In the same way, if we use X?(t) instead of the output process Y(t) and calculate the cross 
bispectrum Sy2xy(W1,@2) , 
Ryoygdt1. 2) = El[X%0) — my 2} -X(e—1y)(¢-72)] 
= E[X()X(t) X@—1) X¢—1)] — my EXC) XC-79)] 
= Ryx(0) Rxx(t2—71) + 2Rxx(T1) Ryx(t2) — my 2Ryx(T2 — 71) 
= 2 Rxx(t1) Rxx(t2), (11.56) 
2, 
(27)? 
thus Sx 2xx (M1, @2) = | | Ryxx(71)Rxx(T2) exp[- i{w yt, + @2T>}] dt \dt7 
= 2 sxx(@1)Sxx(w2). (11.57) 
Inserting into Eq. 11.49 gives 
SYXX(W 1, @ 
H2(@1,@2) = ue) (11.58) 
Sx2xx(M 1,02) 
and inverting the variables as in Eq. 11.55 gives 
SYXX(W1 — @2,@1 + 2) 
le KONO) Fre (11.59) 
Sy2xx(W1—W2,W1 + W2) 
Both Eqs. 11.49 or 11.55 and Eq. 11.58 or 11.59 can be used for the computation of 
H>(@,@ 2). However in actual computation, due to the consideration of the window, 
Eq. 11.58 or 11.59 is much better than Eq. 11.49 or Eq. 11.55. 
5. Application of bispectrum analysis. 
J. F. Dalzell9> 56 57, n4 58 applied these theoretical relations (as compiled above in 
items 1 through 4) to the problem of added resistance of ships in waves. The quadratic 
process Y(r) is the resistance D(t) of ships in waves and the input process x(t) is the wave 
height 7(t) in Eq. 11.1’ (when n =0 — 2) and ho = Do, as 
D(t) = Do+ | rue W(t —T)at + | | totes.22) n(t—T1) H(t—T2) at\dt2. 
Then Egs. 11.33 and 11.33’ are the expression of mean added resistance in waves 
foe) 
E(D()] = | Hx@,-)s, ,(@) do, (11.60) 
-o 
and the linear and nonlinear response functions of resistance to waves are Eq. 11.36, 
308 
