Next week: Kalpesh Jaykar, 17-Feb-26, 12:20pm Central
0 views
Skip to first unread message
Ellad Tadmor
Feb 10, 2026, 10:00:20 AM (4 days ago) Feb 10
to aem-solid...@umn.edu
AEM Mechanics Research Seminar
Tuesday 17-Feb-2026, 12:20pm Central
Mr. Kalpesh Jaykar
Department of Aerospace Engineering and Mechanics, University of Minnesota
Title: Integral representation of time-harmonic wave fields: A unified framework for electromagnetism, acoustic, and elasticity
Abstract: Many physical systems in solid mechanics and wave physics—ranging from electromagnetic radiation to acoustic pressure fields and elastic waves in solids—admit time-harmonic solutions governed by scalar or vector Helmholtz equations. A familiar solution to these equations is the plane wave, which serves as a fundamental building block for modeling disturbances in electromagnetism, sound, and linear elasticity. In this talk, we show that plane waves form a complete basis for constructing all smooth, propagating time-harmonic solutions of the Helmholtz equations. This observation leads to a powerful integral representation formula that expresses any such wave field as a continuous superposition of rotated and phase-shifted plane waves. A single smooth, assignable amplitude function parameterizes the representation. The same representation applies uniformly to Maxwell fields in source-free media, scalar pressure waves in acoustics, and both longitudinal and transverse elastic waves in isotropic solids.
A key feature of this framework is its exponentially fast numerical convergence: when the amplitude function satisfies mild smoothness conditions, the integral can be approximated using the trapezoidal rule with exponential convergence. Each discrete term in the approximation corresponds to a classical plane wave, making the approximation computationally efficient and physically realizable. Numerical examples—including generalized twisted X-ray fields and Gaussian-type beams—illustrate this convergence behavior. Finally, we show how this representation naturally recovers well-known solutions in wave physics when the propagation vector is extended to be complex. This includes spherical waves in acoustics, far-field radiation from antennas in electromagnetism, and the full family of longitudinal and shear modes in linear elasticity. Together, these results provide a unified framework for understanding and constructing time-harmonic wave fields across a broad range of physical systems.
Tuesday 17-Feb-2026, 12:20pm Central
In-person: Room 321, Mechanical Engineering, University of Minnesota
Online: Webinar registration link
Department of Aerospace Engineering and Mechanics, University of Minnesota
Title: Integral representation of time-harmonic wave fields: A unified framework for electromagnetism, acoustic, and elasticity
Abstract: Many physical systems in solid mechanics and wave physics—ranging from electromagnetic radiation to acoustic pressure fields and elastic waves in solids—admit time-harmonic solutions governed by scalar or vector Helmholtz equations. A familiar solution to these equations is the plane wave, which serves as a fundamental building block for modeling disturbances in electromagnetism, sound, and linear elasticity. In this talk, we show that plane waves form a complete basis for constructing all smooth, propagating time-harmonic solutions of the Helmholtz equations. This observation leads to a powerful integral representation formula that expresses any such wave field as a continuous superposition of rotated and phase-shifted plane waves. A single smooth, assignable amplitude function parameterizes the representation. The same representation applies uniformly to Maxwell fields in source-free media, scalar pressure waves in acoustics, and both longitudinal and transverse elastic waves in isotropic solids.
A key feature of this framework is its exponentially fast numerical convergence: when the amplitude function satisfies mild smoothness conditions, the integral can be approximated using the trapezoidal rule with exponential convergence. Each discrete term in the approximation corresponds to a classical plane wave, making the approximation computationally efficient and physically realizable. Numerical examples—including generalized twisted X-ray fields and Gaussian-type beams—illustrate this convergence behavior. Finally, we show how this representation naturally recovers well-known solutions in wave physics when the propagation vector is extended to be complex. This includes spherical waves in acoustics, far-field radiation from antennas in electromagnetism, and the full family of longitudinal and shear modes in linear elasticity. Together, these results provide a unified framework for understanding and constructing time-harmonic wave fields across a broad range of physical systems.
For more information, visit the AEM Mechanics Research Seminar website:
---
Ellad B. Tadmor, Ph.D.
Russell J. Penrose Professor
Department of Aerospace Engineering and Mechanics
University of Minnesota
https://dept.aem.umn.edu/~tadmor/
Russell J. Penrose Professor
Department of Aerospace Engineering and Mechanics
University of Minnesota
https://dept.aem.umn.edu/~tadmor/
Reply all
Reply to author
Forward
0 new messages