Helmholtz equations and their applications in solving physical problems
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Abstract
The Helmholtz equation is a well-known concept in the field of physics, particularly when studying problems involving partial differential equations (PDEs) in both space and time. It is a time-independent form of the wave equation and is derived using the method of separation of variables to simplify the analysis. In this research article, we explore the Helmholtz equation and delve into its physical significance. The Helmholtz equation plays a crucial role in solving various physics problems, such as seismology, electromagnetic radiation, and acoustics. It encompasses a wide range of scenarios encountered in electromagnetics and acoustics and is equivalent to the wave equation under the assumption of a single frequency. In the context of water waves, it emerges when the dependence on depth is removed. This often leads to a transition from the study of water waves to more general scattering problems. By employing a cylindrical eigenfunction expansion, we observe that the modes related to the Helmholtz equation decay rapidly as distance approaches infinity. This property enables us to derive asymptotic results in linear water waves based on findings in acoustic or electromagnetic scattering.
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