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The vibration of a string in one dimension can be understood through the standard wave equation, given by
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
or equivalently, the first order matrix equation
We are especially interested in the eigenvalues $\lambda $ and associated eigenfunctions of the wave equation, such that
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
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
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 $N$ basis functions ${\phi}_{i}\left(x\right)$ that span the space on which the solution is defined. We then calculate the best approximation
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 $\u27e8{u}_{i},{u}_{j}\u27e9\equiv {\int}_{0}^{\ell}{u}_{i}(x,t){u}_{j}(x,t)\phantom{\rule{0.166667em}{0ex}}dx$ . Then, $A$ is
$A$ 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 $c={[{c}_{1},{c}_{2},...,{c}_{N}]}^{T}$ , 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 $[0,\ell ]$
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