Abstract

A simple analytic model is derived for describing the attenuation of a shock wave in a Plexiglas plate. At the same time, experimental measurements are presented with a well-designed optical-fiber sensor based on detection-beam deflection. The amplitude of the shock-wave pressure is measured experimentally and calculated numerically for analytic expressions at different distances from the region of the surface breakdown by the radiation of a single-pulse Nd:YAG laser. Good agreement between the experimental and the calculated values of the shock-wave pressure is established.

© 2006 Optical Society of America

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  1. F. John and Y. Read, "Laser-produced shocks and their relation to material damage," IEEE J. Quantum Electron. 14, 79-84 (1978).
    [Crossref]
  2. L. Berthe and R. Fabbro, "Shock wave from a water-confined laser-generated plasma," J. Appl. Phys. 82, 2826-2832 (1997).
    [Crossref]
  3. Y. Gu, Y. L. Ni, Y. G. Wang, C. S. Mao, F. C. Wu, J. Wu, J. Zhu, and B. G. Wang, "Experimental observation of laser driven high pressure shock waves," Acta Phys. Sin. 37, 1690-1693 (1988).
  4. X. W. Ni and J. Lu, "The study of laser-produced-plasma on dielectric thin films," Opt. Commun. 74, 185-189 (1989).
    [Crossref]
  5. B. Steverdung, "Ignition of laser detonation waves," J. Appl. Phys. 45, 3507-3511 (1974).
    [Crossref]
  6. A. H. Clauer, "Laser shocking of 2024 and 7075 aluminum alloys," NASA Rep. CR-145132 (NASA, 1977).
  7. G. Banas, "Laser shock induced mechanical and microstructural modification of welded marging steel," J. Appl. Phys. 67, 2389-2392 (1990).
    [Crossref]
  8. O. Calin, "Investigation involving shock waves generation and shock pressure measurement in direct ablation regime and confined ablation regime," Shock Waves 11, 393-397 (2002).
    [Crossref]
  9. A. Caruso, S. Yu. Gus'kov, I. Ya. Doskach, and N. V. Zmitrenko, "Laser-generated weak shock wave propagation dynamics in the solids," Proc. SPIE 4424, 508-511 (2001).
    [Crossref]
  10. S. Eliezer, Y. Gazit, and I. Gilath, "Shock wave decay and spall strength in laser-matter interaction," J. Appl. Phys. 68, 356-358 (1990).
    [Crossref]
  11. X. Hong, S. B. Wang, D. H. Guo, H. X. Wu, and J. L. Yu, "An investigation of parameters on the front of laser plasma shock wave," Chin. J. Lasers A25, 743-747 (1998).
  12. V. K. Senecha, M. Shukla, B. K. Godwal, and H. C. Pant, "Laser generated high-pressure shock wave experiments and their simulations," Proc. SPIE 5228, 33-42 (2003).
    [Crossref]
  13. S. S. Chen, A. Boswald, R. Sigel, and S. Witkoski, "Experiments on the wavelength and intensity dependence of laser-plasma interaction using high-speed shadowgraphy," Acta Phys. Sin. 36, 1395-1693 (1987).
  14. B. M. Bian, F. Hou, J. P. Chen, X. W. Ni, and J. Lu, "An investigation of parameters on the front of laser plasma shock wave," Chin. J. Lasers 28, 154-159 (2001).
  15. Y. Q. Li, Blast Mechanics (Science Press, 1992).
  16. R. J. Trainor and Y. T. Lee, "Analytic model for design of laser-generated shock-wave experiments," Phys. Fluids 25, 1898-1907 (1982).
    [Crossref]
  17. G. I. Taylor, The Scientific Papers of Sir Geoffrey Ingram Taylor--Collected Works, G.K.Batchelor, ed. (Cambridge U. Press, 1963), Vol. 3, p. 221.
  18. G. Yuan, G. Y. Zhou, Z. P. Tang, and X. Z. Li, "Propagation and decay of shock duration pulsed shock wave with high pressure," Explosion Shock Waves 12, 307-312 (1992).
  19. F. Cottet and J. P. Romain, "Formation and decay of laser-generated shock waves," Phys. Rev. A 25, 576-579 (1982).
    [Crossref]

2003 (1)

V. K. Senecha, M. Shukla, B. K. Godwal, and H. C. Pant, "Laser generated high-pressure shock wave experiments and their simulations," Proc. SPIE 5228, 33-42 (2003).
[Crossref]

2002 (1)

O. Calin, "Investigation involving shock waves generation and shock pressure measurement in direct ablation regime and confined ablation regime," Shock Waves 11, 393-397 (2002).
[Crossref]

2001 (2)

A. Caruso, S. Yu. Gus'kov, I. Ya. Doskach, and N. V. Zmitrenko, "Laser-generated weak shock wave propagation dynamics in the solids," Proc. SPIE 4424, 508-511 (2001).
[Crossref]

B. M. Bian, F. Hou, J. P. Chen, X. W. Ni, and J. Lu, "An investigation of parameters on the front of laser plasma shock wave," Chin. J. Lasers 28, 154-159 (2001).

1998 (1)

X. Hong, S. B. Wang, D. H. Guo, H. X. Wu, and J. L. Yu, "An investigation of parameters on the front of laser plasma shock wave," Chin. J. Lasers A25, 743-747 (1998).

1997 (1)

L. Berthe and R. Fabbro, "Shock wave from a water-confined laser-generated plasma," J. Appl. Phys. 82, 2826-2832 (1997).
[Crossref]

1992 (2)

Y. Q. Li, Blast Mechanics (Science Press, 1992).

G. Yuan, G. Y. Zhou, Z. P. Tang, and X. Z. Li, "Propagation and decay of shock duration pulsed shock wave with high pressure," Explosion Shock Waves 12, 307-312 (1992).

1990 (2)

G. Banas, "Laser shock induced mechanical and microstructural modification of welded marging steel," J. Appl. Phys. 67, 2389-2392 (1990).
[Crossref]

S. Eliezer, Y. Gazit, and I. Gilath, "Shock wave decay and spall strength in laser-matter interaction," J. Appl. Phys. 68, 356-358 (1990).
[Crossref]

1989 (1)

X. W. Ni and J. Lu, "The study of laser-produced-plasma on dielectric thin films," Opt. Commun. 74, 185-189 (1989).
[Crossref]

1988 (1)

Y. Gu, Y. L. Ni, Y. G. Wang, C. S. Mao, F. C. Wu, J. Wu, J. Zhu, and B. G. Wang, "Experimental observation of laser driven high pressure shock waves," Acta Phys. Sin. 37, 1690-1693 (1988).

1987 (1)

S. S. Chen, A. Boswald, R. Sigel, and S. Witkoski, "Experiments on the wavelength and intensity dependence of laser-plasma interaction using high-speed shadowgraphy," Acta Phys. Sin. 36, 1395-1693 (1987).

1982 (2)

R. J. Trainor and Y. T. Lee, "Analytic model for design of laser-generated shock-wave experiments," Phys. Fluids 25, 1898-1907 (1982).
[Crossref]

F. Cottet and J. P. Romain, "Formation and decay of laser-generated shock waves," Phys. Rev. A 25, 576-579 (1982).
[Crossref]

1978 (1)

F. John and Y. Read, "Laser-produced shocks and their relation to material damage," IEEE J. Quantum Electron. 14, 79-84 (1978).
[Crossref]

1977 (1)

A. H. Clauer, "Laser shocking of 2024 and 7075 aluminum alloys," NASA Rep. CR-145132 (NASA, 1977).

1974 (1)

B. Steverdung, "Ignition of laser detonation waves," J. Appl. Phys. 45, 3507-3511 (1974).
[Crossref]

1963 (1)

G. I. Taylor, The Scientific Papers of Sir Geoffrey Ingram Taylor--Collected Works, G.K.Batchelor, ed. (Cambridge U. Press, 1963), Vol. 3, p. 221.

Banas, G.

G. Banas, "Laser shock induced mechanical and microstructural modification of welded marging steel," J. Appl. Phys. 67, 2389-2392 (1990).
[Crossref]

Berthe, L.

L. Berthe and R. Fabbro, "Shock wave from a water-confined laser-generated plasma," J. Appl. Phys. 82, 2826-2832 (1997).
[Crossref]

Bian, B. M.

B. M. Bian, F. Hou, J. P. Chen, X. W. Ni, and J. Lu, "An investigation of parameters on the front of laser plasma shock wave," Chin. J. Lasers 28, 154-159 (2001).

Boswald, A.

S. S. Chen, A. Boswald, R. Sigel, and S. Witkoski, "Experiments on the wavelength and intensity dependence of laser-plasma interaction using high-speed shadowgraphy," Acta Phys. Sin. 36, 1395-1693 (1987).

Calin, O.

O. Calin, "Investigation involving shock waves generation and shock pressure measurement in direct ablation regime and confined ablation regime," Shock Waves 11, 393-397 (2002).
[Crossref]

Caruso, A.

A. Caruso, S. Yu. Gus'kov, I. Ya. Doskach, and N. V. Zmitrenko, "Laser-generated weak shock wave propagation dynamics in the solids," Proc. SPIE 4424, 508-511 (2001).
[Crossref]

Chen, J. P.

B. M. Bian, F. Hou, J. P. Chen, X. W. Ni, and J. Lu, "An investigation of parameters on the front of laser plasma shock wave," Chin. J. Lasers 28, 154-159 (2001).

Chen, S. S.

S. S. Chen, A. Boswald, R. Sigel, and S. Witkoski, "Experiments on the wavelength and intensity dependence of laser-plasma interaction using high-speed shadowgraphy," Acta Phys. Sin. 36, 1395-1693 (1987).

Clauer, A. H.

A. H. Clauer, "Laser shocking of 2024 and 7075 aluminum alloys," NASA Rep. CR-145132 (NASA, 1977).

Cottet, F.

F. Cottet and J. P. Romain, "Formation and decay of laser-generated shock waves," Phys. Rev. A 25, 576-579 (1982).
[Crossref]

Doskach, I. Ya.

A. Caruso, S. Yu. Gus'kov, I. Ya. Doskach, and N. V. Zmitrenko, "Laser-generated weak shock wave propagation dynamics in the solids," Proc. SPIE 4424, 508-511 (2001).
[Crossref]

Eliezer, S.

S. Eliezer, Y. Gazit, and I. Gilath, "Shock wave decay and spall strength in laser-matter interaction," J. Appl. Phys. 68, 356-358 (1990).
[Crossref]

Fabbro, R.

L. Berthe and R. Fabbro, "Shock wave from a water-confined laser-generated plasma," J. Appl. Phys. 82, 2826-2832 (1997).
[Crossref]

Gazit, Y.

S. Eliezer, Y. Gazit, and I. Gilath, "Shock wave decay and spall strength in laser-matter interaction," J. Appl. Phys. 68, 356-358 (1990).
[Crossref]

Gilath, I.

S. Eliezer, Y. Gazit, and I. Gilath, "Shock wave decay and spall strength in laser-matter interaction," J. Appl. Phys. 68, 356-358 (1990).
[Crossref]

Godwal, B. K.

V. K. Senecha, M. Shukla, B. K. Godwal, and H. C. Pant, "Laser generated high-pressure shock wave experiments and their simulations," Proc. SPIE 5228, 33-42 (2003).
[Crossref]

Gu, Y.

Y. Gu, Y. L. Ni, Y. G. Wang, C. S. Mao, F. C. Wu, J. Wu, J. Zhu, and B. G. Wang, "Experimental observation of laser driven high pressure shock waves," Acta Phys. Sin. 37, 1690-1693 (1988).

Guo, D. H.

X. Hong, S. B. Wang, D. H. Guo, H. X. Wu, and J. L. Yu, "An investigation of parameters on the front of laser plasma shock wave," Chin. J. Lasers A25, 743-747 (1998).

Gus'kov, S. Yu.

A. Caruso, S. Yu. Gus'kov, I. Ya. Doskach, and N. V. Zmitrenko, "Laser-generated weak shock wave propagation dynamics in the solids," Proc. SPIE 4424, 508-511 (2001).
[Crossref]

Hong, X.

X. Hong, S. B. Wang, D. H. Guo, H. X. Wu, and J. L. Yu, "An investigation of parameters on the front of laser plasma shock wave," Chin. J. Lasers A25, 743-747 (1998).

Hou, F.

B. M. Bian, F. Hou, J. P. Chen, X. W. Ni, and J. Lu, "An investigation of parameters on the front of laser plasma shock wave," Chin. J. Lasers 28, 154-159 (2001).

John, F.

F. John and Y. Read, "Laser-produced shocks and their relation to material damage," IEEE J. Quantum Electron. 14, 79-84 (1978).
[Crossref]

Lee, Y. T.

R. J. Trainor and Y. T. Lee, "Analytic model for design of laser-generated shock-wave experiments," Phys. Fluids 25, 1898-1907 (1982).
[Crossref]

Li, X. Z.

G. Yuan, G. Y. Zhou, Z. P. Tang, and X. Z. Li, "Propagation and decay of shock duration pulsed shock wave with high pressure," Explosion Shock Waves 12, 307-312 (1992).

Li, Y. Q.

Y. Q. Li, Blast Mechanics (Science Press, 1992).

Lu, J.

B. M. Bian, F. Hou, J. P. Chen, X. W. Ni, and J. Lu, "An investigation of parameters on the front of laser plasma shock wave," Chin. J. Lasers 28, 154-159 (2001).

X. W. Ni and J. Lu, "The study of laser-produced-plasma on dielectric thin films," Opt. Commun. 74, 185-189 (1989).
[Crossref]

Mao, C. S.

Y. Gu, Y. L. Ni, Y. G. Wang, C. S. Mao, F. C. Wu, J. Wu, J. Zhu, and B. G. Wang, "Experimental observation of laser driven high pressure shock waves," Acta Phys. Sin. 37, 1690-1693 (1988).

Ni, X. W.

B. M. Bian, F. Hou, J. P. Chen, X. W. Ni, and J. Lu, "An investigation of parameters on the front of laser plasma shock wave," Chin. J. Lasers 28, 154-159 (2001).

X. W. Ni and J. Lu, "The study of laser-produced-plasma on dielectric thin films," Opt. Commun. 74, 185-189 (1989).
[Crossref]

Ni, Y. L.

Y. Gu, Y. L. Ni, Y. G. Wang, C. S. Mao, F. C. Wu, J. Wu, J. Zhu, and B. G. Wang, "Experimental observation of laser driven high pressure shock waves," Acta Phys. Sin. 37, 1690-1693 (1988).

Pant, H. C.

V. K. Senecha, M. Shukla, B. K. Godwal, and H. C. Pant, "Laser generated high-pressure shock wave experiments and their simulations," Proc. SPIE 5228, 33-42 (2003).
[Crossref]

Read, Y.

F. John and Y. Read, "Laser-produced shocks and their relation to material damage," IEEE J. Quantum Electron. 14, 79-84 (1978).
[Crossref]

Romain, J. P.

F. Cottet and J. P. Romain, "Formation and decay of laser-generated shock waves," Phys. Rev. A 25, 576-579 (1982).
[Crossref]

Senecha, V. K.

V. K. Senecha, M. Shukla, B. K. Godwal, and H. C. Pant, "Laser generated high-pressure shock wave experiments and their simulations," Proc. SPIE 5228, 33-42 (2003).
[Crossref]

Shukla, M.

V. K. Senecha, M. Shukla, B. K. Godwal, and H. C. Pant, "Laser generated high-pressure shock wave experiments and their simulations," Proc. SPIE 5228, 33-42 (2003).
[Crossref]

Sigel, R.

S. S. Chen, A. Boswald, R. Sigel, and S. Witkoski, "Experiments on the wavelength and intensity dependence of laser-plasma interaction using high-speed shadowgraphy," Acta Phys. Sin. 36, 1395-1693 (1987).

Steverdung, B.

B. Steverdung, "Ignition of laser detonation waves," J. Appl. Phys. 45, 3507-3511 (1974).
[Crossref]

Tang, Z. P.

G. Yuan, G. Y. Zhou, Z. P. Tang, and X. Z. Li, "Propagation and decay of shock duration pulsed shock wave with high pressure," Explosion Shock Waves 12, 307-312 (1992).

Taylor, G. I.

G. I. Taylor, The Scientific Papers of Sir Geoffrey Ingram Taylor--Collected Works, G.K.Batchelor, ed. (Cambridge U. Press, 1963), Vol. 3, p. 221.

Trainor, R. J.

R. J. Trainor and Y. T. Lee, "Analytic model for design of laser-generated shock-wave experiments," Phys. Fluids 25, 1898-1907 (1982).
[Crossref]

Wang, B. G.

Y. Gu, Y. L. Ni, Y. G. Wang, C. S. Mao, F. C. Wu, J. Wu, J. Zhu, and B. G. Wang, "Experimental observation of laser driven high pressure shock waves," Acta Phys. Sin. 37, 1690-1693 (1988).

Wang, S. B.

X. Hong, S. B. Wang, D. H. Guo, H. X. Wu, and J. L. Yu, "An investigation of parameters on the front of laser plasma shock wave," Chin. J. Lasers A25, 743-747 (1998).

Wang, Y. G.

Y. Gu, Y. L. Ni, Y. G. Wang, C. S. Mao, F. C. Wu, J. Wu, J. Zhu, and B. G. Wang, "Experimental observation of laser driven high pressure shock waves," Acta Phys. Sin. 37, 1690-1693 (1988).

Witkoski, S.

S. S. Chen, A. Boswald, R. Sigel, and S. Witkoski, "Experiments on the wavelength and intensity dependence of laser-plasma interaction using high-speed shadowgraphy," Acta Phys. Sin. 36, 1395-1693 (1987).

Wu, F. C.

Y. Gu, Y. L. Ni, Y. G. Wang, C. S. Mao, F. C. Wu, J. Wu, J. Zhu, and B. G. Wang, "Experimental observation of laser driven high pressure shock waves," Acta Phys. Sin. 37, 1690-1693 (1988).

Wu, H. X.

X. Hong, S. B. Wang, D. H. Guo, H. X. Wu, and J. L. Yu, "An investigation of parameters on the front of laser plasma shock wave," Chin. J. Lasers A25, 743-747 (1998).

Wu, J.

Y. Gu, Y. L. Ni, Y. G. Wang, C. S. Mao, F. C. Wu, J. Wu, J. Zhu, and B. G. Wang, "Experimental observation of laser driven high pressure shock waves," Acta Phys. Sin. 37, 1690-1693 (1988).

Yu, J. L.

X. Hong, S. B. Wang, D. H. Guo, H. X. Wu, and J. L. Yu, "An investigation of parameters on the front of laser plasma shock wave," Chin. J. Lasers A25, 743-747 (1998).

Yuan, G.

G. Yuan, G. Y. Zhou, Z. P. Tang, and X. Z. Li, "Propagation and decay of shock duration pulsed shock wave with high pressure," Explosion Shock Waves 12, 307-312 (1992).

Zhou, G. Y.

G. Yuan, G. Y. Zhou, Z. P. Tang, and X. Z. Li, "Propagation and decay of shock duration pulsed shock wave with high pressure," Explosion Shock Waves 12, 307-312 (1992).

Zhu, J.

Y. Gu, Y. L. Ni, Y. G. Wang, C. S. Mao, F. C. Wu, J. Wu, J. Zhu, and B. G. Wang, "Experimental observation of laser driven high pressure shock waves," Acta Phys. Sin. 37, 1690-1693 (1988).

Zmitrenko, N. V.

A. Caruso, S. Yu. Gus'kov, I. Ya. Doskach, and N. V. Zmitrenko, "Laser-generated weak shock wave propagation dynamics in the solids," Proc. SPIE 4424, 508-511 (2001).
[Crossref]

Acta Phys. Sin. (2)

Y. Gu, Y. L. Ni, Y. G. Wang, C. S. Mao, F. C. Wu, J. Wu, J. Zhu, and B. G. Wang, "Experimental observation of laser driven high pressure shock waves," Acta Phys. Sin. 37, 1690-1693 (1988).

S. S. Chen, A. Boswald, R. Sigel, and S. Witkoski, "Experiments on the wavelength and intensity dependence of laser-plasma interaction using high-speed shadowgraphy," Acta Phys. Sin. 36, 1395-1693 (1987).

Chin. J. Lasers (2)

B. M. Bian, F. Hou, J. P. Chen, X. W. Ni, and J. Lu, "An investigation of parameters on the front of laser plasma shock wave," Chin. J. Lasers 28, 154-159 (2001).

X. Hong, S. B. Wang, D. H. Guo, H. X. Wu, and J. L. Yu, "An investigation of parameters on the front of laser plasma shock wave," Chin. J. Lasers A25, 743-747 (1998).

Explosion Shock Waves (1)

G. Yuan, G. Y. Zhou, Z. P. Tang, and X. Z. Li, "Propagation and decay of shock duration pulsed shock wave with high pressure," Explosion Shock Waves 12, 307-312 (1992).

IEEE J. Quantum Electron. (1)

F. John and Y. Read, "Laser-produced shocks and their relation to material damage," IEEE J. Quantum Electron. 14, 79-84 (1978).
[Crossref]

J. Appl. Phys. (4)

L. Berthe and R. Fabbro, "Shock wave from a water-confined laser-generated plasma," J. Appl. Phys. 82, 2826-2832 (1997).
[Crossref]

B. Steverdung, "Ignition of laser detonation waves," J. Appl. Phys. 45, 3507-3511 (1974).
[Crossref]

G. Banas, "Laser shock induced mechanical and microstructural modification of welded marging steel," J. Appl. Phys. 67, 2389-2392 (1990).
[Crossref]

S. Eliezer, Y. Gazit, and I. Gilath, "Shock wave decay and spall strength in laser-matter interaction," J. Appl. Phys. 68, 356-358 (1990).
[Crossref]

Opt. Commun. (1)

X. W. Ni and J. Lu, "The study of laser-produced-plasma on dielectric thin films," Opt. Commun. 74, 185-189 (1989).
[Crossref]

Phys. Fluids (1)

R. J. Trainor and Y. T. Lee, "Analytic model for design of laser-generated shock-wave experiments," Phys. Fluids 25, 1898-1907 (1982).
[Crossref]

Phys. Rev. A (1)

F. Cottet and J. P. Romain, "Formation and decay of laser-generated shock waves," Phys. Rev. A 25, 576-579 (1982).
[Crossref]

Proc. SPIE (2)

A. Caruso, S. Yu. Gus'kov, I. Ya. Doskach, and N. V. Zmitrenko, "Laser-generated weak shock wave propagation dynamics in the solids," Proc. SPIE 4424, 508-511 (2001).
[Crossref]

V. K. Senecha, M. Shukla, B. K. Godwal, and H. C. Pant, "Laser generated high-pressure shock wave experiments and their simulations," Proc. SPIE 5228, 33-42 (2003).
[Crossref]

Shock Waves (1)

O. Calin, "Investigation involving shock waves generation and shock pressure measurement in direct ablation regime and confined ablation regime," Shock Waves 11, 393-397 (2002).
[Crossref]

Other (3)

A. H. Clauer, "Laser shocking of 2024 and 7075 aluminum alloys," NASA Rep. CR-145132 (NASA, 1977).

G. I. Taylor, The Scientific Papers of Sir Geoffrey Ingram Taylor--Collected Works, G.K.Batchelor, ed. (Cambridge U. Press, 1963), Vol. 3, p. 221.

Y. Q. Li, Blast Mechanics (Science Press, 1992).

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Figures (5)

Fig. 1
Fig. 1

Variation of the pressure applied on the surface target and resulting pressure evolution.

Fig. 2
Fig. 2

Shock-wave pressure as a function of distance (filled circles, experimental data; open circles, numerical simulations).

Fig. 3
Fig. 3

Schematic diagram of an experimental setup based on OBD: 1, Q-switched Nd:YAG laser; 2, beam splitter; 3, attenuator group; 4, expanded device; 5, convex lens; 6, Plexiglas; 7, He–Ne laser; 8, convex lens; 9, microscope objective (20×, L2, f2 = 4 mm); 10, interference filter photomultipler (Hamamatsu H5773 with 2 ns rise time); 11, five-axis fiber-regulating stand; 12, single-mode optical-fiber p-i-n photodiode (with 0.1 ns rise time); 13, photomultiplier; 14, digital oscilloscope (Tektronix TDS340); 15, p-i-n photodiode (with 0.1 ns rise time).

Fig. 4
Fig. 4

Characteristic shock-wave sequence induced by 200 mJ laser pulses (the scale of the longitudinal coordinates is different).

Fig. 5
Fig. 5

Shock-wave velocity distribution versus distance.

Tables (1)

Tables Icon

Table 1 Hugoniot Parameters of the Plexiglas

Equations (29)

Equations on this page are rendered with MathJax. Learn more.

u t + u u x + 1 ρ p x = 0 ,
ρ t + u ρ x + ρ u x + 2 u ρ x = 0 ,
C 2 = ( p ρ ) s .
C = ( U u ) [ 2 ( U c 0 1 ) ( 1 r 0 2 u U ) + 1 ] 1 / 2 ,
d p d t = p t + U p x .
U = d x d t ,
p t + u p x + C 2 ρ u x + 2 C 2 u ρ x = 0.
d p d x d x d t = p t + U p x ,
d p d x = p x + 1 U p t .
d u d x = u x + u t d t d x ;
d u d x = u x + 1 U u t .
p t = U { 2 ρ u C 2 x ( U u ) + d p d x [ C 2 + u ( U u ) ] + d u d x C 2 ρ U } [ C 2 ( U u ) 2 ] 1 ,
d u d x = d u d p d p d x .
d p d x = p t C 2 ( U u ) 2 U 2 ρ u C 2 ( U u ) x C 2 + u ( U u ) + d u d p C 2 U ρ .
d p d x = p x ( u + C U 1 ) .
p t = p x x t , x t = ( C + u ) p .
d p d x = p t ( u + C U 1 ) 1 ( C + u ) p .
p p 0 = ρ 0 ( u u 0 ) ( U u 0 ) ,
p = ρ 0 U u .
U = C 0 + s u ,
p = ρ 0 u ( C 0 + s u ) .
d u d p = [ ρ 0 ( 2 U C 0 ) ] 1 .
p t | 0 = p m T 1 .
t i = x i ( C + u ) p i + t i     0 = x i ( C + u ) p i + τ p m p i ,
t i     0 = τ p m p i .
t i + Δ t = x i ( C + u ) p i + Δ p + τ p m ( p i + Δ p ) .
Δ p Δ t | x i = [ x i ( C + u ) p i + Δ p ( C + u ) p i ( C + u ) p i ( C + u ) p i + Δ p Δ p + τ p m ] 1 .
p t | x i = lim Δ p 0 Δ p Δ t | x i = [ τ p m x i ( C + u ) p i       2 ( d u d p + d C d p ) ] 1 .
p = ρ 0 U ( U C 0 S ) .

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