進階搜尋


下載電子全文  
系統識別號 U0026-1907201912021300
論文名稱(中文) 由波方程的解模擬銅管樂音
論文名稱(英文) Simulating brass instrument sound with the solution of wave equation
校院名稱 成功大學
系所名稱(中) 數學系應用數學碩博士班
系所名稱(英) Department of Mathematics
學年度 107
學期 2
出版年 108
研究生(中文) 陳毓德
研究生(英文) Yu-Te Chen
學號 L16051116
學位類別 碩士
語文別 中文
論文頁數 66頁
口試委員 指導教授-沈士育
口試委員-王進猷
口試委員-廖炳松
中文關鍵字 波方程式  數學模型  傅立葉轉換  銅管樂音 
英文關鍵字 Wave equations  Mathematical model  Fourier transformation  Brass instruments 
學科別分類
中文摘要 銅管樂器的發聲原理是藉由吹嘴的振動與送入聲波,在閉管式的空氣管內進行共振而形成的。本文的研究方法是運用偏微分方程裡的波方程式,探討符合其混合邊界條件的數學模型。最後,本文將此數學模型模擬的音色藉由歌曲呈現銅管樂音。
本文的第一章將從了解銅管樂器與基礎樂理開始,重點在確認其發聲原理,還有主要影響音調的原因,並且藉由一些樂理的知識,將未來章節的內容與音樂方面作結合。
第二章是本文的重點,其為整個數學模型需要的數學背景,運用數學理論來解釋其銅管樂器的物理現象。數學模型的整個聲波傳遞也將在此章節詳細描述。
至於本文的第三章,將第二章的數學模型運用電腦來進行模擬。經由 Visual Studio 的應用程式,用 C 與 C++ 的程式執行出需要的數值。運用簡譜來呈現歌曲,並且以 wav 檔作處存,方便之後傅立葉轉換,轉成頻譜圖進行深入觀察。
最後,本文的第四章為討論與結論,銅管樂器的音調主要是藉由管身的共振而呈現,音調的共振頻率與管身的長度有關。因此,吹奏銅管樂器的指法是將管身的長度進行改變,來達到音調產生共振頻率所需的長度。然而,銅管樂音不單只有管身的長度影響,其吹嘴型號也會影響整體音色。期許未來能準確模擬不同吹嘴型號的聲波振動,再帶入本文的數學模型,達成銅管樂器的絕對音色。
英文摘要 The principle of the sound produced by a brass instrument is the vibration and sound waves produced from the mouthpiece, which will generate resonance in the closed tube. Using the wave equation of partial differential equations, the study investigates the boundary value condition of a mathematical model presented in the study. The mathematical model is then used to simulate the tone of the brass instrument sound.
By the application of Visual Studio, simulation is undertaken with C and C++ program languages, and simulation output is produced in the form of a song with notations, which may be saved as a document. By Fourier transformation, the document is transformed into a frequency map to facilitate observation.
As shown by the results of the study, the tone of the sound produced by a brass instrument may be attributable to the vibration of the tube. The resonance frequency of the tone is connected with the length of the tube. The fingering of playing a brass instrument is aimed at changing the length of the tube, so that the tube can meet the requirement for the resonance frequency of the tone. The sound of the brass instrument depends not only on the length of the tube, but on the type of the mouthpiece as well. Hopefully, the sound waves of different types of mouthpieces can be accurately simulated and exact brass instrument sound can be created by computer programs in the future research.
論文目次 中文摘要(roman 1)
英文延伸摘要(roman 2)
誌謝(roman 6)
目錄(roman 7)
圖目錄(roman 9)
第一章-緒論(1)
1.1 銅管樂器介紹(1)
1.2 小號的發聲原理(6)
1.3 空氣管的共振(8)
1.3.1 諧音與泛音(10)
1.4 十二平均律(12)
1.5 其它章節簡介(13)
第二章-理論分析(15)
2.1 波方程式的推導(15)
2.2 不同介質的波方程(21)
2.3 邊界質問題(26)
2.3.1 狄利克雷邊界條件(27)
2.3.2 諾伊曼邊界條件(30)
2.3.3 混合邊界條件(32)
2.4 初始值問題(34)
2.5 數學模型(37)
第三張-銅管樂器的模擬(45)
3.1 電腦模擬(45)
3.2 聲波的初始值(46)
3.2.1 例子(50)
3.3 聲波振動的加入(52)
3.4 歌曲(57)
第四章-討論與結論(59)
4.1 延伸討論(59)
4.2 結論(63)
參考文獻(64)
附錄(66)
參考文獻 [1] Ely, C., Mark, and Amy E. Van Deuren, (2009). "Wind Talk for Brass: A Practical Guide to Understanding and Teaching Brass Instruments." Oxford University Press. United States of America.
[2] Serway, A., Raymond, and John W. Jewett, (2012). "Principles of Physics: A Calculus-Based Text, 5th." Brooks/Cole-Cengage Learning. United States of America.
[3] Prestini, Elena, (2016). "The Evolution of Applied Harmonic Analysis: Models of the Real World, 2nd." Birkhauser.
[4] Christensen, Thomas, (2006). "The Cambridge History of Western Music Theory." Cambridge University Press.
[5] Miller, Michael, (2005). "The complete idiot's guide to music theory, 2nd." ALPHA-A member of Penguin Group Inc.
[6] Serway, A., Raymond, and John W. Jewett, (2004). "Physics for Scientists and Engineers, 6th." Thomson Brooks/Cole.
[7] Strauss, A., Walter. "Partial Differential Equations: An Introduction, 2nd." John Wiley & Sons.
[8] Walker, Jearl, David Halliday, and Robert Resnick, (2010). "Fundamentals of Physics, 9th." John Wiley & Sons.
[9] Salas, L., Saturnino, Garret J. Etgen, and Einar Hille, (2006). "Calculus: One and Several Variables, 10th." John Wiley & Sons.
論文全文使用權限
  • 同意授權校內瀏覽/列印電子全文服務,於2019-08-09起公開。
  • 同意授權校外瀏覽/列印電子全文服務,於2019-08-09起公開。


  • 如您有疑問,請聯絡圖書館
    聯絡電話:(06)2757575#65773
    聯絡E-mail:etds@email.ncku.edu.tw