在這篇論文中, 我們討論的是圓柱活塞問題。在柱對稱系統中, 當在靜止均勻氣體中的圓柱活塞, 均勻地向外擴張時, 在外圍會產生圓柱震波。由解的自我相似性的假設, 此活塞問題能夠被簡化為起始條件在震波上的常微分方程系統的起始值問題。另外, 我們考慮活塞周圍的氣體速度等於活塞速度的邊界條件。代入不同的震波馬赫數, 我們可以由對常微分方程系統的數值積分得到活塞位置以及活塞無因次速度。我們的目標是找出震波馬赫數與活塞無因次速度的關係。除此之外, 我們也整理了S. C. Lin[6]所討論的在流星軌跡周圍所產生的圓柱震波的論文。

In this thesis, we discuss a cylindrical piston problem. In the case of cylindrical symmetry, as a cylindrical piston uniformly expands and pushes out undisturbed uniform polytropic gas ahead of itself, a cylindrical shock wave occurs outside the piston. Resulting from the assumption of self-similarity of the flow, this piston problem is simplified as the initial-value problem of a system of ordinary differential equations with the initial condition at the shock locus. The kinematic condition is considered as the boundary condition at the piston. For a different shock Mach number, we can obtain the position of the inner surface of a piston and the nondimensional velocity of a piston by the numerical integration of the system of ordinary differential equations. Our goal is to find the relationship between the shock Mach number and the nondimensional velocity of a piston. Additionally, we summarize S. C. Lin’s paper[6] regarding the cylindrical shock waves which occur around the luminous trails of meteors.

[1] Courant, R. and Friedrichs, K. O., Supersonic Flow and Shock Waves, Springer-Verlag, New York, 1976.
[2] Evans, L. C., Partial Differential Equations, American Mathematical Society, Providence, R.I., 1998.
[3] John, James E. A., Gas Dynamics, Allyn and Bacon, Boston, 1984.
[4] LaPaz, Lincoln and LaPaz, Jean, Space Nomads: Meteorites in Sky, Field and Laboratory, Holiday House, New York, 1961.
[5] LeVeque, R. J., Numerical Methods for Conservation Laws, Birkh¨auser Verlag, Basel, 1992.
[6] Lin, S. C., Cylindrical Shock Waves Produced by Instantaneous Energy Release, J. Appl. Phys., 25, 54–57, 1954.
[7] Norton, O. R., The Cambridge Encyclopedia of Meteorites, Cambridge University Press, Cambridge, 2002.
[8] Pinchover, Yehuda and Rubinstein, Jacob, An Introduction to Partial Differential Equations, Cambridge University Press, Cambridge, 2005.
[9] Rogers, M. H., Similarity Flows behind Strong Shock Waves, Quart. J. Mech. Appl. Math., 11, 411–422, 1958.
[10] Sachdev, P. L., Shock Waves and Explosions, Chapman & Hall/CRC, Boca Raton, 2004.
[11] Serre, Denis, Systems of Conservation Laws 1: Hyperbolicity, Entropies, Shock Waves, Cambridge University Press, Cambridge, 1999.
[12] Smoller, Joel, ShockWaves and Reaction-Diffusion Equations, Springer-Verlag, New York, 1994.
[13] Strauss, W. A., Partial Differential Equations: An Introduction, Wiley, New York, 1992.
[14] Taylor, G. I., The Air Wave Surrounding an Expanding Sphere, Pro. Roy. Soc. A, 186, 273–292, 1946.
[15] Whitham, G. B., Linear and Nonlinear Waves, Wiley, New York, 1999.
[16] International Meteor Organization, http://www.imo.net/
[17] Digital Images of the Sky, http://www.allthesky.com/