string-pluck-v1: plucked damped string, pickup velocity#

This is the hello-world challenge of the collection. The model is a linear damped string with fixed ends, released from rest from its static equilibrium under a finger load of finite width. The deliverable is a two second mono wav file containing the transverse velocity of the string at a fixed pickup point. Every quantity below is defined in the continuum, so the true solution is a specific function of time that does not depend on any discretization. A modal series solution is given in the appendix; it converges to the true solution and can be evaluated to machine precision, so this challenge is mainly a test of the submission and scoring pipeline.

The machine-readable version of this specification is challenge.json.

Model#

Let $u(x, t)$ denote the transverse displacement in meters of the string at position $x \in [0, L]$ and time $t \ge 0$. The string satisfies

with fixed ends

and initial conditions

Here $c$ is the wave speed, $\sigma_0$ is a frequency-independent loss rate, and $\kappa$ is a Kelvin-Voigt viscosity that damps high frequencies faster than low ones.

The initial shape $u_0$ is the static equilibrium of the string under a transverse load of finite width centered at the pluck point $x_p$, which is how a finger or plectrum deforms the string before release. The load profile is a raised cosine of half-width $w$,

and $u_0$ solves the static problem $u_0'' = -g$ on $(0, L)$ with $u_0(0) = u_0(L) = 0$. Here $Q = \int_0^L g\,dx$ is the total load expressed as a slope (force divided by tension), and it is fixed by requiring the pluck height $u_0(x_p) = h$. In closed form,

where $H$ is the Heaviside step ($H(y) = 1$ for $y > 0$, else $0$). $\tau$ is the triangle produced by a point load, and $\kappa$ is the correction from spreading the load over the contact region; it vanishes outside $|y| \le w$. Note that $u_0$ is three times continuously differentiable, since $g$ is continuously differentiable, and that $w < x_p < L - w$ so the fixed-end conditions hold.

Parameters#

All values are exact decimal numbers, not rounded measurements.

Output signal#

The audio signal is the transverse velocity of the string at the pickup point,

The deliverable is a wav file with the following properties.

The first sample is at $t = 0$, where $s(0) = 0$. Samples are pointwise values of the continuum signal. No anti-aliasing filter is applied, even though $s(t)$ contains frequency content above the Nyquist frequency; that content is heavily damped by the Kelvin-Voigt term and is part of the exact answer. The scale $s_\text{ref}$ is a fixed constant chosen so that the peak sample magnitude is roughly 0.5. Do not normalize the signal; absolute amplitude is compared.

Error measures#

Let $a_k$ and $r_k$, $k = 0, \dots, N-1$, be the samples of a submitted solution and of the reference solution. Two error measures are computed. Both are relative and dimensionless, and both are reported for every solution.

Waveform error (primary):

Spectrogram error. Let $M = 2048$ be the frame length and $H = 512$ the hop. The periodic Hann window is $g_j = \tfrac{1}{2}\left(1 - \cos(2\pi j / M)\right)$ for $j = 0, \dots, M-1$. Frame $m$ of a signal $x$ is $x_{mH + j}\, g_j$, for $m = 0, \dots, \lfloor (N - M) / H \rfloor$, with no zero padding at either end. Let $X_m[q]$, $q = 0, \dots, M/2$, be its discrete Fourier transform, $X_m[q] = \sum_{j=0}^{M-1} x_{mH+j}\, g_j\, e^{-2\pi i q j / M}$. Then

The spectrogram error ignores phase within a frame and is less sensitive than the waveform error to small frequency errors that accumulate into phase drift over the two seconds.

Reference solution#

There is no externally supplied ground truth. The reference is chosen automatically from the submitted solutions by the procedure described in the collection README, unless reference_solution in challenge.json names one explicitly. For this particular challenge the modal series in the appendix gives an essentially exact answer, which is expected to be selected.

Optional field export#

A submission may additionally provide the displacement field for visualization. This is not scored. The file is HDF5 with three datasets: x, float64, shape (201,), the positions $x_i = i L / 200$ in meters; t, float64, shape (4001,), the times $t_j = j / 2000$ in seconds; and u, float32, shape (4001, 201), with u[j, i] $= u(x_i, t_j)$ in meters. Root attributes challenge_id, solution_id, quantity ("transverse displacement") and units ("m") are recommended. The grid is stated under fields in challenge.json. A file of this size is about 3 MB uncompressed and can live in the solution repository; gzip compression inside the file is welcome.

Appendix: modal series solution#

This appendix is provided for convenience and verification; it is not part of the challenge definition, and a submission may use any method.

Expanding $u(x, t) = \sum_{n \ge 1} a_n(t) \sin(k_n x)$ with $k_n = n\pi / L$, each coefficient satisfies

with $a_n(0) = A_n$ and $a_n'(0) = 0$. Integrating by parts twice, using $u_0'' = -g$ and the fixed ends,

With the parameters above, $(\pi/w)^2 - k_n^2 \ne 0$ for every integer $n$, since $L / w$ is not an integer. For every mode that matters $\alpha_n < c k_n$ (the modes are underdamped up to roughly $n = 14800$), and in that case, with $\omega_n = \sqrt{c^2 k_n^2 - \alpha_n^2}$,

The pickup velocity is then

Since $|A_n| = O(n^{-5})$ and $|a_n'|$ carries an extra factor of $n$, the terms decay like $n^{-4}$ before damping is taken into account, and for $t > 0$ the exponential factor makes the series converge rapidly. In practice a few hundred modes give the signal to single precision for all $t \ge 1 / f_s$; at $t = 0$ the exact value is $s(0) = 0$.