Wave Equation 1D

anypinn create my-project --template wave-1d

1D hyperbolic PDE inverse problem. Recovers wave speed \(c\) from sparse displacement measurements.

Background

The wave equation governs the propagation of disturbances in elastic media such as vibrating strings, acoustic pressure waves, and electromagnetic fields. As a hyperbolic PDE it preserves the shape of initial disturbances (unlike the heat equation, which smooths them), transporting energy at speed \(c\) without dissipation. The template uses a standing-wave exact solution (product of spatial and temporal modes) so that the recovered wave speed can be verified analytically.

Governing Equations

\[ \frac{\partial^2 u}{\partial t^2} = c^2 \frac{\partial^2 u}{\partial x^2} \]

where:

• \(u(x, t)\): displacement field
• \(c\): wave propagation speed (to recover)

Exact solution used for data generation:

\[ u(x, t) = \sin(\pi x)\,\cos(c\,\pi\,t) \]

Default Configuration

The generated template uses the following values.

Parameters to recover:

Symbol	Code constant	True value
\(c\)	TRUE_C	\(1.0\)

Initial conditions: \(u(x, 0) = \sin(\pi x), \quad \dfrac{\partial u}{\partial t}(x, 0) = 0\)

Boundary conditions: \(u(0, t) = u(1, t) = 0\) (homogeneous Dirichlet).

Domain: \(x \in [0, 1], \quad t \in [0, 1]\)

Features Demonstrated

• Hyperbolic PDE handling
• Scalar parameter recovery (wave speed)
• Sparse observation data

Results