The network wave equation  (Page 2/11)

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The wave equation

The vibration of a string in one dimension can be understood through the standard wave equation, given by

$\frac{{\partial }^{2}u}{\partial {t}^{2}}={c}^{2}\frac{{\partial }^{2}u}{\partial {x}^{2}},\phantom{\rule{2.em}{0ex}}u\left(0,t\right)=u\left(\ell ,t\right)=0$

where $u$ describes string displacement, $c$ is a constant describing wave speed and $\ell$ is the length of the string. The string is fixed at displacement 0 at the endpoints and assume without loss of generality $c=1$ . This equation is derived in much more detail in . This second order partial differential equation can likewise be rewritten as a system of two ordinary differential equations in time

$\begin{array}{ccc}\hfill \frac{\partial u}{\partial t}& =& v\hfill \\ \hfill \frac{\partial v}{\partial t}& =& {c}^{2}\frac{{\partial }^{2}u}{\partial {x}^{2}}\hfill \end{array}$

or equivalently, the first order matrix equation

$\frac{\partial }{\partial t}\left[\begin{array}{c}u\\ v\end{array}\right]=\left[\begin{array}{cc}0& I\\ {c}^{2}\frac{{\partial }^{2}}{\partial {x}^{2}}& 0\end{array}\right]\left[\begin{array}{c}u\\ v\end{array}\right]$

Eigenvalues, eigenfunctions, and their significance

We are especially interested in the eigenvalues $\lambda$ and associated eigenfunctions of the wave equation, such that

$\left[\begin{array}{cc}0& I\\ \frac{{\partial }^{2}}{\partial {x}^{2}}& 0\end{array}\right]\left[\begin{array}{c}u\\ v\end{array}\right]=\lambda \left[\begin{array}{c}u\\ v\end{array}\right],\phantom{\rule{2.em}{0ex}}\frac{{\partial }^{2}u}{\partial {x}^{2}}={\lambda }^{2}u$

Since only trigonometric functions satisfy both our equation and our boundary conditions, our eigenfunctions take the form $u\left(x\right)=Asin\left(\lambda x\right)+Bcos\left(\lambda x\right)$ . Applying our boundary condition at $x=0$ to $u\left(x\right)$ reveals that $B=0$ . Since we can then set $A$ as an arbitrary scaling factor, our eigenfunction $u\left(x\right)$ is simply $sin\left(\lambda ,x\right)$ . By applying our second boundary condition at $x=\ell$ , we can see that $\lambda$ is of the form $\frac{i\pi n}{\ell }$ for any nonzero integer $n$ . We then get the eigenpairs

${\lambda }_{n}=\frac{i\pi n}{\ell },\phantom{\rule{2.em}{0ex}}{u}_{n}\left(x\right)=sin\left({\lambda }_{n}x\right)$

These eigenfunctions constitute an infinite-dimensional basis for any solution to the wave equation, with ${u}_{i}\left(x\right)$ orthogonal to ${u}_{j}\left(x\right)$ for $i\ne j$ with respect to the inner product

$⟨{u}_{i},{u}_{j}⟩\equiv {\int }_{0}^{\ell }{u}_{i}\left(x,t\right){u}_{j}\left(x,t\right)\phantom{\rule{0.166667em}{0ex}}dx$

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 ${\phi }_{i}\left(x\right)$ that span the space on which the solution is defined. We then calculate the best approximation

$u\left(x,t\right)=\sum _{j=1}^{N}{c}_{j}\left(t\right){\phi }_{j}\left(x\right)$

to the solution from the span of these basis functions via the solution to a matrix equation $Ac=f$ . Recall the definition of our inner product $⟨{u}_{i},{u}_{j}⟩\equiv {\int }_{0}^{\ell }{u}_{i}\left(x,t\right){u}_{j}\left(x,t\right)\phantom{\rule{0.166667em}{0ex}}dx$ . Then, $A$ is

$A=\left[\begin{array}{cccc}〈{\phi }_{1},,,{\phi }_{1}〉& 〈{\phi }_{1},,,{\phi }_{2}〉& ...& 〈{\phi }_{1},,,{\phi }_{N}〉\\ 〈{\phi }_{2},,,{\phi }_{1}〉& 〈{\phi }_{2},,,{\phi }_{2}〉& ...& 〈{\phi }_{2},,,{\phi }_{N}〉\\ ⋮& & \ddots & \\ 〈{\phi }_{N},,,{\phi }_{1}〉& 〈{\phi }_{n},,,{\phi }_{2}〉& ...& 〈{\phi }_{n},,,{\phi }_{N}〉\end{array}\right],\phantom{\rule{2.em}{0ex}}f=\left[\begin{array}{c}〈f,,,{\phi }_{1}〉\\ 〈f,,,{\phi }_{2}〉\\ ⋮\\ 〈f,,,{\phi }_{N}〉\end{array}\right]$

$A$ is called the Gramian matrix - a matrix whose $ij$ th entry is the inner product between the $i$ and $j$ th basis functions. After solving for the vector $c={\left[{c}_{1},{c}_{2},...,{c}_{N}\right]}^{T}$ , we can reconstruct our best approximation to the solution.

We first rearrange our PDE into a more flexible form. Given a function $v\left(x\right)$ obeying the same boundary conditions as $u$ , multiply both sides of our wave equation by this function and integrate over the interval $\left[0,\ell \right]$

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