分享一个神奇的故事好了,我建议台湾应该多开一点dimension analysis的课程
可惜台湾量纲分析(dimension analysis)最强的大师林琦焜教授退休了@@
稍微讲一下量纲分析故事好了,傅立叶先提出量纲分析,后来Lord Rayleigh把他发扬
光大,量纲分析最重要的就是Buckingham π theorem
这个定理是量纲分析的核心定理,概念很简单,用线性代数就可以证明了
https://en.wikipedia.org/wiki/Rayleigh%27s_method_of_dimensional_analysis
https://en.wikipedia.org/wiki/Buckingham_%CF%80_theorem
其实研究生做实验更需要学到dimension analysis的方法,因为实验上面通常并
没有固定的物理定律可以用,但是天才的应用数学家和物理学家总是可以得到物理规律
流体力学很多公式都是用dimension analysis猜出来的,比如著名的雷诺数,
还有Prandl的边界层理论都是dimension analysis,天文物理学电浆的很多PDE的公式都
是用dimension analysis分析的
有一个著名的故事就是英国剑桥数学家物理家流体力学大师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月初时候知道美国在试爆原子弹的时候,Taylor在纸上用dimension analysis
猜出这个原子弹爆炸的威力公式,假设空中的爆炸波的强度足够高的范围时,可以忽略原子
弹的大小和初始大气压力的影响,把这问题理想化为一个点的突然爆炸
paper推导如下
http://www3.nd.edu/~powers/ame.90931/taylor.blast.wave.I.pdf
这个推导我是在一本dimension analysis知道怎么Taylor怎么估的XD,真他妈超天才
paper后面我根本懒得看,高中生就可以学会Taylor的想法了
Geoffrey Taylor在1941年6月27日把这结果寄给了天才数学家John von Neumann,
过了三天John von Neumann改进了这方法得到一个封闭式的解,后来苏联的一个数学家
Leonid I. Sedov也独自得到相同的结果
https://en.wikipedia.org/wiki/Leonid_I._Sedov
后来1945年Geoffrey Taylor后来就用J.E. Mack拍摄的原子弹结果估算原子弹爆炸
的威力,我们看第一段就好关键是那个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
结论
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
二次世界大战期间英国流体力学大师 G. I. Taylor(3/7/1886-6/27/1975) 为
著研究原子弹爆炸, 从量纲分析的角度引进新的变量(similar variables)将 Euler 方程
化为常微分方程, 从而得出自相似解 (self-similar solution), 这结果甚至比美国国防
部的机密资料还精确, 害得美国国防部官员要调查是否有泄密事件。