# Adjoint characteristic decomposition of one-dimensional waves

## TL;DR

This paper explores the connection between continuous and discrete adjoint characteristic decompositions in one-dimensional acoustic wave propagation, demonstrating their relationship via a similarity transformation in the Laplace domain.

## Contribution

It establishes a theoretical link between continuous and discrete adjoint methods through a similarity transformation, applied to acoustic wave problems in the Laplace domain.

## Key findings

- Adjoint characteristic decompositions are connected by a similarity transformation.
- The framework is demonstrated on a one-dimensional acoustic resonator.
- The approach enables future adjoint-based design of acoustic networks.

## Abstract

Adjoint methods enable the accurate calculation of the sensitivities of a quantity of interest. The sensitivity is obtained by solving the adjoint system, which can be derived by continuous or discrete adjoint strategies. In acoustic wave propagation, continuous and discrete adjoint methods have been developed to compute the eigenvalue sensitivity to design parameters and passive devices (Aguilar, J. G. et al, 2017, J. Computational Physics, vol. 341, 163-181). In this short communication, it is shown that the continuous and discrete adjoint characteristic decompositions, and Riemann invariants, are connected by a similarity transformation. The results are shown in the Laplace domain. The adjoint characteristic decomposition is applied to a one-dimensional acoustic resonator, which contains a monopole source of sound. The proposed framework provides the foundation to tackle larger acoustic networks with a discrete adjoint approach, opening up new possibilities for adjoint-based design of problems that can be solved by the method of characteristics.