Impact of fuel supply impedance and fuel staging on gas turbine ...

Acoustics
In case <strong>of</strong> time-harmonic plane waves, the acoustic pressure <strong>and</strong> velocity can
be written in complex form as a function <strong>of</strong> the angular frequency <strong>of</strong> oscillation
(ω=2π f ):
p ′ (x, t ), u ′ (x, t )∼exp(i ω t− i kx), (3.25)
where k = ω/c denotes the wave number with direction normal to the wave
front <strong>of</strong> a plane wave. Using the latter relation Eqn. (3.21) <strong>and</strong> (3.22) can be
rewritten in the following form:
p ′ (x, t )
= exp(i ω t ) ( f exp(−i k x)+ g exp(+i k x) ) , (3.26)
ρ 0 c
u ′ (x, t )=exp(i ω t ) ( f exp(−i k x)− g exp(+i k x) ) . (3.27)
3.2.2 Sound propagation in a moving medium
In the presence <strong>of</strong> an uniform mean flow the convective wave equation can
be obtained by performing a transformation <strong>of</strong> coordinates from the system
moving with the fluid to the laboratory system
x ′ = x+ u 0 t , t ′ = t , u 0 = const., (3.28)
<strong>and</strong> yields
( ) ∂
∂t + u ∂ 2
0 − c 2∂2 p ′
= 0, (3.29)
∂x ∂x2 where ∂ ∂t +u 0 ∂ denotes a time derivative moving with the mean flow. The wave
∂x
number vector k is the same in the laboratory system as it is in the co-moving
system. Therefore, the wave number considering mean flow can be written as:
k ± =± ω/c
1± M . (3.30)
Similar to Eqn. (3.26) <strong>and</strong> (3.27) the following solution for the convective wave
equation can be found:
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