本論文的研究目的是探討平面聲波入射于彈性海床上，載以具聲速及密度皆為連續變化的沉積層上，粗糙界面反射與散射之現象與機制。沉積層的性質將視為流體介質，並考慮其密度隨深度呈指數型態作連續遞增，而聲速分別以三種不同分佈模式為主要研究方向的聲學性質去探討;包括常數，k^2-linear，或者是inverse square連續變化三種聲速分佈，其中以後者最為接近實際海床。選擇上述三種不同聲速分佈的原因是在於沉積層中的聲波方程式都將會得到解析解的結果，並且各有其適用時機，以利從事數值模擬，求解出反射場及散射場。

針對沉積層的不同聲速及密度分佈，分別依不同的頻率、粗糙度、分佈梯度等條件因子，採用數值方法來求解反射係數及散射場的空間能量分佈，並由其結果分析對於聲場所造成的影響;而所有的結果都將以物理意義來作合理地解釋。本文所研究的海洋環境中所構建的海床有三個主要特點，包含了具有粗糙亂度的海床界面、非均勻性的沉積層、及海洋底床的剪力特性。而所考慮的空間為二維平面，粗糙界面則為一維的隨機分佈。因此，整體而言，此環境模式提供了研究海床聲學的典型模式。

沉積層的聲學性質變化不僅只是地質上分佈的差異而已，也同時隱含著過渡層中聲波方程式之解析解將會隨之由不同的特殊函數來闡述。將沉積層中聲波方程式的解析解，依邊界條件的要求，即可求出不同環境下的聲場解。對於散射場中空間能譜的數值演算法則，乃是建構於一階邊界攝動理論的隨機海床環境。聲波入射至沉積層後，所散射出來的能量，僅求取一次反射的結果，因為多重反射後的能量，階次明顯越來越大以致於可以忽略不計，然而對聲場的影嚮並不會造成明顯差別。結果顯示，同調反射聲場的能量分佈主要是受到整體海洋環境中海床粗糙高度的影響，而散射場則是偏重在粗糙度的細部特徵來控制，也就是由粗糙界面的隨機能譜分佈及空間相關長度來決定隨機散射聲場的分佈。

Acoustic plane wave intearctions with a rough seabed with a continuously varying density and sound speed in a fluid-like sediment layer overlying an elastic basement is considered in this thesis. The acoustic properties in the sediment layer possess an exponential type of variation in density and one of the three classes of sound speed profiles, which are constant, k^2-linear, or inverse-square variations. Analytical solutions for the Helmholtz equation in the sediment layer, combined with a formulation based upon boundary perturbation theory, facilitate numerical implementation for the solution of coherent field.

The coherent reflection coefficients corresponding to the aformentioned density and sound speed profiles for various frequencies, roughness parameters, basement stiffness, are numerically generated and analyzed. Physical interpretations are provided for various results. This simple model characterizes three important features of an realistic sea floor, including seabed roughness, sediment inhomogenieties, and basement shear property,%Two dimensions is considered in the seafloor environment and the random roughness is belong to one dimension space.%
, therefore, provides a canonical model for the study of seabed acoustics.

The variation of the acoustic properties takes such a form that it is not only geologically realistic, but also renders analytical solutions for the Helmholtz equation, thus facilitating the formulation of the problem. The computational algorithm for the spatial spectrum of the scattered field due to random seabed has been developed based upon a boundary perturbation method. %About scattering field, only one time reflection from the sediment is taked account of, because the higher numerical order is, the lower scattering energy exist.%
The results have shown that, while the coherent field mainly depends upon the gross structure of the rough seabed represented by the RMS roughness, the scattered field heavily depends upon the details of the roughness structure specialized by the roughness power spectrum and the spatial correlation length of the rough surface. The dependence of the spatial spectrum on the sediment stratification is also carefully examined.

第一章 緒論 1
1.1 研究主題與動機.........................1
1.2 文獻回顧與本研究在文獻上的角色.........3
1.3 研究方法...............................4
1.4 論文範疇...............................5
第二章 理論模式 6
2.1 簡介...................................6
2.2 聲波方程式.............................7
2.2.1 均勻介質.............................7
2.2.2 非均勻介質...........................9
2.3 聲波方程式之解.........................10
2.3.1 均勻介質.............................10
2.3.2 非均勻介質...........................12
2.4 粗糙界面之散射.........................16
2.4.1 平滑界面.............................16
2.4.2 粗糙界面.............................17
2.4.3 平均反射場...........................18
2.4.4 結語.................................21
第三章 平均反射場之分析 23
3.1 線性系統...............................23
3.2 結果與討論.............................27
3.3 結語...................................34
第四章 散射場之分析 36
4.1 頻率對散射能譜密度的影響...............36
4.2 粗糙面性質對散射能譜密度的影響.........41
4.3 結語...................................45
第五章 結論與建議 46
5.1 結論...................................46
5.2 建議...................................48
附錄A 53

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