正激波基本控制方程的推导
音速
能量方程的特殊形式
什么情况下流动是可压缩的?
用于计算通过正激波气体特性变化的方
程的详细推导 ; 物理特性变化趋势的讨论
用皮托管测量可压缩流的流动速度
图 8.2 第八章路线图
8.4 SPECIAL FORMS OF THE ENERGY EQUATIONS
能量方程的特殊形式
本节需要掌握的内容要点,
? 能量方程的各种特殊表达形式
? 总温的计算公式
? 总压、总密度的计算公式
? 临界参数的定义与计算公式
? 特征马赫数（速度系数） M*的定义及计算公式
22
2
2
2
2
1
1
VhVh ???
22
2
2
2
2
1
1
uhuh ???
? 能量方程的各种特殊表达形式
在 7.5节中我们得到了定常、绝热、无粘流动的能量方程,
其中 V1,V2一条三维流线上的任意两点的速度。 对于
我们现在研究的一维流动，能量方程为,
（ 8.28)
（ 8.29)
However,keep in mind that all the subsequent results in this
section hold in general along a streamline and are by no means
limited to just one –dimensional flows,然而，应当记住的是：
这一节中所有的结论对于一般的沿流线的问题都适用，并不
只是局限于一维流动。
22
2
2
2
2
1
1
uTcuTc
pp ???
2121
2
22
2
11 uRTuRT ?
???? ?
?
?
?
2121
2
2
2
2
2
1
2
1 uaua ?
???? ??
(8.30)
(8.31)
RTa ??
(8.32)
以温度表示,
以音速表示,
121
2
0
22
???? ??
aua
12121
2
0
2
2
2
2
2
1
2
1
??????? ???
auaua
Definition of stagnation speed of sound：驻点音速的定义
(8.33)
(8.34)
)1(2
*)1(
21
222
?
???
? ?
?
?
aua
2
*
1
*
21
2222 aaua
????? ??
(8.35)
Definition of a*,a*的定义
7.5节最后一段,As a corollary to the above considerations,we
need another defined temperature,denoted by T*,and defined as
follows,Consider a point in a subsonic flow where the local static
temperature is T,At this point,imagine that the fluid element is
speeded up to sonic velocity,adiabatically,The Temperature it would
have at such sonic conditions is denoted as T*,Similarly,consider a
point in a supersonic flow,where the local static temperature is T,At
this point,imagine that the fluid element is slowed down to sonic
velocity,adiabatically,Again,the Temperature it would have at such
sonic conditions is denoted as T*,
用 *号表示的变量被称为临界参数, 称为临界音速, ** RTa ??
In Equation (8.35),a and u are the speed of sound and velocity,
respectively,at any point of flow,and a* is a characteristic value
associated with that same point,
c o n s tauaua ????????? )1(2 *)1(2121
22
2
2
2
2
1
2
1
?
?
??
对于沿一条流线上的任意两点,有,
(8.36)
c o n s taa ????? 1*)1(2 1
2
02
??
? (8.37)
Clearly,these defined quantities,a0 and a*,are both constants
along a given in a steady,adiabatic,inviscid flow,If all the
streamlines emanate from the same uniform freestream
conditions,then a0 and a* are constants throughout the entire
flow field,很明显,a0 和 a*为定义的量,沿定常、绝热、无粘
流动的给定流线为常数。如果所有流线都来自于均匀自由来
流，则 a0 和 a*在整个流场为常数。
0
2
2 Tc
uTc
pp ??
c o n s tTcuTcuTc ppp ????? 0
2
2
2
2
1
1 22
(8.38)
? 总温的计算公式
回忆 7.5节中总温 T0的定义，有方程 (8.30)可得,
(8.39)
Equation (8.38) provides a formula from which the defined total
temperature T0 can be calculated from the given actual conditions of
T and u at any given points in a general flow field,方程 (8.38) 给出
了由流场中给定点处的实际温度 T和速度 u计算总温 T0的计算公
式。
2
22
0 )(
2
11
)1/(2121 a
u
RT
u
Tc
u
T
T
p
???
?????
?
??
20
2
11 M
T
T ??? ? (8.40)
Equation (8.40) is very important; it states that only M (and,of
course,the value of ) dictates the ratio of total temperature to
static temperature,
方程 (8.40)非常重要； 表明只有马赫数（及 的值）决定总
温与静温的比。
?
For a calorically perfect gas,the ratio of total temperature to
static temperature,is a function of Mach number only,as
follows,（对于量热完全气体，总温和静温的比 是马赫数
的唯一函数，证明如下：）
TT0
TT0
?
)1(120
)1(20
)
2
1
1(
)
2
1
1(
?
?
?
??
?
??
?
??
?
?
?
?
M
M
p
p
)1(
000
?
?
?
??
?
??
???
?
???
?? ???
?
?
T
T
p
p
? 总压、总密度的计算公式,
回忆 7.5节总压和总密度的定义,在定义中包含了将气流速度
等熵地压缩为零速度。由（ 7.32）式,
我们有,
)1(
1
2
1
2
1
2
?
???
?
???
??
???
?
???
?? ???
?
?
T
T
p
p
（ 8.41)
（ 8.42)
（ 8.43)
方程（ 8.42)和（ 8.43）表明：总压静压比,总密度静密度
比 只由 M 和 决定。因此，对于给定气体，即给
定,, 只依赖于马赫数。
?
?
pp0
??0
pp0 ??0
Equation (8.40),(8.42)and (8.43) are very important; they should be
branded on your mind,They provided formulas from which the
defined,and can be calculated from the actual conditions of
M,T,p and at a given point in general flow field (assuming
calorically perfect gas),They are so important that values of
and obtained from Eqs,(8.40),(8.42),and (8.43),respectively,
are tabulated as functions of M in App.A for (which
corresponds to air at standard conditions),
方程 (8.40),(8.42)和 (8.43) 非常重要；应牢记于心。他们给出了对于
量热完全气体的任意流场，由某一给定点实际的 M,T,p 和 的
值来计算定义的量 和 的公式。正因为其重要性，附录 A
列表给出了 随马赫数变化的函数关系。（对
应 的标准大气条件）
000,?pT
00,0 ?pT
?
?
ppTT 00,
4.1??
?? 000,,ppTT
??0
4.1??
)1(120
)1(20
)
2
1
1(
)
2
1
1(
?
?
?
??
?
??
?
??
?
?
?
?
M
M
p
p
20
2
11 M
T
T ??? ?
)1(1
0
)1(
0
0
)
1
2
(
*
)
1
2
(
*
1
2*
?
?
?
?
?
?
?
?
?
??
??
?
?
?
p
p
T
T
634.0* 528.0* 833.0*
000
??? ??ppTT
? 临界参数的定义与计算公式
临界参数的定义,
Consider a point in a general flow where the velocity is exactly
sonic,i.e,where M=1,Denote the static temperature,pressure,an
density at this sonic condition as T*,p*,and ρ*,respectively,
考虑流场中速度为准确的音速这一点，即 M=1 的点。我们定
义这一点（音速条件）的静温、静压、静密度定义为临界参
数，用 T*,p*和 ρ*表示。
(8.44)
(8.45)
(8.46)
** a
uM ?
? 特征马赫数（速度系数） M*的定义及计算公式
In the theory of supersonic flow,it is sometimes convenient
to introduce a,characteristic” Mach number,M*,defined as,
Where a* is the value of the speed of sound at sonic conditions,not
the actual local value,a* 是音速条件（流动速度 u=a*时 )的音速值。
)1(2
)/*)(1(
2
1
1
)/( 22
?
???
? ?
?
?
uaua
2
1)
*
1(
)1(2
)1(
1
)/1( 22 ?
?
??
? M
M
?
?
?
)1(*/)1(
2
2
2
???? ?? MM
2
2
2
)1(2
)1(*
M
MM
??
??
?
?
下面利用能量方程 (8.35)得到 M与 M*的关系,
)1(2
*)1(
21
222
?
???
? ?
?
?
aua （ 8.35）
（ 8.47）
（ 8.48）
??
?
?
?
??
??
??
MM
MM
MM
MM
if
1
1
*
1 if 1*
1 if 1*
1 if 1*
?
?
There,M* acts qualitatively in the same fashion as M except M*
approaches a finite value when the actual Mach number approaches
infinity,
小结,
In summary,a number of equations have been derived in this
section,all of which stem in one fashion or another from the
basic energy equation for steady,inviscid,adiabatic flow,
8.5 WHEN IS A FLOW COMPRESSIBLE?
什么条件下流动是可压缩的？
We have stated several times in the preceding chapters the rule
of thumb that a flow can be reasonably assumed to be
incompressible when M<0.3,whereas it should be considered
compressible when M>0.3,Why?
)1(120 )
2
11( ???? ??
?
? M
V d Vdp ???
V
dVV
pp
dp 2???
V
dVV
pp
dp 20
0)(
???
00)/(
/
?
??
pdp
pdp
(3.12)
Hence,the degree by which deviates from unity as shown
in Fig.8.5 is related to the same degree by which the fractional
pressure change for a given dV/V,
?
?0
2211 uu ?? ?
22222111 upup ?? ???
22
2
2
2
2
1
1
uhuh ???
22 Tch p?
22 RTp ??
8.6 CALCULATION OF NORMAL SHOCK-WAVE
PROPERTIES
2
22
2
1
11
1 u
u
pu
u
p ???
??
12
22
2
11
1 uu
u
p
u
p ???
??
12
2
2
2
1
2
1 uu
u
a
u
a ???
??
)1(2
*)1(
21
222
?
???
? ?
?
?
aua
2
1
22
1 2
1*
2
1 uaa ???? ??
2
2
22
2 2
1*
2
1 uaa ???? ??
122
2
2
1
1
2
2
1*
2
1
2
1*
2
1 uuu
u
au
u
a ?????????
?
?
?
?
?
?
?
?
1212
2
12
21
)(2 1*)(2 1 uuuuauuuu ??????? ????
12 1*2 1 2
21
???? ???? auu
21
2* uua ?
**1
21
a
u
a
u?
**1 21 MM?
*
1*
1
2 MM ?
2
2
2
)1(2
)1(*
M
MM
??
??
?
? 1
2
1
2
1
2
2
2
2
)1(2
)1(
)1(2
)1(
?
?
?
?
?
?
?
??
??
??
?
M
M
M
M
?
?
?
?
2/)1(
]2/)1[(1
2
1
2
12
2 ??
???
??
?
M
MM
2
12
2
1
21
2
1
2
1
1
2 *
* Ma
u
uu
u
u
u ????
?
?
2
1
2
1
2
1
1
2
)1(2
)1(
M
M
u
u
??
???
?
?
?
?
)1()(
1
22
111211
2
11
2
2212 u
uuuuuuupp ??????? ????
)1()1()1(
1
22
1
1
2
2
1
2
1
1
2
1
2
11
1
12
u
uM
u
u
a
u
u
u
p
u
p
pp ??????? ??
?
??
?
?
?
?
?
?
?
?????
2
1
2
12
1
1
12
)1(
)1(21
M
MM
p
pp
?
??
)1(
1
21 2
1
1
2 ?
?
?? M
p
p
?
?
???
?
???
?
???
?
???
?
?
2
1
1
2
1
2
?
?
p
p
T
T
2
1
2
12
1
1
2
1
2
)1(
)1(2)1(
1
21
M
MM
h
h
T
T
?
??
??
?
??
? ?
?
???
?
?
?
?
3 7 8.02 12lim
1
???
?? ?
?M
M
6
1
1
1
2
1
2
1
2
limlim
lim
11
1
????
?
?
?
?
????
??
T
T
p
p
MM
M ?
?
?
?
1
2
1
2
12 lnln p
pR
T
Tcss
p ???
?
?
?
?
?
?
?
?
??
?
?
?
?
?
?
?
??
?
?
?
?
?
?
?
?
???
)1(
)1(
2
1ln
)1(
)1(2
)1(
1
2
1ln
2
1
2
1
2
12
112
MR
M
M
Mcss
p
?
?
?
?
?
?
22
2
2
2
2
1
1
uTcuTc
pp ???
2
2
0
uTcTc
pp ??
2,01,0 TcTc pp ?
2,01,0 TT ?
a
a
a
a
paa p
pR
T
Tcss
1
2
1
2
12 lnln ???
1,0
2,0
1,0
2,0
12 lnln p
p
R
T
T
css p ???
1,0
2,0
12 ln p
p
Rss ???
Rsse
p
p /)(
1,0
2,0 12 ???
)1/(2
1
1
1,0 )
2
11( ???? ??? M
p
p
??
?
??
? ?
??
? 1)(
1
2 /)1(
1
1,02
1
??
? p
pM
??
?
??
? ?
??
? 1)(
1
2 /)1(
1
1,0
2
12
1
??
? p
pau
1
21
)1(24
)1( 21
)1/(
2
1
2
1
2
1
2,0
?
??
???
?
???
?
??
??
?
?
??
??
?
??
M
M
M
p
p
1
2
2
2,0
1
2,0
p
p
p
p
p
p ?
)1/(2
2
2
2,0 )
2
11( ???? ??? M
p
p
2/)1(
]2/)1[(1
2
1
2
12
2 ??
???
??
?
M
MM
)1(
1
21 2
1
1
2 ?
?
?? M
p
p
?
?
音速
能量方程的特殊形式
什么情况下流动是可压缩的?
用于计算通过正激波气体特性变化的方
程的详细推导 ; 物理特性变化趋势的讨论
用皮托管测量可压缩流的流动速度
图 8.2 第八章路线图
8.4 SPECIAL FORMS OF THE ENERGY EQUATIONS
能量方程的特殊形式
本节需要掌握的内容要点,
? 能量方程的各种特殊表达形式
? 总温的计算公式
? 总压、总密度的计算公式
? 临界参数的定义与计算公式
? 特征马赫数（速度系数） M*的定义及计算公式
22
2
2
2
2
1
1
VhVh ???
22
2
2
2
2
1
1
uhuh ???
? 能量方程的各种特殊表达形式
在 7.5节中我们得到了定常、绝热、无粘流动的能量方程,
其中 V1,V2一条三维流线上的任意两点的速度。 对于
我们现在研究的一维流动，能量方程为,
（ 8.28)
（ 8.29)
However,keep in mind that all the subsequent results in this
section hold in general along a streamline and are by no means
limited to just one –dimensional flows,然而，应当记住的是：
这一节中所有的结论对于一般的沿流线的问题都适用，并不
只是局限于一维流动。
22
2
2
2
2
1
1
uTcuTc
pp ???
2121
2
22
2
11 uRTuRT ?
???? ?
?
?
?
2121
2
2
2
2
2
1
2
1 uaua ?
???? ??
(8.30)
(8.31)
RTa ??
(8.32)
以温度表示,
以音速表示,
121
2
0
22
???? ??
aua
12121
2
0
2
2
2
2
2
1
2
1
??????? ???
auaua
Definition of stagnation speed of sound：驻点音速的定义
(8.33)
(8.34)
)1(2
*)1(
21
222
?
???
? ?
?
?
aua
2
*
1
*
21
2222 aaua
????? ??
(8.35)
Definition of a*,a*的定义
7.5节最后一段,As a corollary to the above considerations,we
need another defined temperature,denoted by T*,and defined as
follows,Consider a point in a subsonic flow where the local static
temperature is T,At this point,imagine that the fluid element is
speeded up to sonic velocity,adiabatically,The Temperature it would
have at such sonic conditions is denoted as T*,Similarly,consider a
point in a supersonic flow,where the local static temperature is T,At
this point,imagine that the fluid element is slowed down to sonic
velocity,adiabatically,Again,the Temperature it would have at such
sonic conditions is denoted as T*,
用 *号表示的变量被称为临界参数, 称为临界音速, ** RTa ??
In Equation (8.35),a and u are the speed of sound and velocity,
respectively,at any point of flow,and a* is a characteristic value
associated with that same point,
c o n s tauaua ????????? )1(2 *)1(2121
22
2
2
2
2
1
2
1
?
?
??
对于沿一条流线上的任意两点,有,
(8.36)
c o n s taa ????? 1*)1(2 1
2
02
??
? (8.37)
Clearly,these defined quantities,a0 and a*,are both constants
along a given in a steady,adiabatic,inviscid flow,If all the
streamlines emanate from the same uniform freestream
conditions,then a0 and a* are constants throughout the entire
flow field,很明显,a0 和 a*为定义的量,沿定常、绝热、无粘
流动的给定流线为常数。如果所有流线都来自于均匀自由来
流，则 a0 和 a*在整个流场为常数。
0
2
2 Tc
uTc
pp ??
c o n s tTcuTcuTc ppp ????? 0
2
2
2
2
1
1 22
(8.38)
? 总温的计算公式
回忆 7.5节中总温 T0的定义，有方程 (8.30)可得,
(8.39)
Equation (8.38) provides a formula from which the defined total
temperature T0 can be calculated from the given actual conditions of
T and u at any given points in a general flow field,方程 (8.38) 给出
了由流场中给定点处的实际温度 T和速度 u计算总温 T0的计算公
式。
2
22
0 )(
2
11
)1/(2121 a
u
RT
u
Tc
u
T
T
p
???
?????
?
??
20
2
11 M
T
T ??? ? (8.40)
Equation (8.40) is very important; it states that only M (and,of
course,the value of ) dictates the ratio of total temperature to
static temperature,
方程 (8.40)非常重要； 表明只有马赫数（及 的值）决定总
温与静温的比。
?
For a calorically perfect gas,the ratio of total temperature to
static temperature,is a function of Mach number only,as
follows,（对于量热完全气体，总温和静温的比 是马赫数
的唯一函数，证明如下：）
TT0
TT0
?
)1(120
)1(20
)
2
1
1(
)
2
1
1(
?
?
?
??
?
??
?
??
?
?
?
?
M
M
p
p
)1(
000
?
?
?
??
?
??
???
?
???
?? ???
?
?
T
T
p
p
? 总压、总密度的计算公式,
回忆 7.5节总压和总密度的定义,在定义中包含了将气流速度
等熵地压缩为零速度。由（ 7.32）式,
我们有,
)1(
1
2
1
2
1
2
?
???
?
???
??
???
?
???
?? ???
?
?
T
T
p
p
（ 8.41)
（ 8.42)
（ 8.43)
方程（ 8.42)和（ 8.43）表明：总压静压比,总密度静密度
比 只由 M 和 决定。因此，对于给定气体，即给
定,, 只依赖于马赫数。
?
?
pp0
??0
pp0 ??0
Equation (8.40),(8.42)and (8.43) are very important; they should be
branded on your mind,They provided formulas from which the
defined,and can be calculated from the actual conditions of
M,T,p and at a given point in general flow field (assuming
calorically perfect gas),They are so important that values of
and obtained from Eqs,(8.40),(8.42),and (8.43),respectively,
are tabulated as functions of M in App.A for (which
corresponds to air at standard conditions),
方程 (8.40),(8.42)和 (8.43) 非常重要；应牢记于心。他们给出了对于
量热完全气体的任意流场，由某一给定点实际的 M,T,p 和 的
值来计算定义的量 和 的公式。正因为其重要性，附录 A
列表给出了 随马赫数变化的函数关系。（对
应 的标准大气条件）
000,?pT
00,0 ?pT
?
?
ppTT 00,
4.1??
?? 000,,ppTT
??0
4.1??
)1(120
)1(20
)
2
1
1(
)
2
1
1(
?
?
?
??
?
??
?
??
?
?
?
?
M
M
p
p
20
2
11 M
T
T ??? ?
)1(1
0
)1(
0
0
)
1
2
(
*
)
1
2
(
*
1
2*
?
?
?
?
?
?
?
?
?
??
??
?
?
?
p
p
T
T
634.0* 528.0* 833.0*
000
??? ??ppTT
? 临界参数的定义与计算公式
临界参数的定义,
Consider a point in a general flow where the velocity is exactly
sonic,i.e,where M=1,Denote the static temperature,pressure,an
density at this sonic condition as T*,p*,and ρ*,respectively,
考虑流场中速度为准确的音速这一点，即 M=1 的点。我们定
义这一点（音速条件）的静温、静压、静密度定义为临界参
数，用 T*,p*和 ρ*表示。
(8.44)
(8.45)
(8.46)
** a
uM ?
? 特征马赫数（速度系数） M*的定义及计算公式
In the theory of supersonic flow,it is sometimes convenient
to introduce a,characteristic” Mach number,M*,defined as,
Where a* is the value of the speed of sound at sonic conditions,not
the actual local value,a* 是音速条件（流动速度 u=a*时 )的音速值。
)1(2
)/*)(1(
2
1
1
)/( 22
?
???
? ?
?
?
uaua
2
1)
*
1(
)1(2
)1(
1
)/1( 22 ?
?
??
? M
M
?
?
?
)1(*/)1(
2
2
2
???? ?? MM
2
2
2
)1(2
)1(*
M
MM
??
??
?
?
下面利用能量方程 (8.35)得到 M与 M*的关系,
)1(2
*)1(
21
222
?
???
? ?
?
?
aua （ 8.35）
（ 8.47）
（ 8.48）
??
?
?
?
??
??
??
MM
MM
MM
MM
if
1
1
*
1 if 1*
1 if 1*
1 if 1*
?
?
There,M* acts qualitatively in the same fashion as M except M*
approaches a finite value when the actual Mach number approaches
infinity,
小结,
In summary,a number of equations have been derived in this
section,all of which stem in one fashion or another from the
basic energy equation for steady,inviscid,adiabatic flow,
8.5 WHEN IS A FLOW COMPRESSIBLE?
什么条件下流动是可压缩的？
We have stated several times in the preceding chapters the rule
of thumb that a flow can be reasonably assumed to be
incompressible when M<0.3,whereas it should be considered
compressible when M>0.3,Why?
)1(120 )
2
11( ???? ??
?
? M
V d Vdp ???
V
dVV
pp
dp 2???
V
dVV
pp
dp 20
0)(
???
00)/(
/
?
??
pdp
pdp
(3.12)
Hence,the degree by which deviates from unity as shown
in Fig.8.5 is related to the same degree by which the fractional
pressure change for a given dV/V,
?
?0
2211 uu ?? ?
22222111 upup ?? ???
22
2
2
2
2
1
1
uhuh ???
22 Tch p?
22 RTp ??
8.6 CALCULATION OF NORMAL SHOCK-WAVE
PROPERTIES
2
22
2
1
11
1 u
u
pu
u
p ???
??
12
22
2
11
1 uu
u
p
u
p ???
??
12
2
2
2
1
2
1 uu
u
a
u
a ???
??
)1(2
*)1(
21
222
?
???
? ?
?
?
aua
2
1
22
1 2
1*
2
1 uaa ???? ??
2
2
22
2 2
1*
2
1 uaa ???? ??
122
2
2
1
1
2
2
1*
2
1
2
1*
2
1 uuu
u
au
u
a ?????????
?
?
?
?
?
?
?
?
1212
2
12
21
)(2 1*)(2 1 uuuuauuuu ??????? ????
12 1*2 1 2
21
???? ???? auu
21
2* uua ?
**1
21
a
u
a
u?
**1 21 MM?
*
1*
1
2 MM ?
2
2
2
)1(2
)1(*
M
MM
??
??
?
? 1
2
1
2
1
2
2
2
2
)1(2
)1(
)1(2
)1(
?
?
?
?
?
?
?
??
??
??
?
M
M
M
M
?
?
?
?
2/)1(
]2/)1[(1
2
1
2
12
2 ??
???
??
?
M
MM
2
12
2
1
21
2
1
2
1
1
2 *
* Ma
u
uu
u
u
u ????
?
?
2
1
2
1
2
1
1
2
)1(2
)1(
M
M
u
u
??
???
?
?
?
?
)1()(
1
22
111211
2
11
2
2212 u
uuuuuuupp ??????? ????
)1()1()1(
1
22
1
1
2
2
1
2
1
1
2
1
2
11
1
12
u
uM
u
u
a
u
u
u
p
u
p
pp ??????? ??
?
??
?
?
?
?
?
?
?
?????
2
1
2
12
1
1
12
)1(
)1(21
M
MM
p
pp
?
??
)1(
1
21 2
1
1
2 ?
?
?? M
p
p
?
?
???
?
???
?
???
?
???
?
?
2
1
1
2
1
2
?
?
p
p
T
T
2
1
2
12
1
1
2
1
2
)1(
)1(2)1(
1
21
M
MM
h
h
T
T
?
??
??
?
??
? ?
?
???
?
?
?
?
3 7 8.02 12lim
1
???
?? ?
?M
M
6
1
1
1
2
1
2
1
2
limlim
lim
11
1
????
?
?
?
?
????
??
T
T
p
p
MM
M ?
?
?
?
1
2
1
2
12 lnln p
pR
T
Tcss
p ???
?
?
?
?
?
?
?
?
??
?
?
?
?
?
?
?
??
?
?
?
?
?
?
?
?
???
)1(
)1(
2
1ln
)1(
)1(2
)1(
1
2
1ln
2
1
2
1
2
12
112
MR
M
M
Mcss
p
?
?
?
?
?
?
22
2
2
2
2
1
1
uTcuTc
pp ???
2
2
0
uTcTc
pp ??
2,01,0 TcTc pp ?
2,01,0 TT ?
a
a
a
a
paa p
pR
T
Tcss
1
2
1
2
12 lnln ???
1,0
2,0
1,0
2,0
12 lnln p
p
R
T
T
css p ???
1,0
2,0
12 ln p
p
Rss ???
Rsse
p
p /)(
1,0
2,0 12 ???
)1/(2
1
1
1,0 )
2
11( ???? ??? M
p
p
??
?
??
? ?
??
? 1)(
1
2 /)1(
1
1,02
1
??
? p
pM
??
?
??
? ?
??
? 1)(
1
2 /)1(
1
1,0
2
12
1
??
? p
pau
1
21
)1(24
)1( 21
)1/(
2
1
2
1
2
1
2,0
?
??
???
?
???
?
??
??
?
?
??
??
?
??
M
M
M
p
p
1
2
2
2,0
1
2,0
p
p
p
p
p
p ?
)1/(2
2
2
2,0 )
2
11( ???? ??? M
p
p
2/)1(
]2/)1[(1
2
1
2
12
2 ??
???
??
?
M
MM
)1(
1
21 2
1
1
2 ?
?
?? M
p
p
?
?
