A Tutorial on the Analytical Solution of Lamb’s Problem and Its Numerical Implementation

Sangin Cho; Sukjoon Pyun

doi:10.7582/GGE.2026.29.2.131

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2026 Vol.29, Issue 2 Preview Page Next Page

A Tutorial on the Analytical Solution of Lamb’s Problem and Its Numerical Implementation Lamb 문제 해석해와 수치 구현에 대한 해설

31 May 2026. pp. 131-142

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Abstract

Lamb’s problem is a core fundamental topic for understanding wave phenomena in seismology and geophysical exploration. This paper details the derivation of the analytical solution and its numerical implementation for a 2D semi-infinite medium subjected to a vertical point load on the surface. We derive the solution in the frequency-wavenumber domain using Helmholtz decomposition and Fourier transforms and present a step-by-step algorithm to reconstruct the time-space wavefield via the discrete Fast Fourier Transform (FFT). Comparing the implemented analytical wavefields with finite element method (FEM) modeling results confirmed that the travel times and waveforms of major seismic events match with high accuracy. This paper will serve as a useful guide for beginners to practically understand the basic concepts of elastic analytical solutions and their numerical implementation.

Keywords

Lamb’s Problem

Analytical Solution

Elastic Wave Equation

Numerical Implementation

Finite Element Method

Lamb 문제는 지진학 및 물리탐사 분야에서 파동 현상을 이해하기 위한 핵심적인 기초 주제이다. 이 해설 논문은 2차원 반무한 매질 표면에 수직 점하중이 작용하는 경우에 대한 해석해 유도와 수치적 구현 과정을 상세히 다룬다. 헬름홀츠 분해와 푸리에 변환을 이용해 주파수-파수 영역의 해를 유도하고, 이산 고속 푸리에 변환(FFT)을 통해 시간-공간 영역 파동장으로 복원하는 알고리즘을 단계별로 제시한다. 구현된 해석해 파동장을 유한요소법(FEM) 모델링 결과와 비교한 결과, 주요 파동의 도달 시간과 파형이 매우 정확히 일치함을 확인했다. 이 논문은 입문자들이 탄성파 해석해의 기본 개념과 수치적 구현 과정을 실무적으로 이해하는 데 유용한 지침이 될 것이다.

키워드

Lamb 문제

해석해

탄성파동방정식

수치구현

유한요소법

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Information

Publisher :Korean Society of Earth and Exploration Geophysicists
Publisher(Ko) :한국지구물리물리탐사학회
Journal Title :Geophysics and Geophysical Exploration
Journal Title(Ko) :지구물리와 물리탐사
Volume : 29
No :2
Pages :131-142
Received Date : 2026-05-15
Revised Date : 2026-05-21
Accepted Date : 2026-05-26
DOI :https://doi.org/10.7582/GGE.2026.29.2.131

Geophysics and Geophysical Exploration ISSN:1229-1064(Print) 2384-051X(Online) 지구물리와 물리탐사

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