German Aerospace Center (DLR)
Institute of Aerodynamics and Flow Technology
Lilienthalplatz 7, 38108 Braunschweig
TU Braunschweig, WS 2014/2015
Lecture:
Seminar:
Prof. Dr.Ing. Jan Delfs
Dipl.Ing. Nils Reiche
The root mean square (RMS) of sound pressure is doubled. How does the sound pressure level (SPL) change?
The absolute value of sound intensity is doubled.
a) How does the sound intensity level change?
b) Assuming a simple plane sound wave, the absolute value of sound intensity
I like I p
2
/Z
0
. (Constant Z
0
=
%
0 a
0 is the socalled free field impedance, refer to the lecture notes.) How and the sound pressure level L p change when doubling sound intensity I ?
Calculate the sound pressure root mean square of the pure tone p
0
( t ) = A sin( ωt ) !
1
Following pressure signal is given: p ( t ) = C + A
1 sin( ω
1 t ) + A
2 sin( ω
2 t + ϕ ) .
a) What is this signal’s sound pressure RMS?
b) What is this signal’s sound pressure level?
c) Assume that following conditions hold:
Calculate the difference ∆ L p
ω
1
= ω
2
= ω , A
1
= A
2
= A to the sound pressure level of a pure tone
.
with the same amplitude A (refer to exercise 3)! Sketch ∆ L p as a function of phase angle ϕ !
What is the perceived difference in loudness between a pure tone of frequency f
1
= 3000 Hz and a pure tone of frequency the same sound pressure level of L p 1
= L p 2 f
2
= 300 Hz, when both tones have
= 100 dB?
What is the sound pressure level L p 1 of a pure tone of frequency that is perceived as loud as a pure tone at f
2 f
1
= 100 Hz
= 1000 Hz having a sound pressure level of L p 2
= 70 dB?
Compute the average of the three sound pressure levels L p 1 and L p 3
= 90 dB
= 80 dB, L p 2
= 85 dB, a) approximately as arithmetic mean!
b) exactly by averaging the sound pressure mean square values (assuming uncorrelated signals)!
Following octave band levels were measured: f
L i c i in Hz: p 1 / 1 in dB:
125
68.0
250
70.5
500
72.5
What is the overall sound pressure level ?
1000
77.0
2000
75.5
4000
71.5
8000
68.5
.
2
A signal has a constant power spectral density ˆ ( f ) = P
0
(ideal white noise).
a) Compute this signal’s third octave and octave band levels L i pm
!
b) How do these band levels change from band i to band i + 1?
c) Sketch the octave band spectrum L i p 1 / 1
!
d) Why is this signal called “white noise”?
e) Does ideal white noise exist in reality? What is its mean square value?
Ideal pink noise is a stochastic signal with constant mband levels L i pm
.
a) Compute (for m = 1) the ratio
ˆ
( f )! Hints:
ˆ
( f u i
)
ˆ
( f i l
) and sketch the power spectral density
–
–
L i pm
2 i
= const.
=
R b a
⇒
2 i
= const.
⇒ d ˜ df
2 i c i
= 0,
ˆ
( f ) df , where a := f i l
= 2
− m/ 2 f i c and b := f i u
= 2 m/ 2 f i c
.
b) Why is this signal called “pink noise”?
c) Can ideal pink noise exist in reality? What is its mean square value?
The governing equations of fluid dynamics
1 are given by the continuity equation
(31), the momentum equations (32) and the energy equation (33). Equations (31)
(33) are in conservation form. However, the primitive form is more appropriate as a starting point for acoustics.
a) Derive the continuity equation (35) in primitive form!
b) Derive the momentum equations (36) in primitive form (lecture notes may be helpful)!
1 same numbering as in lecture, hopefully
3
Under standard conditions air is characterized by: p
%
T
0
0
0
=
=
=
1 .
01325
·
10
5
1 .
292 kg / m
3
273.15 K.
Pa
What is the speed of sound a) for these standard conditions?
b) when the temperature increases to 25
◦
C or to 350
◦
C?
c) for the same temperature, but twice the ambient pressure?
Derive the momentum perturbation equations (55) from the momentum equations
(48) by linearization around a time independent base flow! Use the standard perturbation approach from the lecture notes:
1. Decompose the flow variables using parameter ε (0
≤
ε 1), e.g.
p = p
0
+ εp
0
;
2. Insert the decomposed variables into (48). This gives a function obeys
~
( ε ) = 0;
~
( ε ) that
3. Develop
~
( ε ) into a Taylor series around
~
( ε = 0);
4. Linearize by assuming ε 1 (small perturbations) in the Taylor series.
Derive the momentum perturbation equations (55) from the momentum equations (48) by linearization around a time independent base flow! Do not use the standard perturbation approach as in exercise 13, but use the following technique instead:
1. Decompose the flow variables without using a parameter where b
∈ {
%, v , p, f
}
;
ε , i.e., b = b
0
+ b
0
,
2. Linearize by assuming
 b
0
  b
0

(neglect “prime products”).
4
The following equation describes a sound wave travelling through air: p
0
( x, t ) = 6 .
25 exp[ i (471 .
2 t
−
1 .
37 x )] .
Following holds: p
0 in Pa, t in s, x in m, %
0
= 1 .
292 kg / m
3
. Calculate a) the frequency f , b) the wave length λ , c) the speed of sound a
0
, d) the ambient temperature T
0
, of sound pressure, of sound velocity, g) the sound intensity I , h) the sound pressure level L p
, i) the physical solution p
0
( x, t ) phys
!
Derive the Helmholtz equation (69) from the wave equation (61) for the sound pressure perturbations! Hints: Replace p
0
( x , t ) and Q p
( x , t ) in equation (61) according to equation (23) for the inverse Fourier transform, i.e.: p
0
( x , t ) =
Q p
( x , t ) =
1
2 π
1
2 π
Z
+
∞
Z
−∞
+
∞
Q p
( x , ω ) exp( iωt ) dω
−∞
ˆ ( x , ω ) exp( iωt ) dω
5
Sir Michael James Lighthill published the following famous inhomogeneous wave equation (251) for density perturbations in 1952:
∂
2
%
0
∂t
2
− a
2
∞
∆ %
0
=
∇ · ∇ ·
( % vv + ( p
0
− a
2
∞
%
0
) I
−
τ ) .
He derived it through simple rearrangement of the NavierStokes equations. Derive the inhomogeneous wave equation (253) for pressure perturbations, being
1 a
2
∞
∂
2 p
0
∂t
2
−
∆ p
0
=
∇ · ∇ ·
( % vv
−
τ ) +
1 a
2
∞
∂
2
∂t
2
( p
0
− a
2
∞
%
0
)!
Hint: As the only difference to the derivation of equation (251), another term than a
2
∞
∆ % has to be subtracted.
The noise of an engine’s jet is supposed to be reduced by enlarging its diameter
D . The original jet speed is U s
, and its thrust is supposed to remain constant.
How does the sound intensity level L
I decrease, when the jet’s diameter increases from D
1 to D
2
= 1 .
5 D
1
? Hints:
•
Use the famous estimate for the sound intensity of a jet, which was derived from Lighthill’s acoustic analogy: I
∝

1 x

2
%
∞ a
5
0
D
2
U
8 s
(270).
•
The jet’s thrust is: F = %U
2 s
A = %U
2 s
π
4
D
2
.
An infinite, twodimensional, wavy wall moves steadily into positive at the speed v > 0, see the figure below.
x direction y v y w
(x,t)
H x
w
6
In complex notation, the wall surface is given by y w
( x, t ) = H exp( i ( α w vt
−
α w x )) ,
α w
= 2 π/λ w
> 0 being its wavenumber. The height
The ambient medium is assumed to be at rest, v
0
H is assumed to be small.
= 0, and to have some constant mean density %
0
.
a) Calculate the resulting velocity potential for the two cases and and
M = v/a
0
< 1
M >
M
1, where a
0 is the speed of sound in the surrounding medium, denotes the wallrelated Mach number! Hints:
1. Start with the following solution approach:
Φ = A exp ( iωt
− iαx
− iβy ) .
2. Set the y component of the wall’s vibration velocity ( v
0 yw to that of the acoustic particle velocity ( v
0 y
). Since H
) locally equal is small, v
0 yw
= v
0 y
( y = 0) approximately holds. Particular solutions for ω and α follow.
3. Calculate particular solutions for β and A by inserting the above solution approach for Φ into the respective twodimensional, homogeneous wave equation, which gives the dispersion relation (75).
Afterwards, insert the solutions for ω and α from step 2 into the dispersion relation. Two possible solutions for β follow. Choose the physically meaningful one for both of the cases M < 1 and
You should end up with the following results:
M > 1.
Φ = i
√
1 vH
−
M
2 exp (
−
α w y
√
1
−
M
2
) exp ( iα w vt
− iα w x ); M < 1 ,
Φ =
− vH
√
M
2 −
1 exp (
− iα w y
√
M
2 −
1) exp ( iα w vt
− iα w x ); M > b) Calculate the physical part case M > 1 using the respective result from subtask a) ! Does sound waves
2 p
0 phys of the pressure perturbation field for the
? Sketch the pressure field!
p
0 phys describe
1 .
c) Now also calculate the physical part for the case M < p
0 phys of the pressure perturbation field
1 using the respective result from subtask a) ! Does p
0 phys describe sound waves? Sketch the pressure field!
2
Definition of sound wave: wavelike propagation of small pressure and density perturbations at the speed of sound a
0
(ambient medium at rest).
7
A sphere oscillates radially about its average radius R with a small amplitude
A
R
R at a circular frequency of Ω, see the figure below. In complex notation, the timedependent radius of the sphere’s surface is thus given by δ ( t ) = R +
A
R exp( i Ω t ). The ambient medium is assumed to be at rest, some constant mean density %
0
.
v
0
= 0, and to have a) Calculate the radiated sound pressure field in complex representation!
1. Begin by calculating the velocity potential using following solution approach:
A
Φ
Φ = exp ( iωt
− ikr ) , r
≥
δ ( t ) .
r
2. Calculate the particular solution of Φ, i.e., calculate A
Φ and ω ! To this end, calculate the generic radial component v
0 r of the acoustic particle velocity from the above solution approach for Φ as well as the sphere’s radial vibration velocity v
0
K from δ ( t ). These two velocities are the same on the sphere’s surface, i.e., making use of v
0
K
( r = δ ( t )) = v r
0
( r = R ) approximately holds.
A
R
R ,
3. Finally calculate the sound pressure field tion of the velocity potential Φ!
p
0 from the particular solu
You should end up with: p
0
( r, t ) = r
%
0
Ω
2
A
R
R
(
− ik
−
1
R
) exp ( i Ω t
− ik ( r
−
R )) , r
≥
δ ( t ) .
b) A breathing sphere can model sound generation through pulsation or pulsating outflow. Which sound sources may thus concretely be replicated? Give examples!
8
A rigid, massless sphere of radius small amplitude A z
R
R oscillates harmonically in z direction with a at a circular frequency of Ω, see the figure below.
In complex notation, the sphere’s timedependent displacement in z direction is: δ zK
( t ) = A z exp( i Ω t ). Again, the ambient medium is assumed to be at rest, v
0
= 0, and to have some constant mean density coordinates ( x, y, z ), (fixed) spherical coordinates (
%
0
. Next to (fixed) Cartesian r, θ, ϕ ), 0
≤
θ
≤
π , 0
≤ ϕ
≤
2 π , are employed to describe the problem.
a) The z derivative of the velocity potential’s solution approach Φ
∗ from the breathing sphere (see exercise 20) is used to calculate the oscillating sphere’s velocity potential Φ. The solution approach for the breathing sphere was:
Φ
∗
=
A
Φ exp ( iωt
− ikr ) , r r
≥
R.
Derive the the solution approach Φ for the velocity potential of the oscillating sphere (in spherical coordinates)!
b) Reasoning similar to exercises 19 and 20 finally leads to the following particular solution of the oscillating sphere’s velocity potential:
Φ =
− r ( i Ω A z
( ik
2
R
3
− k
2
R
+
+
1 r
)
2 ik
R
2
) exp ( i Ω t
− ik ( r
−
R )) cos θ, r
≥
R.
What is the acoustic particle velocity field v
0 generated by the oscillating sphere (spherical coordinates)? How does it simplify for large distance r ?
c) An oscillating sphere can model sound generation through an external disturbance force. Which sound sources may thus concretely be replicated?
Give examples!
9
a) From equation (144), calculate and sketch the normalized directivities
˜ ( θ,ϕ ) of a monopole and of a dipole! Hints:
˜ ( θ,ϕ ) max
D ( θ, ϕ ) =
– substitute some general, undetermined, monopole moment and dipole moment point to be at ξ
0
= m
000
( τ
0
) m
001
( τ
0
) into eq. (144), and assume the reference
0 .
– coordinates are chosen like in exercise 21, and holds.
x = x
1
, y = x
2
, z = x
3 b) Compare the normalized directivities from subtask a) to those resulting from the breathing and oscillating sphere from exercises 20 and
21, respectively!
A function f n
( x ) is given by:
f n
( x ) =
0
2 n x
1
+
1
2
− x <
1
≤ n x > x
−
1 n
≤
1 n
.
1 n f n
(x)
1 n = 1 , 2 , . . .
x
−1/n 0 1/n
This function allows to describe the unit step function function series
{ f n
( x )
} for n
→ ∞
, i.e., lim n →∞ f n
( x ) =
H ( x ) as the limit of the
H ( x ).
a) Show that the function perty lim n →∞ g n
( x ) = g n
( x ), being the
δ ( x ) !
x derivative of f n
( x ), has the prob) Show the validity of the sifting property (108) for the Delta function δ ( x ) by way of function g n
( x ) from subtask a) !
Hints:
•
According to the monotone convergence theorem, integration and calculation of the limit can be interchanged here:
R b a lim n
→∞ g n
( x ) dx = lim n
→∞
R b a g n
( x ) dx.
• mean value theorem for integration: if the interval [
R b a s ( x ) dx = s a, b
( ξ
], then there exists at least one value
)( b
− a ) .
s ( x ) is a continuous function on
ξ
∈
[ a, b ], such that
10
Differential operators and arithmetic operations on matrices and vectors have usually been represented in vector notation so far. However, the socalled index notation is often encountered in aeroacoustic literature, too.
In that case, indices identify a certain entry of a matrix or vector. For example,
T ij denotes the entry in row i and column j of matrix T , i.e., in the case of two indices, the first identifies the row, and the second the column, where the respective entry is to be found in a matrix. Moreover, Einstein’s summation convention applies for repeated subscripts. For example, assuming i
∈ {
1 , 2
}
, a i b i
= a
1 b
1
+ a
2 b
2 holds.
a) Show the validity of
∂
2
T ij
∂x i
∂x j
=
∇ · ∇ ·
T .
Assume twodimensional Cartesian coordinates, i.e., i, j
∈ {
1 , 2
}
, and
∇
=
∂
∂x
1
∂
∂x
2
.
b) Let T denote Lighthill’s stress tensor:
T = ( % vv + ( p
0
− a
2
∞
%
0
) I
−
τ ) .
What does entry T ij look like?
Calculate the real mechanical power P mech
, which is required to make the massless sphere from exercise 21 oscillate, and compare it to the sound power P ac in the farfield. Which results are obtained for the special case of an incompressible medium? Use the particular solution of the velocity potential Φ from exercise 21.
11