Introduction The purpose of the Physics of Strings seminar has traditionally been to study the motion of a vibrating string by analyzing its eigenfunctions and eigenvalues, equivalent to the string's fundamental modes and fundamental frequencies, respectively. The progression of these eigenvalues and eigenvectors tell us a great deal about the string; for example, given eigenvalues of a string, we can determine how quickly its vibrations decay, and whether the frequency of a vibration affects how quickly it's damped. The properties of the string, likewise, can tell us something about the eigenvalues. Physical constants, such as the length of the string, are proportionally related to the eigenvalues. Given data on the vibration of a string, there are also methods to reverse-engineer the eigenvalues of that string. There are several models of a vibrating string, and the most detailed ones can reproduce eigenvalues that accurately match the reverse-engineered string eigenvalues. However, while much research has been done on several models of a single string, the behavior of networks of strings is less well understood. We seek to mathematically model and investigate the motion of networks of strings, specifically by understanding eigenvalues and the corresponding modes of vibration. We study these behaviors within the context of the tritar (a guitar-like instrument based upon a Y-shaped network of 3 strings) and in the vibrations of more complex networks such as spiderwebs.

The wave equation The vibration of a string in one dimension can be understood through the standard wave equation, given by ∂ 2 u ∂ t 2 = c 2 ∂ 2 u ∂ x 2 , u ( 0 , t ) = u ( ℓ , t ) = 0 where c is a constant describing wave speed and ℓ is the length of the string. We take Dirichlet boundary conditions at the ends and assume without loss of generality c=1. This second order partial differential equation can likewise be rewritten as a system of two ordinary differential equations in time ∂ u ∂ t = v ∂ v ∂ t = c 2 ∂ 2 u ∂ x 2 or equivalently, the first order matrix equation ∂ ∂ t u v = 0 I c 2 ∂ 2 ∂ x 2 0 u v

Eigenvalues, eigenfunctions, and their significance We are especially interested in the eigenvalues λ and associated eigenfunctions of the wave equation, such that 0 I ∂ 2 ∂ x 2 0 u v = λ u v , ∂ 2 u ∂ x 2 = λ 2 u Since only trigonometric functions satisfy both our equation and our boundary conditions, our eigenfunction to take the form u(x)=Asin(λx)+Bcos(λx). Applying our boundary condition at x=0 to u(x) reveals that B=0. Since we can then set A as an arbitrary scaling factor, our eigenfunction u(x) is simply sinλx. By applying our second boundary condition at x=ℓ, we can see that λ is of the form iπnℓ for any nonzero integer n. We then get the eigenpairs λ n = i π n ℓ , u n ( x ) = sin ( λ n x ) These eigenfunctions constitute an infinite-dimensional basis for any solution to the wave equation, with ui(x) orthogonal to uj(x) for i≠j with respect to the inner product 〈ui,uj〉≡∫0ℓui(x,t)uj(x,t)dx . Intuitively, these correspond to the fundamental modes of a string - any vibration of the string can be decomposed into a linear combination of the fundamentals. The magnitude of each eigenvalue, likewise, is related to the frequency at which the corresponding fundamental mode vibrates - in other words, each eigenvalue is tied to a note in the progression of the Western scale. As we will see, this linear progression of the eigenvalues is lost when a single string is replaced by a network of strings, leading to more of a dissonant sound when a network is plucked.

Finite element solution method In this report, we use the finite element method to numerically solve for solutions to the wave equation. The idea behind this method is based on picking a finite-dimensional set of N basis functions φi(x) that span the space on which the solution is defined. We then calculate the best approximation uN=∑j=1Ncjφj to the solution from the span of these basis functions via the solution to a matrix equation Ac=f, where A = φ 1 , φ 1 φ 1 , φ 2 ... φ 1 , φ N φ 2 , φ 1 φ 2 , φ 2 ... φ 2 , φ N ⋮ ⋱ φ N , φ 1 φ n , φ 2 ... φ n , φ N , f = f , φ 1 f , φ 2 ⋮ f , φ N A is called the Gramian matrix - a matrix whose ijth entry is the inner product between the i and jth basis functions. After solving for the vector c=[c1,c2,...,cN]T, we can reconstruct our best approximation to the solution. We first derive what is called the “weak form" of our PDE. Given a function v(x) obeying the same boundary conditions as u, multiply both sides of our wave equation by this function and integrate over the interval [0,ℓ] ∫ 0 ℓ u t t v d x = ∫ 0 ℓ u x x v d x If we integrate the right hand side by parts and apply Dirichlet boundary conditions, we get ∫ 0 ℓ u t t v d x = - ∫ 0 ℓ u x v x d x This form of the wave equation is called the weak form. We now expand u in the space spanned by our basis functions u N ( x , t ) = ∑ j = 1 N c j ( t ) φ j ( x ) Let v(x)=φi(x) for i∈{1,2,...,N}. Plugging this into the wave equation's weak form, we get the relation ∑ j = 1 N c j ' ' ( t ) ∫ 0 ℓ φ i ( x ) φ j ( x ) d x = ∑ j = 1 N c j ( t ) ∫ 0 ℓ φ i ' ( x ) φ j ' ( x ) d x Note that if we define a new “energy" inner product au,v≡ux,vx, we can then rewrite our whole relation as ∑ j = 1 N c i ' ' ( t ) φ i , φ j = ∑ j = 1 N c i ( t ) a φ i , φ j for i=1,2,...,N. Thus, we have N unknowns along with N linear equations; we can now formulate our problem as the matrix equation M c ' ' = K c where M is the Gramian matrix created using regular inner products, and K is the Gramian matrix resulting from energy inner products. Using the finite element method, we choose these basis functions to be piecewise linear “hat" functions. If we partition the space [0,ℓ] into n segments of the form [xk-1,xk], with x1<x2<...<xN, we can define these hat functions as φ k ( x ) = x - x k - 1 x k - x k - 1 if x ∈ [ x k - 1 , x k ] , x k + 1 - x x k + 1 - x k if x ∈ [ x k , x k + 1 ] , 0 otherwise for k=1,...,N. Since the support of φi and φj overlap only if |i-j|≤1, most of the entries of M and K are automatically zero. For the rest of the terms, the inner products are easy to compute. If we take a uniform discretization of [0,1] into these n segments, with h=1/(N+1) and xk=kh, then for |i-j|=1, φi,φi=2h/3, φi,φj=h/6, aφi,φj=1/h, and aφi,φi=-2/h. M and K are just M = h 6 4 1 1 4 ⋱ ⋱ ⋱ 1 1 4 , K = 1 h - 2 1 1 - 2 ⋱ ⋱ ⋱ 1 1 - 2 We can solve for our coefficients c by rewriting Mc''=Kc as a system of equations c ' = d d ' = M - 1 K c ∂ ∂ t c d = 0 I M - 1 K 0 c d We can see the relation to the continuous system, ∂ ∂ t u v = 0 I ∂ 2 ∂ x 2 0 u v where ∂2∂x2 is approximated by M-1K. With this discretization, we can numerically calculate the time solution of the wave equation given some initial condition, as well as approximate the eigenvalues λ.

Damping A closely related equation is the wave equation with viscous damping (resulting from a viscous medium in which the string vibrates, i.e. air). To simulate this effect, a velocity-dependent damping function a(x) is added to the equation ∂ 2 u ∂ t 2 = ∂ 2 u ∂ x 2 - 2 a ( x ) ∂ u ∂ t For the cases we consider here, we shall take a(x)=a, some constant. Thankfully, the finite element discretization of this equation doesn't involve much new work; all we do is reuse some of our calculations. If we make the substitution for u u N = ∑ j = 1 N c j ( t ) φ j ( x ) we get ∑ j = 1 N c j ' ' ( t ) φ j ( x ) = ∑ j = 1 N c j ( t ) φ j ' ' ( x ) - 2 a ∑ j = 1 N c j ' ( t ) φ j ( x ) Taking an inner product with φk for k=1,2,...,N leads us to the following discretization M c ' ' = K c - 2 a M c ' We usually refer to the matrix -2aM as the damping matrix G. Again, we can solve this by writing it out as a system of ordinary differential equations c ' = d d ' = M - 1 K c - M - 1 G ∂ ∂ t c d = 0 I M - 1 K M - 1 G c d Should I write about eigenvalues here?

Networks of strings Unlike our simple one dimensional case, it is much more difficult to determine the closed form eigenvalues and eigenfunctions of a network of strings. To this end, we apply the finite element method to numerically simulate the behavior of a network wave equation.

Network wave equation Let the ith string in a network of strings be defined on an interval from [0,ℓi], where ℓi is the length of that particular string. To generalize the wave equation to a network of strings in three dimensions, we reference Schmidt's [Find REFERENCE ] system of equations for the planar displacement ui(xi,t) of the ith string, where xi∈[0,ℓi] . We define the stensor matrix P i = k i [ ( s i - 1 ) I + v i v i T ] where ki is stiffness, si>1 is prestress (string tension), and vi is a unit vector specifying 3-dimensional orientation of the ith string. We characterize network movement by ρ i I ∂ 2 u i ∂ t 2 = P i ∂ 2 u i ∂ x i 2 where ρi is the ith strings density. I is the 3-by-3 identity matrix. Our boundary conditions are Dirichlet at endpoints and a condition enforcing force balance laws and connectivity of each leg at the joint. We define an end of the first string to have position 0, and for the other endpoints, we consider them to be at position ℓk on their respective kth string. Our Dirichlet conditions can be written as u 1 ( 0 , t ) = 0 , u k ( ℓ k , t ) = 0 If we define the set Si to be the set of integer indices of all strings incident to a joint at the end of the ith string, the force-balance joint conditions connecting strings in the set {i,Si} can be described by P i ∂ u i ∂ x i ( ℓ i , t ) = ∑ j ∈ S i P j ∂ u j ∂ x j ( 0 , t ) This network wave equation stensor matrix can also be mathematically derived from the nonlinear model of Antman; the linear, one dimensional wave equation is derived by taking the orientation vector v to be a standard basis vector.

An example of the notation for the simple tritar case. Here, xi denotes position on the ith string, and S1={2,3}. The network wave equation is much more tractable for a concrete example. We begin by covering the network wave equation for the simplest net - a Y-shaped net called a “tritar", in honor of the http://www.tritare.com guitar with Y-shaped strings. For our simple case, then, we have the boundary conditions u 1 ( 0 , t ) = 0 , u 2 ( ℓ 2 , t ) = 0 , u 3 ( ℓ 3 , t ) = 0 with the force balance equation P 1 ∂ u 1 ∂ x 1 ( ℓ 1 , t ) = P 2 ∂ u 2 ∂ x 2 ( 0 , t ) + P 3 ∂ u 3 ∂ x 3 ( 0 , t ) We will investigate this example further using a discretization of the network.

Finite element discretization of the network wave equation To model behavior and structure of a continuous network, we discretize and solve our equations using the finite element method. For the most part, applying FEM to our network model is the same as applying it to a simple string - the hat functions overlap and form a basis for the structure of each leg. The exception is at a joint, which has a new type of hat function, with its support spanning a small section of each string connected at that joint.

Finite element discretization of a tritar, with a pyramidal hat function φn1 at the joint. r1,r2 and r3 denote the first, second, and third strings, respectively, which are discretized into n1,n2, and n3 parts, respectively.

The tritar example case Let us write out the discretization for the example net in . If we take a uniform discretization of each string into n1,n2, and n3 pieces (with N=n1+n2+n3), respectively, we can again derive a system of differential equations to describe the evolution of the coefficients ck(t) over time. Define the N basis hat functions as being . Consider first the kth hat function on string i, where k≠n1. We multiply each side of the network wave equation by the non-joint hat functions φk and integrate over the support of that function. After integration by parts, we have the relation ρ i I ∫ 0 ℓ i ∂ 2 u i ∂ t 2 ( x i , t ) φ k ( x i ) d x i = - P i ∫ 0 ℓ i ∂ u i ∂ x i ∂ φ k ∂ x i d x i analagous to the one dimensional finite element discretization of a string. If we substitute in our approximation from the basis of hat functions u N = ∑ j = 1 N c j ( t ) φ j ( x ) we arrive at the relation ρ i I ∑ j = 1 N ∂ 2 c j ( t ) ∂ t 2 ∫ 0 ℓ i φ j ( x i ) φ k ( x i ) d x i = - P i ∑ j = 1 N c j ( t ) ∫ 0 ℓ i ∂ φ j ∂ x i ∂ φ k ∂ x i d x i Let L be the number of connections in our web; L=3 for our tritar. Defining our inner products ·,· and a·,· as u , v = ∑ i = 1 L ∫ 0 ℓ i u ( x i ) v ( x i ) d x i , a u , v = ∑ i = 1 L ∫ 0 ℓ i ∂ u ( x i ) ∂ x i ∂ v ( x i ) ∂ x i d x i we see these inner products behave much like the simple string inner products on the topology our network. This gives the relation ρ i I ∑ j = 1 N ∂ 2 c j ( t ) ∂ t 2 φ j , φ k = - P i ∑ j = 1 N c j ( t ) a φ j , φ k The joint is a different case. Let us our joint hat function be φn1(x). Then, since integration by parts moves a derivative from one function to another with the addition of a boundary value, we get ρ 1 I ∫ 0 ℓ 1 ∂ 2 u 1 ∂ t 2 ( x 1 , t ) φ n 1 ( x 1 ) d x 1 = P 1 u 1 ( ℓ 1 , t ) - P 1 ∫ 0 ℓ 1 ∂ u 1 ∂ x 1 ∂ φ n 1 ∂ x 1 d x 1 ρ 2 I ∫ 0 ℓ 2 ∂ 2 u 2 ∂ t 2 ( x 2 , t ) φ n 1 ( x 2 ) d x 2 = - P 2 u 2 ( 0 , t ) - P 2 ∫ 0 ℓ 2 ∂ u 2 ∂ x 2 ∂ φ n 1 ∂ x 2 d x 2 ρ 3 I ∫ 0 ℓ 3 ∂ 2 u 3 ∂ t 2 ( x 3 , t ) φ n 1 ( x 3 ) d x 3 = - P 3 u 3 ( 0 , t ) - P 3 ∫ 0 ℓ 3 ∂ u 3 ∂ x 3 ∂ φ n 1 ∂ x 3 d x 3 after integrating over each string where the joint hat function is nonzero. If we recall that our force balance equation was P1∂x1u1(ℓ1,t)-P2∂x2u2(0,t)-P3∂x3u3(0,t)=0, however, we can sum these equations together to achieve the relation ∑ i = 1 3 ρ i I ∫ 0 ℓ i ∂ 2 u i ∂ t 2 ( x i , t ) φ n 1 ( x i ) d