Structural Acoustics

Linear Systems Equations 179

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where A is some decay constant giving the amount that the wave has

decreased in amplitude from point 1 to point 2, and τ is the time of travel

from 1 to 2. Forming the cross correlation C(x

1

, x

2

, τ) gives

C x x

T

x t Ax t dt

AC x

T

T

( , , ) ( ) ( )

(

1 2 1 1 1

1

2

τ τ τ= + −

=

−

+

∫

11 2 1

, , )x τ τ−

(9.40)

Thus, the cross correlation of a random wave train in a nondispersive sys-

tem is exactly the same form as the autocorrelation of the wave at the start-

ing point, except that the peak occurs at a time delay corresponding to the

time necessary for the wave to travel between the points.