Study on the Scattering Problem of SH Waves by an Underground Semielliptic Cavity

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Project title

Code/計畫編號

NSC101-2221-E019-051

Translated Name/計畫中文名

地表下半橢圓形孔洞SH波散射問題之研究

Project Coordinator/計畫主持人

Deng-How Tsaur

Funding Organization/主管機關

National Science and Technology Council

Department/Unit

Department of Harbor and River Engineering

Website

https://www.grb.gov.tw/search/planDetail?id=2646251

Year

2012

Start date/計畫起

01-08-2012

Expected Completion/計畫迄

01-07-2013

Bugetid/研究經費

486千元

ResearchField/研究領域

土木水利工程

"關於地下孔洞對 SH 波所產生的散射問題，過去學者們以理論解過的孔洞形狀，都是“圓形”或 “半圓形”的，而以理論求解“橢圓形”或“半橢圓形”孔洞的，則非常少見。這是因為“圓形” 及“半圓形”孔洞的SH 波散射問題，僅需用到Bessel 及Hankel 函數等大家較熟悉的特殊函數，搭配三角函數，即可求解。但是，“橢圓形”或“半橢圓形”孔洞的問題，則需用大家較不熟悉的 Mathieu 函數才可以處理，而且，在座標轉換時，會使用到較為複雜的Mathieu 函數加法公式，因此，用理論求解“橢圓形”或“半橢圓形”孔洞的SH 波散射問題，較“圓形”及“半圓形”孔洞的少。再者，“橢圓形”斷面可以退化成“圓形”斷面，而“半橢圓形”斷面則可退化成“半圓形”斷面。因而，本年度(101 年)計畫擬以《地表下半橢圓形孔洞SH 波散射問題之研究》為主軸，採用理論解析的方式，配合Mathieu 函數之加法公式(或適當的座標變換公式)，求解此問題，並探討孔洞形狀及埋置深度不同時，入射波以不同頻率及角度入射，對地表反應特性的影響。本計畫之成果可提供給其他數值方法作為比對之用，亦可作為地震工程相關研究之參考。" "For researchers using theoretical methods to deal with the SH-waves scattering problems induced by underground cavities, circular and semicircular shapes are the most interested geometries. However, problems concerning elliptic or semielliptic shapes are very rare. This is because, when tackling problems pertaining to circular and semicircular shapes, only familiar special functions (i.e. Bessel and Hankel functions) and trigonometric functions are needed. Inevitably, for problems related to elliptic or semielliptic shapes, uncommon functions (i.e. Mathieu functions) are involved. Furthermore, the use of a more complicated addition formula of Mathieu functions is required in performing a coordinate transformation. This is why the number of literature citations regarding elliptic or semielliptic shapes is significantly less than the number of references associated with circular and semicircular shapes. Besides, the elliptic cross-section can be degenerated into the circular one, while the semielliptic cross-section into the semicircular one. Thus, the purpose of this project is to investigate the problem of SH-waves scattering induced by an underground semielliptic cavity. A theoretical analysis, applying the addition formula of Mathieu functions (or an appropriate formula on coordinate transformation), is proposed to attack the problem under consideration. Herein, the effect of wave frequency, angle of incidence, cavity shape and depth on the characteristic of ground motions will be evaluated and discussed. In addition, computed results presented in this project may serve as a cross-reference for other numerical methods and for researchers in the field of earthquake engineering."

Keyword(s)

散射
SH 波
孔洞
Mathieu 函數
加法公式
scattering
SH waves
cavity
Mathieu function
addition formula

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Study on the Scattering Problem of SH Waves by an Underground Semielliptic Cavity

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