Recover an arbitrary sound speed from one incident direction
Fix a direction $\theta_0 \in S^2$ and $R > 0$. For a positive $c \in C^\infty(\mathbb{R}^3)$ with $c = 1$ outside the ball $B_R$ and a wave number $k > 0$, let $u = e^{ik x \cdot \theta_0} + u^s$ solve $$\bigl(\Delta + k^2 c(x)^{-2}\bigr) u = 0 \ \text{ in } \mathbb{R}^3,$$ with $u^s$ outgoing, and define $a_c$ by $$u^s(r\omega) = r^{-1} e^{ikr} a_c(\omega, k) + O(r^{-2}), \qquad r \to \infty, \ \omega \in S^2.$$ The data $a_c(\omega, k)$, $(\omega, k) \in S^2 \times (0, \infty)$, come from a single incident direction and depend on three variables, as $c$ does. For the time-domain version of the problem, in which one impulsive plane wave travelling in direction $\theta_0$ is sent in and the wave is measured on a sphere enclosing the support of $c - 1$, Ma, Potenciano-Machado and Salo proved that a sound speed close to $1$ whose data agree with those of the constant speed $1$ is identically $1$, with a stability estimate; and Oksanen, Rakesh and Salo proved this for every smooth positive sound speed equal to $1$ outside a ball, and the analogue for Riemannian metrics, up to diffeomorphism, with finitely many incident waves.
Problem. Let $c_1, c_2$ be two such sound speeds. Does $a_{c_1}(\omega, k) = a_{c_2}(\omega, k)$ for all $\omega \in S^2$ and all $k > 0$ imply $c_1 = c_2$? Both speeds may be far from constant, and only the one incident direction $\theta_0$ is used.
Source: The fixed angle inverse scattering problem for a sound speed, called longstanding by Oksanen, Rakesh and Salo (2026), §1.1; for sound speeds close to constant, Ma, Potenciano-Machado and Salo (2023). Stated in this form, in three dimensions with far-field data, by AIM — Open Applied Problems.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- AP — Analysis of PDEs
- Secondary
- SP
- Keywords
- AIM Open Applied Problems, fixed angle scattering, inverse scattering, sound speeds
References
@article{oksanen2026,
author = {Oksanen, Lauri and Rakesh and Salo, Mikko},
title = {Rigidity for fixed angle inverse scattering for {R}iemannian metrics},
journal = {SIAM J. Math. Anal.},
volume = {58},
number = {1},
year = {2026},
pages = {308--334},
doi = {10.1137/24M1702623},
eprint = {2410.06864},
archivePrefix = {arXiv},
}
@article{ma2023,
author = {Ma, Shiqi and Potenciano-Machado, Leyter and Salo, Mikko},
title = {Fixed angle inverse scattering for sound speeds close to constant},
journal = {SIAM J. Math. Anal.},
volume = {55},
number = {4},
year = {2023},
pages = {3420--3456},
doi = {10.1137/22M147640X},
eprint = {2109.07761},
archivePrefix = {arXiv},
}
@misc{aim-fixed-angle-sound-speed,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Recover an arbitrary sound speed from one incident direction}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/053-fixed-angle-sound-speed.md},
note = {Entry 053 at commit a153f46, 2026-10-13},
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00521, "Recover an arbitrary sound speed from one incident direction", clopen.solutions.