#
KUSSNER FUNCTION

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Solution for Prescribed Freestream Fluctuations

__Airfoil encountering a transverse gust__

Assume that the gust velocity is a unit step function
of magnitude w0 which is located at the leading edge at t=0, and is convected
at the freestream speed.

where

is the unit step function.

**Gust velocity at midchord becomes:**

. Its Fourier transform is:

To obtain this, set
; apply the Fourier transform to the normalwash at midchord:
.

To use this, we must first develop an expression for cl
for a case of simple harmonic gust velocity variation, then substitute
the indicial gust expression n the frequency domain. As before, we seek
a generalized expression where the complicated parts of the integral are
computed as a general function.

**Simple harmonic transverse gust:**

In flow-fixed coordinates,
, a simple harmonic spatial function. In terms of
, non-dimensional body-fixed coordinates,
; periodic in space and time.

Substitute in our expression for Cp in the frequency domain.
Now

and
. Substituting for Cp, we get (skipping some math here),
, and
.

Define
. Thus .
Transform cl(k,t) into the frequency domain:
and

and
. Substituting for
, with
(semi-chords traveled in time t),

; .

is the Kussner Function.

**Approximate Expressions:**

, or

, both appropriate near s=0, and

, which is more appropriate away from s=0.