Apply a unit step force 1(t) to the mass m inthe +ve x direction:
The system was initially in rest at unloaded equilibrium:
; 
.
The motion is described by 
General solution: 
where 
Applying the initial conditions, we get: 
; 
.
Thus the response of the linear system to a unit step function is:
. This is called "indicial admittance". The particular functional form
of A depends on the linear system being considered. The response to an
arbitrary force f(t) can be built up from this.
Response at time to a step load Df
applied at time t+Dt is:
Summing up,
As 
, 
This is the Duhamel integral, in terms of the indicial admittance and the derivatives of the forcing function. The forcing function may not be available in analytic form, so we need an alternate form:
Integrate by parts, using:
where 
, so that 
, so that 
Note that:
Substituting,
Thus if the indicial admittance function is known, the
system response can be determined from this.
. Take the Fourier transform of w:
. Circulatory lift coefficient is: 
(taking the inverse Fourier transform). Define 
as the number of semichords traveled in time t. Then
. This can be writen as 
where f(s) is the Wagner function.
Also, 
.
Note: The Wagner function is the indicial admittance for the normalized circulatory lift associated with a step change in normalwash at the 3/4-chord.
This is approximated by the following expressions:
.
These expressions are compared in the linked
table and figure.
