Here, we consider Case I of an imploding strong shock in the neighborhood of implosion. For the problem of a converging shock collapsing at the axis, the origin of time

*t* is taken to be the instant at which the shock reaches the axis so that

$t\le 0$ in (

28). Therefore, the definition of the similarity variable is slightly modified by setting

$X=A{(-t)}^{\delta},\phantom{\rule{2em}{0ex}}\xi =x/A{(-t)}^{\delta},$

(42)

so that the intervals of the variables are

$-\mathrm{\infty}<t\le 0$,

$X\le x<\mathrm{\infty}$ and

$1\le \xi <\mathrm{\infty}$. At the instant of collapse (

$t=0$), the gas velocity, pressure, density and the sound speed at any finite axial distance

*x* are bounded, but with

$t=0$ and finite

*x*,

$\xi =\mathrm{\infty}$. In order for the quantities

*u*,

*p*,

*ρ* and

*h* to be bounded when

$t=0$ and

*x* is finite, we must have the following boundary conditions at

$\xi =\mathrm{\infty}$:

$U(\mathrm{\infty})=0,\phantom{\rule{2em}{0ex}}\frac{P(\mathrm{\infty})}{S(\mathrm{\infty})}=0,\phantom{\rule{2em}{0ex}}E(\mathrm{\infty})=0.$

(43)

In the matrix notation, system (

28) can be written as

where

$W={(U,S,P,E)}^{tr}$, and the matrix

*C* and the column vector

*B* can be identified by observing system (

28). In system (

28), there is an unknown parameter

*δ*, which cannot be obtained from an energy balance or the dimensional considerations; it is computed only by solving a non-linear eigenvalue problem for a system of ordinary differential equations. The range of similarity variable is

$1\le \xi <\mathrm{\infty}$ for the implosion problem, and system (

44) can be solved for the derivatives

${U}^{\prime}$,

${S}^{\prime}$,

${P}^{\prime}$ and

${E}^{\prime}$ in the following form:

${U}^{\prime}=\frac{{\mathrm{\Delta}}_{1}}{\mathrm{\Delta}},\phantom{\rule{2em}{0ex}}{S}^{\prime}=\frac{{\mathrm{\Delta}}_{2}}{\mathrm{\Delta}},\phantom{\rule{2em}{0ex}}{P}^{\prime}=\frac{{\mathrm{\Delta}}_{3}}{\mathrm{\Delta}},\phantom{\rule{2em}{0ex}}{E}^{\prime}=\frac{{\mathrm{\Delta}}_{4}}{\mathrm{\Delta}},$

(45)

where Δ, defined as the determinant of the matrix

*C*, is given by

$\mathrm{\Delta}={(U-\xi )}^{2}[{(U-\xi )}^{2}-\frac{(\gamma P+2E)}{S}],$

(46)

and

${\mathrm{\Delta}}_{k}$ (

$k=1,2,3,4$) are the determinants obtained from Δ by replacing the

*k* th column by the column vector

*B*. It can be observed that

$U<\xi $ in the interval

$[1,\mathrm{\infty})$, while Δ is positive at

$\xi =1$ and negative at

$\xi =\mathrm{\infty}$ indicating that there exists a

$\xi \in [1,\mathrm{\infty})$ at which Δ vanishes, and consequently the solutions become singular. In order to get a non-singular solution of (

44) in the interval

$[1,\mathrm{\infty})$, we select the value of the exponent

*δ* such that Δ vanishes only at the points where the determinant

${\mathrm{\Delta}}_{1}$ is zero too. It can be checked that at points where Δ and

${\mathrm{\Delta}}_{1}$ vanish, the determinants

${\mathrm{\Delta}}_{2}$,

${\mathrm{\Delta}}_{3}$ and

${\mathrm{\Delta}}_{4}$ also vanish simultaneously. To find the exponent

*δ* in such a manner, we introduce the variable

*Z* as follows:

$Z(\xi )={(U-\xi )}^{2}-\frac{\gamma P(\xi )+2E(\xi )}{S(\xi )}.$

(47)

This, in view of (

45), implies

${Z}^{\prime}=\{2(U-\xi )({\mathrm{\Delta}}_{1}-\mathrm{\Delta})-\frac{\gamma {\mathrm{\Delta}}_{3}+2{\mathrm{\Delta}}_{4}}{S}+\frac{\gamma P+2E}{{S}^{2}}{\mathrm{\Delta}}_{2}\}/\mathrm{\Delta}.$

(48)

Equations (

45), in view of (

48), become

$\begin{array}{r}\frac{dU}{dZ}=\frac{{\mathrm{\Delta}}_{1}}{{\mathrm{\Delta}}_{5}},\phantom{\rule{2em}{0ex}}\frac{dS}{dZ}=\frac{{\mathrm{\Delta}}_{2}}{{\mathrm{\Delta}}_{5}},\\ \frac{dP}{dZ}=\frac{{\mathrm{\Delta}}_{3}}{{\mathrm{\Delta}}_{5}},\phantom{\rule{2em}{0ex}}\frac{dE}{dZ}=\frac{{\mathrm{\Delta}}_{4}}{{\mathrm{\Delta}}_{5}},\end{array}$

(49)

where

${\mathrm{\Delta}}_{5}=2(U-\xi )({\mathrm{\Delta}}_{1}-\mathrm{\Delta})-\frac{\gamma {\mathrm{\Delta}}_{3}+2{\mathrm{\Delta}}_{4}}{S}+\frac{\gamma P+2E}{{S}^{2}}{\mathrm{\Delta}}_{2}$, with

$\xi =U+{\{Z+\frac{\gamma P+2E}{S}\}}^{1/2}$.