1# plate-strike-v1: struck free aluminum plate, radiated pressure
3This is the tier 1 challenge: a thin square aluminum plate, free on all faces, struck once by a smooth, short force pulse. The deliverable is a three second mono wav file containing the acoustic pressure at a fixed listener point, computed from the plate's surface acceleration by the Rayleigh integral. The model is three-dimensional linear elastodynamics, so thin-plate theories are approximations to it and their error relative to a converged three-dimensional solution includes the modeling error of plate theory as well as discretization error. 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.
5The machine-readable version of this specification is [challenge.json](challenge.json).
7## Model
9The plate occupies the box
11$$
12\Omega = (0, L_x) \times (0, L_y) \times (-h/2, h/2),
13$$
15with the mid-plane at $z = 0$ and the top face $S^+ = \{ z = h/2 \}$. Let $\mathbf{u}(\mathbf{x}, t) \in \mathbb{R}^3$ be the displacement in meters. The material is a homogeneous isotropic Hookean solid with Kelvin-Voigt viscosity and an additional frequency-independent loss:
17$$
18\rho\, \frac{\partial^2 \mathbf{u}}{\partial t^2} = \nabla \cdot \boldsymbol{\sigma} - 2 \sigma_0 \rho\, \frac{\partial \mathbf{u}}{\partial t}, \qquad
19\boldsymbol{\sigma} = \lambda\, \mathrm{tr}(\boldsymbol{\varepsilon} + \eta\, \dot{\boldsymbol{\varepsilon}})\, \mathbf{I} + 2 \mu\, (\boldsymbol{\varepsilon} + \eta\, \dot{\boldsymbol{\varepsilon}}), \qquad
20\boldsymbol{\varepsilon} = \tfrac{1}{2} \left( \nabla \mathbf{u} + \nabla \mathbf{u}^{\mathsf{T}} \right),
21$$
23where $\dot{\boldsymbol{\varepsilon}} = \partial \boldsymbol{\varepsilon} / \partial t$, and the Lamé constants follow from Young's modulus $E$ and Poisson's ratio $\nu$ in the usual way, $\lambda = E\nu / ((1+\nu)(1-2\nu))$ and $\mu = E / (2(1+\nu))$. The constant $\eta$ has units of time; the Kelvin-Voigt term damps a mode of angular frequency $\omega$ at the rate $\eta \omega^2 / 2$, and $\sigma_0$ adds a rate that is the same for every mode.
25All faces are traction-free except for the applied strike on the top face:
27$$
28\boldsymbol{\sigma}\, \mathbf{n} = \mathbf{0} \text{ on } \partial\Omega \setminus S^+, \qquad
29\boldsymbol{\sigma}\, \mathbf{n} = -\, p(x, y, t)\, \mathbf{e}_z \text{ on } S^+,
30$$
32with $\mathbf{n}$ the outward normal, so that $p$ is a pressure pushing the plate in the $-z$ direction. The plate starts at rest:
34$$
35\mathbf{u}(\mathbf{x}, 0) = \mathbf{0}, \qquad \frac{\partial \mathbf{u}}{\partial t}(\mathbf{x}, 0) = \mathbf{0}.
36$$
38The plate is free, so rigid-body translation and rotation are possible and the strike imparts net momentum. This is intended; the output depends on acceleration and is unaffected by uniform drift.
40### The strike
42The strike is a raised-cosine bump in space, of radius $R$ centered at $(x_s, y_s)$, times a Hann pulse in time, of duration $T_s$:
44$$
45p(x, y, t) = \frac{F_{\max}}{c_\phi\, \pi R^2}\; \phi\!\left( \frac{r}{R} \right) \psi\!\left( \frac{t}{T_s} \right), \qquad r = \sqrt{(x - x_s)^2 + (y - y_s)^2},
46$$
48$$
49\phi(s) = \begin{cases} \tfrac{1}{2}(1 + \cos \pi s), & 0 \le s \le 1, \\ 0, & s > 1, \end{cases} \qquad
50\psi(\tau) = \begin{cases} \tfrac{1}{2}(1 - \cos 2\pi \tau), & 0 \le \tau \le 1, \\ 0, & \text{otherwise,} \end{cases} \qquad
51c_\phi = \frac{1}{2} - \frac{2}{\pi^2}.
52$$
54The constant $c_\phi$ is $\int_0^1 2 s\, \phi(s)\, ds$, so that the total force $\int_{S^+} p\, dS = F_{\max}\, \psi(t / T_s)$ peaks at $F_{\max}$ at $t = T_s / 2$, and the total impulse is $F_{\max} T_s / 2$. Both factors are continuously differentiable and compactly supported, and the strike footprint lies inside the top face.
56## Parameters
58| Symbol | Value | Unit | Meaning |
59| --- | --- | --- | --- |
60| $L_x, L_y$ | 0.3048 | m | plate side (12 in) |
61| $h$ | 0.00104 | m | plate thickness |
62| $E$ | 68.9e9 | Pa | Young's modulus (aluminum 6061-T6) |
63| $\nu$ | 0.33 | | Poisson's ratio |
64| $\rho$ | 2700 | kg/m^3 | density |
65| $\sigma_0$ | 2.3 | 1/s | frequency-independent loss rate |
66| $\eta$ | 5.0e-8 | s | Kelvin-Voigt viscosity |
67| $(x_s, y_s)$ | (0.09144, 0.10668) | m | strike center ($0.3 L_x$, $0.35 L_y$) |
68| $R$ | 0.01 | m | strike radius |
69| $T_s$ | 0.0005 | s | strike duration |
70| $F_{\max}$ | 10 | N | peak total force |
71| $\rho_a$ | 1.204 | kg/m^3 | air density |
72| $c_a$ | 343 | m/s | speed of sound in air |
73| $\mathbf{x}_L$ | (0.2524, 0.1524, $h/2$ + 0.5) | m | listener point |
75All values are exact decimal numbers, not rounded measurements. The listener is 0.5 m above the top face, offset 0.1 m from the plate center in $x$, so that modes antisymmetric in $x$ radiate to it.
77With these constants the plate's fundamental mode is near 37 Hz, $\sigma_0$ gives the lowest modes a sixty-decibel decay time of about 3 s, and $\eta$ makes a mode at 5 kHz decay in about 0.3 s and one at 20 kHz in about 17 ms.
79## Output signal
81The output is the acoustic pressure at the listener point given by the Rayleigh integral over the top face, using the normal acceleration of the top face and the retarded time,
83$$
84s(t) = \frac{\rho_a}{2\pi} \int_{S^+} \frac{1}{|\mathbf{x}_L - \mathbf{y}|}\; \frac{\partial^2 u_z}{\partial t^2}\!\left( \mathbf{y},\; t - \frac{|\mathbf{x}_L - \mathbf{y}|}{c_a} \right) dS(\mathbf{y}), \qquad \text{in Pa},
85$$
87with $\partial^2 u_z / \partial t^2$ taken as zero for negative arguments. This is the field of a planar radiator in an infinite rigid baffle; it is an idealization of the true radiation from a free plate, adopted here because it is an explicit formula that needs no solution of the acoustic wave equation. A later challenge will replace it with the coupled acoustic problem.
89The deliverable is a wav file with the following properties.
91| Property | Value |
92| --- | --- |
93| channels | 1 |
94| sample rate $f_s$ | 48000 Hz |
95| number of samples $N$ | 144000 (3.0 s) |
96| sample format | 32-bit IEEE float (WAVE_FORMAT_IEEE_FLOAT) |
97| sample $k$, for $k = 0, \dots, N-1$ | $s(k / f_s) / s_\text{ref}$ |
98| $s_\text{ref}$ | see `output.signal_scale` in challenge.json (Pa) |
100The first sample is at $t = 0$, the onset of the strike. The signal is zero until the first sound arrives at the listener, about 1.46 ms after onset. Samples are pointwise values of the continuum signal, with no anti-aliasing filter; the Kelvin-Voigt term makes the content above the Nyquist frequency small after the first milliseconds. 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.
102## Error measures
104The error measures are the same two as for the string challenges, applied to the 144000 samples: the relative waveform $\ell^2$ error (primary) and the relative spectrogram $\ell^2$ error with a periodic Hann window of length 2048 and hop 512, no padding, over $\lfloor (N - 2048) / 512 \rfloor + 1$ frames. See [string-pluck-v1](https://vault1.magland.org/challenges/string/blob/main/challenges/string-pluck-v1/README.md#error-measures) for the formulas.
106## Reference solution
108There is no externally supplied ground truth. The reference is chosen automatically from the submitted solutions by the procedure described in the [collection README](../../README.md), unless `reference_solution` in challenge.json names one explicitly. Until a converged three-dimensional solution is submitted, the reference will be the best thin-plate solution, and the errors of other thin-plate solutions relative to it measure only discretization differences within plate theory.
110## Optional field export
112A submission may provide the normal displacement of the mid-plane for visualization; this is not scored. The file is HDF5 with datasets `x`, float64, shape (61,), $x_i = i L_x / 60$; `y`, float64, shape (61,), $y_j = j L_y / 60$; `t`, float64, shape (3001,), $t_k = k / 1000$; and `u`, float32, shape (3001, 61, 61), with `u[k, j, i]` $= u_z(x_i, y_j, 0, t_k)$ in meters. Root attributes `challenge_id`, `solution_id`, `quantity` and `units` are recommended. The file is about 45 MB and should be stored with git LFS or at an external URL. Thin-plate methods report their transverse displacement $w$ as $u_z$.
114## Appendix: thin-plate reduction
116For reference, the Kirchhoff-Love reduction of the model above, which is an approximation valid when the bending wavelength is large compared with $h$, is
118$$
119\rho h\, \frac{\partial^2 w}{\partial t^2} + 2 \sigma_0 \rho h\, \frac{\partial w}{\partial t} + D\, \nabla^4 \left( w + \eta\, \frac{\partial w}{\partial t} \right) = p(x, y, t), \qquad D = \frac{E h^3}{12 (1 - \nu^2)},
120$$
122for the transverse displacement $w(x, y, t)$ of the mid-plane, with the free-edge conditions of vanishing bending moment and effective shear force on all four edges and vanishing twisting moment at the corners. The same constants $\sigma_0$ and $\eta$ apply. Note that the strike pressure enters with a positive sign here because $w$ is measured in the $-z$ direction in the usual plate convention; a submission must report $u_z = -w$ in that case, or use $w$ measured in $+z$ with $-p$ on the right-hand side. In this challenge $D \approx 7.25$ N m and $\rho h \approx 2.81$ kg/m^2.