 3.94102 4.01966 4.02183 4.04196 3.94183 4.01966 4.02759 4.60666 4.94906 4.96263 4.96508 5.01140 4.95094 4.99599 5.00536 5.01140 4.97547 4.99599 5.00998 5.03580 4.99166 5.02941 5.02937 5.03580 4.99228 5.02941 5.03328 5.20610 5.20610 5.97699 5.98064 5.94088 5.93563 5.99154 5.98262 5.94088 5.93990 5.99154 6.01512 6.02062 6.00404 6.00830 6.03161 6.02062 6.03169 6.00830 6.03195 6.03794 6.04174 6.02942 6.04823 6.03794 6.045917 6.02942 6.89622 7.00222 6.94445 6.95229 6.89622 7.03068 6.97099 6.95407 6.99877 7.05143 6.97099 6.95596 6.99877 7.05290 7.03250 7.00613 7.00206 7.06494 7.03250 7.00856 7.00206 7.09823 7.04128 7.04516 7.04581 7.09903 7.04128 7.06688 7.04581 7.86193 8.03331 8.00764 7.97116 7.86193 8.03776 8.00764 7.97405 7.94946 8.07467 8.01114 7.97968 7.94946 8.07940 8.01114 7.99371 8.05393 8.10207 8.02375 8.02289 8.05393 8.10630 8.02375 8.02599 8.06877 8.11436 8.05954 8.06458 8.10869 8.14755 8.05954 8.06885 8.10869 8.85750 8.86712 8.95882 8.95431 8.85750 9.01033 8.96516 8.95730 8.88606 9.01236 8.96516 9.01187 8.88606 9.09808 9.07308 9.01807 8.97474 9.10117 9.07308 9.06084 8.97474 9.10648 9.07648 9.06284 9.10604 9.12118 9.07648 9.09719 9.10604 9.12895 9.11618 9.10624 9.17764 9.13907 9.11618 9.11700 9.17764 9.13907 9.11618 9.11700

Table 2. The optimal value of the harmonicity function (12) and the approximate harmonic relations for some two and three-density membranes.

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Pedro R. S. Antunes

6. Appendix

j 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

Coefficients of the optimal domain (H = 6) x0 (Ω) y0 (Ω) x0 (D) y0 (D) 0 0 -1.0663×10−2 -8.5144×10−2 aΩ bΩ aD bD j j j j 1.248150 0.481949 -2.3994×10−2 -1.2260×10−1 -1.2704×10−2 -3.0406×10−2 1.7615×10−2 6.7664×10−3 -5.1745×10−3 -2.6313×10−3 −2 -1.4800×10 1.0115×10−1 6.0984×10−3 -4.9065×10−2 1.5673×10−2 -1.6291×10−3 -5.6037×10−3 1.6063×10−3 -7.9246×10−4 8.8201×10−4 -5.9911×10−4 -6.7495×10−4 3.4866×10−2 3.7180×10−3 -2.6544×10−3 -2.2115×10−3 9.7241×10−5 -2.8393×10−3 -4.2567×10−4 -1.6805×10−3 -1.6714×10−3 -2.3191×10−4 1.1905×10−2 2.8626×10−3 2.1569×10−3 -7.1754×10−3 7.8660×10−3 3.4602×10−2 −3 −5 −2 -1.2176×10 -7.1608×10 1.2059×10 5.0984×10−4 -4.2675×10−4 1.8935×10−4 -2.5290×10−2 2.2243×10−2 -1.4524×10−3 -2.0619×10−4 -7.2018×10−2 -2.9808×10−2 -8.0827×10−5 3.4489×10−4 -4.8490×10−3 3.3592×10−2 2.0064×10−4 1.3373×10−4 5.8453×10−3 -6.4237×10−3 −4 −4 -2.0377×10 2.1932×10 7.6286×10−3 -2.8221×10−3 2.7438×10−4 4.6968×10−5 -4.1893×10−3 2.2876×10−3 1.1313×10−4 -4.9072×10−5 -1.5421×10−2 1.2170×10−2 2.1013×10−4 4.2399×10−5 -2.7404×10−2 -1.6374×10−2 2.1019×10−5 -4.7080×10−5 -5.9999×10−3 1.9907×10−2 1.6316×10−5 -1.2491×10−5 7.3540×10−3 -3.6058×10−3 -2.6728×10−5 2.7763×10−5 3.5966×10−3 -2.3831×10−3

Table 3. Coefficients of the optimal domain plotted in Figure 4-left.

ˆncias da Universidade de Lisboa, Group of Mathematical Physics, Faculdade de Cie Campo Grande, Edif´ıcio C6 1749-016 Lisboa, Portugal E-mail address: [email protected]

Harmonic membranes

j 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

Coefficients of the optimal domain (H = 6) x0 (Ω) y0 (Ω) x0 (D) y0 (D) 0 0 1.5364×10−2 4.0805×10−2 aΩ bΩ aD bD j j j j 0.992998 0.402587 -8.7065×10−3 3.0819×10−2 -1.6406×10−2 -6.8657×10−3 4.0205×10−3 9.7076×10−4 -2.4398×10−4 -7.7082×10−3 -3.2165×10−2 3.6284×10−2 3.9286×10−2 -3.5866×10−2 −2 −3 -1.4133×10 -5.8114×10 -4.2914×10−3 -1.0013×10−3 -3.8359×10−3 -1.4604×10−3 4.7031×10−4 -9.4072×10−4 -3.3566×10−3 -5.6995×10−4 1.2603×10−3 -1.3913×10−2 -3.9347×10−3 -1.3801×10−3 2.1999×10−3 -1.1196×10−3 -1.9242×10−3 -2.4626×10−3 -2.8130×10−3 -3.3404×10−3 -1.0441×10−3 -1.9985×10−3 4.3578×10−3 -6.8185×10−3 -5.2898×10−4 -1.5545×10−3 7.8238×10−3 -9.4487×10−3 -3.1877×10−4 -1.4573×10−3 1.4720×10−2 -9.6295×10−4 8.5421×10−5 -1.3077×10−3 1.7059×10−2 9.4081×10−3 3.7954×10−4 -1.0589×10−3 1.8936×10−3 1.3160×10−2 −4 −4 −3 5.6304×10 -7.1313×10 -4.2304×10 7.9456×10−3 5.6993×10−4 -4.1167×10−4 -3.7034×10−3 2.6822×10−3 5.0427×10−4 -1.9919×10−4 -2.2210×10−3 1.3578×10−3 4.3455×10−4 -3.8026×10−5 -2.7811×10−3 1.2303×10−3 2.9115×10−4 8.4405×10−5 -2.7526×10−3 1.3534×10−3 1.6406×10−4 7.1738×10−5 -2.2207×10−3 -1.4584×10−3 1.3317×10−4 5.5633×10−5 -7.5877×10−4 -1.5191×10−3 −5 8.5226×10 9.2728×10−5 5.9109×10−4 -3.9745×10−4

Table 4. Coefficients of the optimal domain plotted in Figure 4-right.

19

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Pedro R. S. Antunes

j 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

Coefficients of the optimal domain (H = 9) x0 (Ω) y0 (Ω) x0 (D) y0 (D) 0 0 -1.0980×10−2 -8.4372×10−2 aΩ bΩ aD bD j j j j 1.231909 5.15083 × 10−1 -2.1595×10−2 -1.2230×10−1 -1.2593×10−2 -3.4966×10−2 −2 −3 −3 1.6014×10 5.7362×10 -8.6056×10 -3.6038×10−3 -1.7407×10−2 1.1064×10−1 1.1012×10−2 -8.3202×10−2 −2 −3 −3 2.0049×10 -1.8181×10 -8.6216×10 2.1592×10−3 2.0752×10−3 -7.5998×10−3 -2.1702×10−3 3.9769×10−3 −2 −3 −2 5.3587×10 8.9376×10 -2.2899×10 -6.1999×10−3 −4 −3 −4 -2.5768×10 -8.2047×10 5.8104×10 4.1152×10−3 -8.0908×10−3 -2.0043×10−3 2.1382×10−3 1.6231×10−3 −3 −2 −3 7.2290×10 -1.6186×10 -3.8450×10 9.3562×10−3 -6.0640×10−3 -2.2954×10−4 2.5630×10−3 -9.6989×10−5 -2.5602×10−3 4.8673×10−3 1.1760×10−3 -1.6671×10−3 -8.4425×10−3 -3.9613×10−3 4.1471×10−3 2.6845×10−3 -2.4095×10−4 2.0457×10−3 -2.1606×10−4 -2.2151×10−3 2.7879×10−3 1.4176×10−3 -1.8834×10−3 -2.4027×10−3 -2.1892×10−3 2.4454×10−3 3.1217×10−3 -4.6461×10−3 −4 −5 −4 3.3001×10 4.2328×10 -6.8602×10 -7.9853×10−4 3.3297×10−4 -3.5225×10−4 1.8354×10−3 -3.9520×10−4 −4 −4 −3 2.8037×10 2.6738×10 2.4364×10 2.3081×10−3

Table 5. Coefficients of the optimal domain plotted in Figure 7-left.

j 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

x0 (Ω) 0 aΩ j 1.048565 7.8860 × 10−4 2.6052 × 10−2 −1.2927 × 10−3 1.0670 × 10−4 5.0117 × 10−5 1.0845 × 10−5 2.9027 × 10−6 −1.7167 × 10−6 7.0914 × 10−7 −6.0807 × 10−9 1.0276 × 10−9 −5.6371 × 10−9 −1.6032 × 10−11 4.8195 × 10−11 −4.3964 × 10−12

Coefficients of the optimal domain (H = 9) x0 (D1 ) y0 (D1 ) x0 (D2 ) 1.8734 × 10−2 −1.0230 × 10−2 9.3995 × 10−3 1 1 2 aD bD aD j j j 0.545235 0.374876 −4.3548 × 10−4 −4.4230 × 10−3 −3.5520 × 10−4 4.1653 × 10−3 −1.2116 × 10−4 5.9928 × 10−4 −1.8065 × 10−4 −2.5375 × 10−3 −4 −2 −2.3554 × 10 1.2487 × 10 −1.5643 × 10−3 −1.3922 × 10−2 9.3889 × 10−5 −3.1291 × 10−4 −6.8736 × 10−6 -1.3089 × 10−4 7.4948 × 10−7 −5.2481 × 10−4 2.1864 × 10−3 2.8162 × 10−4 1.6426 × 10−4 −1.2854 × 10−3 4.2579 × 10−4 1.4330 × 10−3 −6 −5 −5 3.0184 × 10 −6.9778 × 10 −6.3990 × 10 −6.5225 × 10−4 −4.6261 × 10−6 −5.3086 × 10−4 −2.8535 × 10−4 5.3473 × 10−4 4.1855 × 10−7 1.0288 × 10−3 −6.3585 × 10−4 −6.2034 × 10−3 1.6418 × 10−8 -5.6622 × 10−6 4.2663 × 10−6 −1.4868 × 10−5 −8.5550 × 10−10 1.1009 × 10−6 2.4825 × 10−6 1.1013 × 10−5 −9 −5 −5 8.6457 × 10 −1.6757 × 10 2.7425 × 10 −2.6035 × 10−5 7.6536 × 10−11 2.9910 × 10−9 −5.0037 × 10−8 −2.3174 × 10−7 −4.2774 × 10−11 8.6339 × 10−9 3.5429 × 10−9 2.8463 × 10−8 −12 −8 −8 −9.4911 × 10 −3.4079 × 10 −4.5194 × 10 −5.4313 × 10−7 y0 (Ω) 0 bΩ j

Table 6. Coefficients of the optimal domain plotted in Figure 7-right.

y0 (D2 ) −9.3209 × 10−3 2 bD j 6.2843 × 10−4 −3.7328 × 10−4 3.3024 × 10−2 6.3550 × 10−4 2.9147 × 10−4 −2.3401 × 10−4 8.0947 × 10−5 3.7842 × 10−4 −1.1050 × 10−3 9.8515 × 10−8 4.8222 × 10−7 2.3977 × 10−5 −1.4357 × 10−7 1.1749 × 10−8 −1.0735 × 10−6