APPENDIX – DERIVATIONS OF THE TFR RESULTS
Branched pipeline system
The transfer matrix of the branched junction has a following form (Chaudhry 1987),
1 Z 
P
,
0 1 
(A1)
in which Z represents the discharge impendence factor due to the branched pipeline. For
example, when the boundary of the pipe branch is constant-head reservoir as shown in Figure
1(b), it has:
Z
U 2b2r
,
U 2b1r
(A2)
in which U ijb r is the element of the lumped transfer matrix for the branched pipeline (which
can be single or series pipeline). Therefore, the TFR of the intact (leak-free) branched pipe
system in Figure 1(b) can be obtained by transfer matrix multiplication principle as (Duan &
Lee 2016):
b
a
1
1




q   cos2l2  i sin 2l2  1 Z c 3   cos1l1  i sin 1l1  q 
Y2
Y1
  
  ,
 0 1  
 h 
h iY sin  l 
cos2l2  
2 2
 2
iY1 sin 1l1  cos1l1  
(A3)
or:
b
d
1
1




q   cos2l2  i sin 2l2  1 Z c1   cos3l3  i sin 3l3  q 
Y3
Y2
  
  ,
 h
 0 1  
h iY sin  l 
cos2l2  
cos3l3  
2 2
 2
iY3 sin 3l3 
(A4)
along the two possible pipe flow directions of a-c-b and d-c-b respectively. That is, in these
two flow directions, pipe 3 and pipe 1 are treated as branched sections respectively (Duan &
Lee 2016). As a result, the transient pressure heads at downstream end b for the above two
cases can be obtained respectively as:
hb  
hb  
iY2 sin  2l2 cos1l1   Z C 3Y2Y1 sin  2l2 sin 1l1   iY1 cos 2l2 sin 1l1 
,
Y1
cos 2l2 cos1l1   sin  2l2 sin 1l1   iY1Z C 3 cos 2l2 sin 1l1 
Y2
(A5)
iY2 sin  2l2 cos3l3   Z C1Y2Y3 sin  2l2 sin 3l3   iY3 cos 2l2 sin 3l3 
,
Y3
cos 2l2  cos3l3   sin  2l2 sin 3l3   iY3 Z C1 cos 2l2 sin 3l3 
Y2
(A6)
Therefore, the singularity (resonance) conditions are given by:
cos 2 l2  cos1l1  
Y1
sin  2 l2  sin 1l1   iY1 Z C 3 cos 2 l2  sin 1l1   0 ,
Y2
(A7)
cos 2 l2  cos 3l3  
Y3
sin  2 l2  sin  3l3   iY3 Z C 1 cos 2 l2  sin  3l3   0 .
Y2
(A8)
According to Duan & Lee (2016):
Z c3 
cos3l3 
cos1l1 
; Z c1 
,
iY3 sin 3l3 
iY1 sin 1l1 
(A9)
which gives the identical result for both cases as:
 Y3Y2 sin  3l3 cos 2 l2 cos1l1  
  Y Y sin  l sin  l sin  l    0
.
3 3
2 2
1 1
 3 1

 Y2Y1 cos 3l3 cos 2 l2 sin 1l1 
(A10)
That is, the resonant condition for this branched system is unique, no matter which main flow
path is considered for the analysis.
Under pipe leakage situation, three possible cases where the single leak is located at each
of the three pipe sections respectively exist in this system, which are considered individually
as below:
(1) Leak is located on pipe 1, gives:
b
1


q   cos 2l2  i sin  2l2  1 Z C 3 
Y2
  
 0 1 
h  iY sin  l 
cos 2l2   
2 2
 2
a
1
1




 cos1 x2  i Y sin 1 x2  1  K   cos1 x1  i Y sin 1 x1  q 
1
1

 0 1  
 h 
iY1 sin 1 x2  cos1 x2  
iY1 sin 1 x1  cos1 x1  
,
(A11)
(2) Leak is located on pipe 3, gives:
b
1


q   cos 2l2  i sin  2l2  1 Z C1 
Y2
  
 0 1 
h  iY sin  l 


cos

l
2 2
2 2
 2

, (A12)
1
1

 1 K 
 q a
  cos3 y1  i sin 3 y1   
 cos3 y2  i Y sin 3 y2  
Y3
 

3

 0 1  
 h 








iY
sin

y
cos

y
iY
sin

y
cos

y
3 2
3 2
3 1
3 1
 3

 3

(3) Leak is located on pipe 2, gives:
b
1
1




q   cos 2 z2  i sin  2 z2  1  K   cos 2 z1  i sin  2 z1 

Y2
Y2
 
 0 1  

h  iY sin  z 




cos

z
iY
sin

z
cos 2 z1  
2 2
2 2
2 1
 2

 2
,
a
1


1 Z C 3   cos1l1  i sin 1l1  q 
Y1
 
0 1  

 iY sin  l  cos l   h 
11
11
 1

(A13)
in which x1+x2=l1; y1+y2=l3; z1+z2=l2; KL = the impendence factor of the leak orifice and KL =
QL0/2HL0 with QL0 and HL0 are steady state discharge and head difference at the leak orifice in
the given system (Duan et al. 2011). Correspondingly, after the mathematical manipulations
and essential simplifications (Lee et al. 2006; Duan et al. 2011; Duan & Lee 2016), the TFR
results at the downstream of the system (e.g. node b) are derived as:


 ,


 ,
(A15)


 ,
(A16)
1
K
 BL 1  cos 21 x1  1B
b
hL1 C1
1
K
 BL 1  cos 23 y1  2B
b
hL 2 C2
1
K
 BL 1  cos 2 2 z2  3B
b
hL 3 C3
(A14)
where C1B, C2B, C3B and 1B, 2B, 3B are intact system based known coefficients for specific
frequencies, and, for the considered branched system in Figure 1(b), 1B = 2B = 0; 3B = /2:
C1B 
2 sin 1l1  
Y2
Y2 cos3l3 sin 2l2 sin 1l1 
cos2l2 sin 1l1   sin 2l2 cos1l1  
;
cos2l2  
Y1
Y3
sin 3l3 

(A17)
C2B 
2 sin 3l3  
Y2
Y2 sin 3l3 sin 2l2 cos1l1 
cos2l2 sin 3l3   sin 2l2 cos3l3  
;
cos2l2  
Y3
Y1
sin 1l1 

(A18)
C3B 
2 cos2l2   Y2
Y2 cos3l3 sin 1l1 sin 2l2 
 sin 2l2 cos1l1   cos2l2 sin 1l1  
.
sin 1l1   Y1
Y3
sin 3l3 

(A19)
Consequently, a general form of the TFR results can be summarized for the branched pipeline
system in Figure 1(b) as expressed in Equation (7) in the text.
Looped pipeline system
For the simple looped system in Figure 1(c), there are two branch junctions in the system, but
without any explicit boundary for describing the branches (e.g. constant-head reservoir or
dead-end as for the branched system in Figure 1(b)). In this study, the loop is considered as a
“loop point” consisting of two parallel pipes, so that its equivalent transfer matrix can be
derived and applied appropriately. Based on the results of single or series pipe section in
Equation (3), the transfer matrix equations for the branched pipes 3 & 4, respectively, are:
R
L
R
L
 v11 v12  qˆ  qˆ 
u11 u12  qˆ 
qˆ 
 ˆ  
 ˆ ;  ˆ  

  ˆ ,
h 3 v21 v22  h 3 h 4 u21 u22  h 4
(B1)
where the transfer matrix elements v, u are known based on the intact system configurations
as shown in Equation (3). Meanwhile, considering the continuity and energy conservation
relationships at the both branched junctions c and d, it has (ignoring minor loss here):
q1R  qˆ3L  qˆ 4L q2L  qˆ3R  qˆ 4R
;
.
h1R  hˆ3L  hˆ4L h2L  hˆ3R  hˆ4R
(B2)
Combining Equations (B1) and (B2) provides:
q2L  q1R  h1R
h2L  q1R  h1R
,
(B3)
where are known coefficients for the given intact system, and:

v  u v  u   u11  v11 u22  v22  ;
v11u21  v21u11
;   12 12 21 21
v21  u21
v21  u21

u 21v21
u v u v
;   22 21 21 22 .
v21  u 21
v21  u21
That is, for the “loop point”, it has the form:
q 


 
h  Loop 
RHS
  q 
  ,
  h  Loop
LHS
(B4)
where superscripts LHS and RHS denote the quantities for the left hand side and right hand
side of the “loop point”. As a result, the transfer matrix equation for the intact looped pipe
system of Figure 1(c) can be described by:
b
1


q   cos2l2  i sin 2l2  
Y2
  
 
h  iY sin  l 
cos 2l2   
2 2
 2
U12  q 
U
  11
 
U 21 U 22  h 
   cos1l1  i sin 1l1  q 
Y1
 
 h 
  iY sin  l  cos
 l 
1

1
11
11

a
,
(B5)
a
where U is the lumped transfer matrix element. With applying similar operations in the
previous studies for series and branched pipeline systems (Duan et al. 2011; Duan & Lee
2016), the resonant conditions for this looped system is governed by:

 

Y1

sin  2 l2 sin 1l1   i Y1 cos 2 l2 sin 1l1   sin  2 l2 cos1l1   0
 cos 2 l2 cos1l1  
Y2
Y2

 

.
(B6)
Under pipe leakage condition for the considered system in Figure 1(c), four possible
leakage cases exist: the leakage is located on pipes 1, 2, 3, 4 respectively. For illustration, the
derivations of the cases for the leakage on pipes 1 and 3 are provided below, and the other
two cases can be achieved with similar procedures.
(1) Leakage is on pipe 1 (or on pipe 2 using similar analysis):
b
1


q   cos 2l2  i sin  2l2    
Y2
  
   
h  iY sin  l 


cos

l
2 2
2 2
 2

, (B7)
1
1

 1 K 
 q a
  cos1 x1  i sin 1 x1   
 cos1 x2  i Y sin 1 x2  
Y1
 

1

 0 1  
 h 
iY1 sin 1 x2  cos1 x2  
iY1 sin 1 x1  cos1 x1  
where x1+ x2=l1;
(2) Leak is within the loop c-d and on pipe 3 (or on pipe 4 using similar analysis):
b
1


q   cos2l2  i sin 2l2   L

Y2
 
  L
h iY sin  l 


cos

l
2 2
2 2
 2



 L   cos1l1  i sin 1l1  q
Y1
  ,

 h
 L  iY sin  l  cos
 l 

1
1
11
11
a

(B8)
with:
RHS
qˆ 
 ˆ
h3
LHS
1
1




cos3 z2  i sin 3 z2  1  K   cos3 z1  i sin 3 z1  qˆ 


Y3
Y3
  ,

 0 1  
 hˆ 3
cos3 z2  
cos3 z1  
iY3 sin 3 z2 
iY3 sin 3 z1 
(B9)
and
RHS
 L
q 

 
 L
h  LoopL 
LHS
 L  q 
,
 
 L  h  LoopL
(B10)
where z1+z2=l3; subscript Loop-L represents the quantities for the “loop point” under
leakage condition.
After mathematical manipulations and re-arrangements, the final TFR results for these
two cases at the downstream are obtained as:
1
1
 O
b
hL1 C1
RO 
 1
S   T 
O 2
1
O 2
1


sin 1l1  21 x2  1O  ,

(B11)
1
1
 O
b
hL 3 C3
RO 
 3
S   T 
O 2
3
O 2
3


sin 3l3  23 z2  3O  ,

(B12)
where: C1O, C3O, R1O, R3O, S1O, S3O, T1O, T3O, 1O, 3O are intact pipeline system based known
coefficients, and:
C1O  
 SO 
 SO 
2 F1
2F
; C3O   3 ; 1O  arctan  1O  ; 3O  arctan  3O  ;
Y1
Y3
 T1 
 T3 
Y sin 2l2 cos1l1   Y1 cos2l2 sin 1l1  
F1  F3   2
;









i

cos

l
cos

l

i

Y
Y
sin

l
sin

l
2 2
11
2 1
2 2
11 

Y1


  cos 2l2 sin 1l1    Y sin  2l2 cos1l1  
2
;
R1O  
 

 i Y sin  2l2 sin 1l1   iY1 cos 2l2 cos1l1 
2


Y3Y1


 Y3 cos 4l4  cos3l3  cos 2l2  cos1l1   Y sin  4l4  cos3l3  cos 2l2 sin 1l1  
4


Y3Y1
 Y4Y3

 Y sin  4l4  cos3l3 sin  2l2  cos1l1   Y cos 4l4  cos3l3 sin  2l2 sin 1l1 
2
2
;
R3O  
Y4Y1


 Y4 sin  4l4 sin 3l3  cos 2l2  cos1l1   Y sin  4l4 sin 3l3 sin  2l2 sin 1l1  
2


 Y4Y1

 Y sin  4l4  cos3l3  cos 2l2 sin 1l1   2Y1 cos 4l4 sin 3l3  cos 2l2 sin 1l1  
3


 Y

S1O   1 sin 2l2   iY1 cos2l2  ;
 Y2



Y3Y1
sin  4l4 cos 2l2 sin 1l1  
Y3 cos 4l4 cos 2l2 cos1l1  
Y4




YY
YY
S3O   4 3 sin  4l4 sin  2l2 cos1l1   4 1 sin  4l4 cos 2l2 sin 1l1  ;
Y3
 Y2

 Y3Y1

cos 4l4 sin  2l2 sin 1l1 


 Y2




T1O   cos2l2   i sin 2l2  ;
Y2


Y Y

T3O   4 1 sin 4l4 sin 2l2 sin 1l1   Y4 sin 4l4 cos2l2 cos1l1  .
 Y2

As a result, the TRF results of the looped pipeline system can be expressed with a general
form of Equation (8) in the text.