Knowing how the solution to time-harmonic wave scattering problems depends on medium properties and boundary conditions is pivotal in wave-based inverse problems, e.g. for imaging. This paper is devoted to the exposition of a computationally efficient method, called the adjoint state method, that allows to quantify the influence of media properties, directly and through boundary conditions, in the study of acoustic, electromagnetic and elastic time-harmonic waves. Firstly, the adjoint state method is derived for general boundary value problems. A continuous (rather than discrete) formalism is adopted in order to highlight the role of the boundary terms. Then, the method is applied systematically to acoustic, electromagnetic and elastic scattering problems with impedance boundary conditions, making use of the similitude between the three problems. Finally, numerical examples solved using the finite element method are presented to demonstrate the validity of the proposed method.

在基于波的逆问题（例如，成像）中，了解时谐波散射问题的解决方案如何取决于介质属性和边界条件至关重要。本文致力于阐述一种计算有效的方法，称为伴随状态方法，该方法可以在研究声，电磁和弹性时谐波时直接并通过边界条件来量化介质属性的影响。首先，针对一般边值问题推导了伴随状态方法。为了强调边界项的作用，采用了连续的（而不是离散的）形式主义。然后，利用这三个问题之间的相似性，将该方法系统地应用于具有阻抗边界条件的声学，电磁和弹性散射问题。