分享一個神奇的故事好了,我建議台灣應該多開一點dimension analysis的課程
可惜台灣量綱分析(dimension analysis)最強的大師林琦焜教授退休了@@
稍微講一下量綱分析故事好了,傅立葉先提出量綱分析,後來Lord Rayleigh把他發揚
光大,量綱分析最重要的就是Buckingham π theorem
這個定理是量綱分析的核心定理,概念很簡單,用線性代數就可以證明了
https://en.wikipedia.org/wiki/Rayleigh%27s_method_of_dimensional_analysis
https://en.wikipedia.org/wiki/Buckingham_%CF%80_theorem
其實研究生做實驗更需要學到dimension analysis的方法,因為實驗上面通常並
沒有固定的物理定律可以用,但是天才的應用數學家和物理學家總是可以得到物理規律
流體力學很多公式都是用dimension analysis猜出來的,比如著名的雷諾數,
還有Prandl的邊界層理論都是dimension analysis,天文物理學電漿的很多PDE的公式都
是用dimension analysis分析的
有一個著名的故事就是英國劍橋數學家物理家流體力學大師Geoffrey Ingram Taylor
https://en.wikipedia.org/wiki/G._I._Taylor
https://en.wikipedia.org/wiki/Blast_wave
The classic flow solution—the so-called "similarity solution"—was
independently devised by John von Neumann[1] [2] and British mathematician
Geoffrey Ingram Taylor[3][4] during World War II. After the war, the similarity
solution was published by three other authors—L. I. Sedov,[5] R. Latter,[6]
and J. Lockwood-Taylor[7]—who had discovered it independently.[8]
Since the early theoretical work more than 50 years ago, both theoretical
and experimental studies of blast waves have been ongoing.[9][10]
1941年6月初時候知道美國在試爆原子彈的時候,Taylor在紙上用dimension analysis
猜出這個原子彈爆炸的威力公式,假設空中的爆炸波的強度足夠高的範圍時,可以忽略原子
彈的大小和初始大氣壓力的影響,把這問題理想化為一個點的突然爆炸
paper推導如下
http://www3.nd.edu/~powers/ame.90931/taylor.blast.wave.I.pdf
這個推導我是在一本dimension analysis知道怎麼Taylor怎麼估的XD,真他媽超天才
paper後面我根本懶得看,高中生就可以學會Taylor的想法了
Geoffrey Taylor在1941年6月27日把這結果寄給了天才數學家John von Neumann,
過了三天John von Neumann改進了這方法得到一個封閉式的解,後來蘇聯的一個數學家
Leonid I. Sedov也獨自得到相同的結果
https://en.wikipedia.org/wiki/Leonid_I._Sedov
後來1945年Geoffrey Taylor後來就用J.E. Mack拍攝的原子彈結果估算原子彈爆炸
的威力,我們看第一段就好關鍵是那個R^5/2*t^-1
Photographs by J.E. Mack of the first atomic explosion in New Mexico were
measured, and the radius, R,of the luminous globe or 'ball of fire' which
spread out from the centre was determined for a large range of values of t,
the time measured from the start of the explosion.
The relationship predicted in part I, namely, that Rs would be proportional
to t, is surprisingly accurately verified over a range from R=20 to 185 m. The
value of R^5/2*t^-1 so found was used in conjunction with the formulae of part
I to estimate the energy E which was generated in the explosion. The amount of
this estimate depends on what value is assumed for y, the ratio of the specific
heats of air.
http://www3.nd.edu/~powers/ame.90931/taylor.blast.wave.II.pdf
結論
It will be seen that if y = 1.40, the T.N.T. equivalent of the energy of the
New Mexico explosion, or more strictly that part of the energy which was not
radiated outside the ball of fire, was 16,800 tons
八卦是
http://www.math.nctu.edu.tw/UPLOAD_FILES/NEWS_FILES/272_3.pdf
二次世界大戰期間英國流體力學大師 G. I. Taylor(3/7/1886-6/27/1975) 為
著研究原子彈爆炸, 從量綱分析的角度引進新的變數(similar variables)將 Euler 方程
化為常微分方程, 從而得出自相似解 (self-similar solution), 這結果甚至比美國國防
部的機密資料還精確, 害得美國國防部官員要調查是否有洩密事件。