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.
We are especially interested in the eigenvalues λλ 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
0
I
∂
2
∂
x
2
0
u
v
=
λ
u
v
,
∂
2
u
∂
x
2
=
λ
2
u
(4)
Since only trigonometric functions satisfy both our equation and our boundary conditions, our eigenfunction to take the form u(x)=Asin(λx)+Bcos(λx)u(x)=Asin(λx)+Bcos(λx). Applying our boundary condition at x=0x=0 to u(x)u(x) reveals that B=0B=0. Since we can then set AA as an arbitrary scaling factor, our eigenfunction u(x)u(x) is simply sinλxsinλx. By applying our second boundary condition at x=ℓx=ℓ, we can see that λλ is of the form iπnℓiπnℓ for any nonzero integer nn. We then get the eigenpairs
λ
n
=
i
π
n
ℓ
,
u
n
(
x
)
=
sin
(
λ
n
x
)
λ
n
=
i
π
n
ℓ
,
u
n
(
x
)
=
sin
(
λ
n
x
)
(5)
These eigenfunctions constitute an infinite-dimensional basis for any solution to the wave equation, with ui(x)ui(x) orthogonal to uj(x)uj(x) for i≠ji≠j with respect to the inner product 〈ui,uj〉≡∫0ℓui(x,t)uj(x,t)dx〈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.
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 NN basis functions φi(x)φi(x) that span the space on which the solution is defined. We then calculate the best approximation uN=∑j=1NcjφjuN=∑j=1Ncjφj to the solution from the span of these basis functions via the solution to a matrix equation Ac=fAc=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
=
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
(6)
AA is called the Gramian matrix - a matrix whose ijijth entry is the inner product between the ii and jjth basis functions. After solving for the vector c=[c1,c2,...,cN]Tc=[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)v(x) obeying the same boundary conditions as uu, multiply both sides of our wave equation by this function and integrate over the interval [0,ℓ][0,ℓ]
∫
0
ℓ
u
t
t
v
d
x
=
∫
0
ℓ
u
x
x
v
d
x
∫
0
ℓ
u
t
t
v
d
x
=
∫
0
ℓ
u
x
x
v
d
x
(7)
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
∫
0
ℓ
u
t
t
v
d
x
=
-
∫
0
ℓ
u
x
v
x
d
x
(8)
This form of the wave equation is called the weak form. We now expand uu in the space spanned by our basis functions
u
N
(
x
,
t
)
=
∑
j
=
1
N
c
j
(
t
)
φ
j
(
x
)
u
N
(
x
,
t
)
=
∑
j
=
1
N
c
j
(
t
)
φ
j
(
x
)
(9)
Let v(x)=φi(x)v(x)=φi(x) for i∈{1,2,...,N}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
∑
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
(10)
Note that if we define a new “energy" inner product au,v≡ux,vxau,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
∑
j
=
1
N
c
i
'
'
(
t
)
φ
i
,
φ
j
=
∑
j
=
1
N
c
i
(
t
)
a
φ
i
,
φ
j
(11)
for i=1,2,...,Ni=1,2,...,N. Thus, we have NN unknowns along with NN linear equations; we can now formulate our problem as the matrix equation
M
c
'
'
=
K
c
M
c
'
'
=
K
c
(12)
where MM is the Gramian matrix created using regular inner products, and KK 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][0,ℓ] into nn segments of the form [xk-1,xk][xk-1,xk], with x1<x2<...<xNx1<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
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
(13)
for k=1,...,Nk=1,...,N. Since the support of φiφi and φjφj overlap only if |i-j|≤1|i-j|≤1, most of the entries of MM and KK 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][0,1] into these nn segments, with h=1/(N+1)h=1/(N+1) and xk=khxk=kh, then for |i-j|=1|i-j|=1, φi,φi=2h/3φi,φi=2h/3, φi,φj=h/6φi,φj=h/6, aφi,φj=1/haφi,φj=1/h, and aφi,φi=-2/haφi,φi=-2/h. MM and KK are just
M
=
h
6
4
1
1
4
⋱
⋱
⋱
1
1
4
,
K
=
1
h
-
2
1
1
-
2
⋱
⋱
⋱
1
1
-
2
M
=
h
6
4
1
1
4
⋱
⋱
⋱
1
1
4
,
K
=
1
h
-
2
1
1
-
2
⋱
⋱
⋱
1
1
-
2
(14)
We can solve for our coefficients cc by rewriting Mc''=KcMc''=Kc as a system of equations
c
'
=
d
d
'
=
M
-
1
K
c
c
'
=
d
d
'
=
M
-
1
K
c
(15)
∂
∂
t
c
d
=
0
I
M
-
1
K
0
c
d
∂
∂
t
c
d
=
0
I
M
-
1
K
0
c
d
(16)
We can see the relation to the continuous system,
∂
∂
t
u
v
=
0
I
∂
2
∂
x
2
0
u
v
∂
∂
t
u
v
=
0
I
∂
2
∂
x
2
0
u
v
(17)
where ∂2∂x2∂2∂x2 is approximated by M-1KM-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 λλ.
