本論文主要目的：
1.探討不同頻率在長空間中傳播並不是以往預測中的呈線性衰減，而是受到頻率的影響，而
呈現非線性的衰減。
2.在波動方程式的理論上運用虛源法為基礎，分別從理論模型、數值模擬和實驗進行長空間
具有旁支及複雜的阻抗情況進行研究。
3.使用聲波理論導出長空間和垂直旁支管音場的理論解，並利用虛源法的相干模型對長空間
音場及噪音衰減進行預測。
對基於波動理論中的虛源法（或稱鏡像源法Mirror Image Source Method，簡稱MISM）做了
一定程度的研究與貢獻，對干涉模型（Coherent Model）的邊界參數進行了改進，研究了不同邊
界條件下某些平面或直線沿長方向的衰減，充分考慮聲波的頻率因素和波動性能，可以瞭解不同
阻抗邊界的長空間音場現象，亦對不同的音源位置進行了多種阻抗邊界條件下的數值分析。另
外，進行了原模型、有限長旁支模型、有限長旁支加寬模型、有限長旁支加長模、無限長旁支模
型及無限長旁支加寬模型的實驗，加上理論與數值模擬計算和比較。
文章安排如下：
第一章文獻回顧。
第二章介紹了長空間中聲傳播的特性以及長空間消音方法，概括了理論求解長空間音場的常
用方法。
第三章首先推導點音源激勵下長空間音場的理論解，再介紹了長空間中基於幾何聲學的虛源
法的運用，然後詳細研究了基於波動理論的虛源模型。不同空間情況對波動方程式的解法基礎
上，將單一表面球面聲波的反射及其在多面空間的擴展引入，綜合得到長空間中基於波動聲學的
複雜虛源法，即「干涉模型」，將其運用於不同阻抗邊界（貼上吸音材料）的長空間音場研究。
在第三章建立模型的基礎上，第四章進行Odeon 數值模擬及對於不同的音源位置進行了多種
阻抗邊界（貼上吸音材料）條件下的數值分析。所討論的音源位置分為三種：音源在長空間正中，
音源靠近側面邊界和音源靠近角落。每種音源位置對於不同的阻抗邊界（貼上吸音材料）進行了
音場衰減的預測和具體的分析。
第五章是具體的實驗。首先進行了，原模型、有限長旁支模型、有限長旁支加寬模型、有限
長旁支加長模、無限長旁支模型及無限長旁支加寬模型的實驗，透過理論與數值模擬計算和比
較，可以發現些規律。數值模擬解隨距離的衰減過於平滑，預測衰減量比實驗值大；而理論解隨
距離的衰減波動較大，預測衰減量比實驗值小。
第六章對於全文提出結論與建議。

The propagation of sound in long enclosures with branches has been studied theoretically and experimentally, and an efficient combined method is proposed to predict the sound field in long enclosures with branches.  Based on the wave-acoustics theory, the theoretical analysis of the sound field of the long enclosures with branches is performed. This paper also investigated the sound field prediction of long enclosures with branches, by using the acoustic modeling program, ODEON. The results obtained by the theoretical analysis and the numerical simulation ODEON are compared with the experimental measurements, and the characteristics of the two methods for predicting the sound field of long enclosures with branches are analyzed. Compared with the experimental results, it is found that: 1) the results predicted by the theoretical analysis fluctuate relatively large with respect to the source-receiver distance, and the sound pressure level (SPL) attenuation obtained is smaller than that measured; and 2) the results predicted by the numerical simulation is smoother, and the calculated SPL attenuation is larger than that measured. To effectively predict the sound field of long enclosures with branches, a combined numerical method is thus proposed. The effectiveness of the proposed combined method is demonstrated by the scale-model experiments. A theoretical model has been developed for the prediction of sound propagation in a rectangular long enclosure. The model is based on the image-source method, and the effect of interference among the infinite number of image sources generated by multiple reflections is incorporated by coherently summing the contributions from the image sources. The various impedances of the boundaries are added in the model. Experiments are carried out to validate the proposed theoretical model, where the enclosure walls are lined with different kinds of sound absorption material to simulate different impedance boundaries. It is shown in the paper that the developed model agrees reasonably better with the experimental data than that of the ODEON software.

第一章 文獻回顧 1
1.1 長空間的定義 1
1.2 長空間聲學研究歷史 1
1.2.1 長空間音場研究的歷史 1
1.2.2 長空間餘響研究的歷史 6
1.2.3 長空間的相關問題 7
1.3 現有研究方法 8
1.3.1 波動聲學處理長空間音場及其限制性 9
1.3.2 幾何聲學處理長空間音場及其限制性 10
1.3.3 統計聲學處理長空間音場及其限制性 12
1.4 現有建立模型方法 16
1.4.1 數值模擬 16
1.4.2 縮尺模型 25
第二章 長空間音場計算 30
2.1 平面波在管中的傳播 31
2.1.1 平面波在突變截面管的傳播 31
2.1.2 平面波在垂直旁支管的傳播 33
2.2 波動理論 38
2.2.1 矩形長空間的模式 38
2.2.2 關於長空間模式的討論 42
2.3 突變截面管的模式解 46
2.4 虛源法求解長空間中音場 48
第三章 理論部分 51
3.1 長空間音場的理論解 51
3.1.1 有限長旁支 51
3.1.2 無限長旁支 53
3.2 基於幾何聲學的虛源法的運用 57
3.2.1 點音源 57
3.2.2 線音源 60
3.3 基於波動理論的虛源法的運用 61
3.3.1 空間不同情況對波動方程式的求解 61
3.3.2 單一表面音波的反射 63
3.3.3 多表面音波的反射 63
3.4 基於波動聲學的複雜虛源法－干涉模型的運用 64
3.5 干涉模型用於求解不同阻抗邊界的長空間音場 65
第四章 數值模擬 69
4.1Odeon 數值模擬 69
4.1.1 原模型 71
4.1.2 有限長旁支模型 72
4.1.3 有限長旁支加寬模型 73
4.1.4 有限長旁支加長模型 74
4.1.5 無限長旁支模型 75
4.1.6 無限長旁支加寬模型 76
4.2 過度衰減係數EA 77
4.3 音源在正中時不同吸音材料鋪設方式下音場衰減的
預測
78
4.3.1 底面鋪設吸音材料時音場衰減的預測 78
4.3.2 側面鋪設吸音材料時音場衰減的預測 79
4.3.3 兩個側面鋪設相同的吸音材料音場衰減的預測 81
4.3.4 一個側面和一個底面鋪設相同的吸音材料音場
衰減的預測
83
4.3.5 兩個側面和底面鋪設相同的吸音材料音場衰減
的預測
85
4.4 音源靠近側面邊界時不同吸音材料鋪設方式下音場
衰減的預測
88
4.4.1 靠近音源的側面鋪設吸音材料音場衰減的預測 88
V
4.4.2 遠離音源的側面鋪設吸音材料音場衰減的預測 89
4.4.3 頂面鋪設吸音材料音場衰減的預測 90
4.5 音源靠近角落時不同吸音材料鋪設方式下音場衰減
的預測
92
4.5.1 遠離音源的側面和底面鋪設相同的吸音材料音
場衰減的預測
92
4.5.2 兩個側面鋪設相同的吸音材料音場衰減的預測 92
4.5.3 靠近音源的側面和底面鋪設相同的吸音材料音
場衰減的預測
95
第五章 數值模擬、理論與實驗結果比較 98
5.1 測量系統說明 98
5.1.1 聲學系統 98
5.1.2 測量設備 99
5.2 音源指向性測試 102
5.3 測量系統和測量可重複性測試 104
5.3.1 背景噪音 104
5.3.2 測量系統和測量可重複性測試 104
5.4 測量結果 106
5.4.1 相對SPL 106
5.4.2 原模型 107
5.4.3 有限長旁支模型 108
5.4.4 有限長旁支加寬模型 109
5.4.5 有限長旁支加長模型 110
5.4.6 無限長旁支模型 111
5.4.7 無限長旁支加寬模型 112
5.5 數值模擬、理論與實驗比較 113
5.5.1 原模型 114
5.5.2 有限長旁支模型 115
5.5.3 有限長旁支加寬模型 116
5.5.4 有限長旁支加長模 117
5.5.5 無限長旁支模型 118
VI
5.5.6 無限長旁支加寬模型 119
5.5.7 干涉模 120
第六章 結論 123
6.1 結論 123
6.2 建議 127
參考文獻
附錄目錄
附錄一、積分參考
附錄二、複雜虛源法干涉模型Matlab 程式說明
附錄三、長空間音場的理論解與Matlab 程式說明
附錄四、ODEON 模擬程式
附錄五、透射係數解
附錄六、以波動理論計算矩形長空間的音場

圖目錄
頁次
圖1.3-1 來自不同方向的音量的反射和接收..................................................12
圖1.3-2 長空間音傳播中散射體的作用..........................................................15
圖1.4-1 音束與不同介面的關係......................................................................19
圖1.4-2 反射音束..............................................................................................19
圖2.1-1 平面波在突變截面管的傳播..............................................................32
圖2.1-2 突變截面管的音壓反射係數和透射係數與面積比的關係..............33
圖2.1-3 垂直旁支管的擴張管模型..................................................................33
圖2.1-4 不同擴張管長度在不同面積比下音壓透射係數的選頻特性..........35
圖2.1-5 垂直旁支管的旁支管模型..................................................................36
圖2.1-6 不同旁支管長度在不同面積比下音壓透射係數的選頻特性..........37
圖2.2-1 矩形長空間的二次元示意圖（x 方向垂直於紙面向內） ...............39
圖2.2-2 正方形橫截面不同模式的分佈..........................................................43
圖2.2-3 單頻點音源在管道中的音壓分布......................................................46
圖2.2-4 管道中1m、2m 和5m 處各個頻率的音壓值....................................46
圖2.3-1 Miles 用等效電路法研究高階理論模式的各種不連續長空間.........48
圖2.4-1 虛源法原理圖......................................................................................49
圖2.4-2 長空間中點音源產生的虛源網格（橫截面圖） ..............................49
圖2.4-3 各階虛源與接收點的距離..................................................................49
圖2.4-4 虛源的可見性示意圖..........................................................................50
圖3.1-1 點音源激發下有限長旁支長空間理論解解示意圖..........................51
圖3.1-2 點音源激發下無限長旁支長空間理論解解示意圖..........................53
圖3.4-1 矩形截面的虛源反射模型..................................................................64
圖3.5-1 邊界條件不同的矩形截面的虛源反射模型......................................66
圖3.5-2 所有的源與接收處的空間位置..........................................................67
圖4.1-1 Odeon 軟體模擬流程 ..........................................................................69
圖4.1-2 長空間模型測量示意...........................................................................70
圖4.1-3 旁支長空間示意圖..............................................................................70
圖4.1-4 加寬旁支長空間示意圖......................................................................70
圖4.1-5 長旁支長空間示意圖..........................................................................70
圖4.1-6 原模型長空間ODEON 圖中點的標號...............................................71
圖4.1-7 無旁支長空間Odeon 模型..................................................................71
圖4.1-8 有限長旁支模型長空間ODEON 圖中點的標號...............................72
圖4.1-9 有限長旁支長空間ODEON 模型......................................................72
圖4.1-10 有限長旁支加寬模型長空間ODEON 圖中點的標號.....................73
圖4.1-11 有限長旁支加寬長空間Odeon 模型.................................................73
圖4.1-12 有限長旁支加長模型長空間Odeon 圖中點的標號.......................74
圖4.1-13 有限長旁支加長長空間Odeon 模型...............................................74
圖4.1-14 無限長旁支長空間Odeon 圖中點的標號.......................................75
圖4.1-15 無限長旁支長空間Odeon 模型........................................................75
圖4.1-16 無限長旁支加寬長模型空間Odeon 圖中點的標號.......................76
圖4.1-17 無限長旁支加寬長模型空間Odeon 模型.......................................76
圖4.3-1 長空間音源、吸音材料和音場預測平面圖......................................78
圖4.3-2 頂面為吸音材料，音源位於(0.225,0,0.35)，z=0.35m平面的因場分佈......................................78
圖4.3-3 長空間音源、吸音材料和音場預測平面圖a)x-z 平面示意圖；
b)x-y 平面示意圖...............................................................................79
圖4.3-4 側面為吸音材料，音源位於(0.225, 0, 0.35)， =0.35m 平面的
音場分佈.............................................................................................80
圖4.3-5 音源位於(0.225, 0, 0.35)，底面為吸音材料情況下與側面為吸
音材料情況下=0.35m 平面的過度衰減係數的吸音差量............80
圖4.3-6 長空間音源、吸音材料和音場預測平面圖a)x-z 平面示意圖；
b)x-y 平面示意圖...............................................................................81
圖4.3-7 兩個側面為吸音材料，音源位於(0.225, 0, 0.35)， =0.35m 平
面的音場分佈.....................................................................................82
圖4.3-8 音源位於(0.225, 0, 0.35)，兩個側面為吸音材料情況下與單個
側面為吸音材料情況下=0.35m 平面的過度衰減係數的吸
音差量.................................................................................................82
圖4.3-9 長空間音源、吸音材料和音場預測平面圖a)x-z 平面示意圖；
b)x-y 平面示意圖...............................................................................83
圖4.3-10 側面和底面為吸音材料，音源位於(0.225, 0, 0.35)， z =0.35
平面的音場分佈.................................................................................83
圖4.3-11 音源位於(0.225, 0, 0.35)，側面和底面都為吸音材料情況下
與側面為吸音材料情況下 z =0.35m 平面的過度衰減係數的
吸音差量.............................................................................................84
圖4.3-12 長空間音源、吸音材料和音場預測平面圖a)x-z 平面示意
圖；b)x-y 平面示意圖.......................................................................85
圖4.3-13 底面和兩個側面為吸音材料，音源位於(0.225, 0, 0.35)，
z =0.35m 平面的音場分佈................................................................85
圖4.3-14 音源位於(0.225, 0, 0.35)，兩個側面和底面都為吸音材料情
況下，與側面和底面為吸音材料情況下=0.35m 平面的過度
衰減係數的吸音差量.........................................................................86
z
圖4.3-15 音源位於(0.225, 0, 0.35)，兩個側面和底面都為吸音材料情
況下，與兩個側面為吸音材料情況下z =0.35m 平面的過度衰
減係數的吸音差量.............................................................................86
圖4.4-1 長空間音源、吸音材料和音場預測平面圖a)x-z 平面示意圖；
b)x-y 平面示意圖...............................................................................88
圖4.4-2 側面為吸音材料，音源位於(0.1, 0, 0.35)， =0.35m 平面的音
場分佈.................................................................................................88
圖4.4-3 長空間音源、吸音材料和音場預測平面圖a)x-z 平面示意圖；
b)x-y 平面示意圖...............................................................................89
圖4.4-4 側面為吸音材料，音源位於(0.1, 0, 0.35)，
z
z =0.35m 平面的音
場分佈.................................................................................................89
圖4.4-5 音源位於(0.1, 0, 0.35)，側面x =0 為吸音材料情況下，與側面
x =0.45 為吸音材料情況下................................................................90
圖4.4-6 長空間音源、吸音材料和音場預測平面圖a)x-z 平面示意圖；
b)x-y 平面示意圖...............................................................................91
圖4.4-7 頂面為吸音材料，音源位於(0.1, 0, 0.35)，z =0.35m 平面的音
場分佈.................................................................................................91
圖4.4-8 音源位於(0.1, 0, 0.35)，側面x =0 為吸音材料情況下，與側面
x =0.45 為吸音材料情況下=0.35m 平面的過度衰減係數的
吸音差量.............................................................................................92
z
圖4.5-1 長空間音源、吸音材料和音場預測平面圖a)x-z 平面示意圖；
b)x-y 平面示意圖...............................................................................93
圖4.5-2 音源在 (0.1 0, 0.1)處，遠離音源的側面和底面為吸音材料，
z =0.35 平面的音場分佈....................................................................93
圖4.5-3 長空間音源、吸音材料和音場預測平面圖a)x-z 平面示意圖；
b)x-y 平面示意圖...............................................................................94
圖4.5-4 音源位於(0.1, 0, 0.1)，兩個側面為吸音材料， z =0.35m 平面
的音場分佈.........................................................................................94
圖4.5-5 音源位於(0.1, 0, 0.1)，x =0 的側面和x =0.45 的側面為吸音材
料情況下，與z =0.7 的頂面和x =0.45 的側面為吸音材料情
況下， =0.35m 平面的過度衰減係數的吸音差量........................95
圖4.5-6 長空間音源、吸音材料和音場預測平面圖a)x-z 平面示意圖；
b)x-y 平面示意圖...............................................................................95
圖4.5-7 音源在 (0.1 0, 0.1)處，靠近音源的側面和底面為吸音材料，
=0.35 平面的音場分佈....................................................................96
圖4.5-8 音源位於(0.1, 0, 0.1)，兩個側面和底面都為吸音材料情況
下，與側面和底面為吸音材料情況下=0.35m 平面的過度衰
減係數的吸音差量.............................................................................97
圖5.1-1 長空間縮尺模型測量示意圖..............................................................98
圖5.1-2 實際測量系統內部結構照片..............................................................99
圖5.1-3 點音源照片..........................................................................................99
圖5.1-4 PULSE (聲學)材料測試系統.............................................................100
圖5.1-5 吸音棉的吸音係數............................................................................101
圖5.2-1 長管型音源指向性測試照片............................................................102
圖5.2-2 長管型音源指向性繪圖....................................................................103
圖5.3-1 多次測量無旁支系統音場分布圖....................................................105
圖5.4-1 無旁支長空間示意圖........................................................................107
圖5.4-2 無旁支長空間照片............................................................................107
圖5.4-3 短旁支長空間示意圖........................................................................108
圖5.4-4 短旁支長空間照片............................................................................108
圖5.4-5 寬旁支長空間示意圖........................................................................109
圖5.4-6 寬旁支長空間照片.............................................................................109
圖5.4-7 長旁支長空間示意圖........................................................................110
圖5.4-8 長旁支長空間照片.............................................................................110
圖5.4-9 無限長旁支長空間示意圖................................................................ 111
圖5.4-10 長旁支長空間照片.......................................................................... 111
圖5.4-11 無限長旁支加寬長空間示意圖......................................................112
圖5.4-12 長旁支長空間照片..........................................................................112
圖5.5-1 長直管道的理論解，實驗測量值與數值模擬解比較....................114
圖5.5-2 原旁支管道的理論解，實驗測量值與數值模擬解比較................115
圖5.5-3 加寬旁支管道的理論解，實驗測量值與數值模擬解比較............116
圖5.5-4 加長旁支管道的理論解，實驗測量值與數值模擬解比較............117
圖5.5-5 無限長旁支管道的理論解，實驗測量值與數值模擬解比較........118
圖5.5-6 加寬無限長旁支管道的理論解、實驗測量值與數值模擬解比
較.......................................................................................................119
圖5.7-7 音源位於(0.2, 0, 0.35)，接收處位於x=0.35, z=0.35 一條直線
上，頻率分別為a)f=500Hz、b) f =1,000Hz、c) f =2,000Hz、
d) f =4,000Hz....................................................................................120
圖5.8-8 音源位於(0.2, 0, 0.35)，接收處位於x=0.1, z=0.35 一條直線
上，頻率分別為a)f=500Hz、b) f =1,000Hz、c) f =2,000Hz、
d) f =4,000Hz....................................................................................121

表目錄
頁次
表1.2-1 長空間音場研究的歷史彙整表............................................................5
表1.3-1 現有研究方法比較分析表..................................................................16
表1.4-1 現有建立模型原理說明表..................................................................25
表1.4-2 現有建立模型方法比較分析表..........................................................28
表1.4-2 現有建立模型方法比較分析表(續) ...................................................29
表3.3-1 三種空間模型的本徵函數和本徵值..................................................62
表4.1-1 無旁支長空間(原模型)相對SPL Odeon 數值模擬結果...................72
表4.1-2 有限長旁支長空間相對SPL Odeon 數值模擬結果..........................73
表4.1-3 有限長旁支加寬長空間相對SPL Odeon 數值模擬結果..................74
表4.1-4 有限長旁支加長模型長空間相對SPL Odeon 數值模擬結果..........75
表4.1-5 無限長旁支長空間相對SPL Odeon 數值模擬結果..........................76
表4.1-6 無限長旁支加寬長空間相對SPL Odeon 數值模擬結果..................77
表5.1-1 測量設備清單......................................................................................99
表5.1-2 吸音棉的導納....................................................................................101
表5.3-1 背景噪音............................................................................................104
表5.4-1 無旁支長空間相對SPL 測量結果...................................................107
表5.4-2 短旁支長空間相對SPL 測量結果...................................................108
表5.4-3 寬旁支長空間相對SPL 測量結果...................................................109
表5.4-4 長旁支長空間相對SPL 測量結果...................................................110
表5.4-5 無限長旁支長空間相對SPL 測量結果........................................... 111
表5.4-6 無限長旁支加寬長空間相對SPL 測量結果...................................112
表5.5-1 長直管道相對SPL 理論解結果.......................................................114
表5.5-2 原旁支管道相對SPL 理論解結果...................................................115
表5.5-3 加寬旁支管道相對SPL 理論解結果...............................................116
表5.5-4 加長旁支管道相對SPL 理論解結果...............................................117
表5.5-5 無限長旁支管道相對SPL 理論解結果...........................................118
表5.5-6 加寬無限長旁支管道相對SPL 理論解結果...................................119

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