Small frequency approximation of (causal) dissipative pressure waves

0.22 MB·English

by  Richard Kowar

Small frequency approximation of (causal)
dissipative pressure waves
R. Kowar∗
∗Department of Mathematics, University of Innsbruck,
Technikerstrasse 21a/2, A-6020,Innsbruck, Austria
2 (e-mail: [email protected])
1
0
2 Abstract:
Inthis paper wediscuss the problemof smallfrequency approximationof the causaldissipative
n
pressure wavemodel proposedin Kowaret al. (2010).We show that for appropriatesituations
a
J the Green function Gc of the causal wave model can be approximated by a noncausal Green
1 function GpMl that has frequencies only in the small frequency range [−M,M] (M ≤ 1/τ0, τ0
3 relaxationtime)andobeysapowerlaw.Forsuchcases,thenoncausalwaveGpl containspartial
wavespropagatingarbitrarilyfastbutthe sumofthe noncausalwavesissmalMlinthe L2 sense.
] −
h
p Keywords: Attenuation, Dispersion, Wave equations, Causality
-
h
at 1. INTRODUCTION: FREQUENCY FRAMEWORK 2p(x,t) D + 1 ∂p 2p(x,t)= f(x,t), (4)
m ∇ − ∗ c ∂t −
In the first section we summarize the general framework (cid:18) 0 (cid:19)
[
of frequency driven wave dissipation.
where D denotes the time convolution operator
∗
2
From the mathematical point of view, in case of a homo- 1
v D (g) (ω)= α (ω) g (ω)
6 geneous and isotropic medium, dissipative pressure waves F{ ∗ } √2π ∗ F{ }
can be modeled by (cf. e.g. Kowar et al. (2010))
7 defined for appropriate functions g = g(t). Here g
7 p(x,t)=(G x,tf)(x,t) x R3, t R denotes the Fourier transform of g with respect to tiFm{e.}
∗ ∈ ∈
5
with
. More details about attenuation-dispersion laws and re-
1 ∂p
p(,t)=0 and (,t)=0 for t<0, (1) spective wave equations can be found in e.g. Nachman
0 · ∂t ·
et al. (1990), Szabo (1995), Waters et al. (2000), Chen
2
1 where G denotes a distribution (Green function), x,t et al. (2004), Patch et al. (2006), Kelly et al. (2008)
: denotesthespace-timeconvolutionandf denotesaforc∗ing and Kowar et al. (2012).
v
term (source term) which models wave generation. We
i
X note that f is the same forcing term as in the absence of 2. ATTENUATION-DISPERSION LAWS AND
r dissipation as long as (1) holds. (This is not true if p(x,t) CAUSALITY
a is calculated for t t (t >0) from initial data at t .)
0 0 0
≥
2.1 What do we mean by causality?
1.1 The Green function
Let c < denote the (constant) speed of the wave
F
The Green function can be modeled by ∞
frontofthewaveG.Causalityrequiresthatthe wave front
Gˆ(x,ω)= e−4βπ∗(|xx|,ω) eicω0 |x| x∈R3, ω ∈R, (2) ixni=tiat0ednaott tbheefoorerigtihne0timatetipmereiotd=T0(xar)riv=es|xa|t pisosoitvieorn.
| | 6 | | cF
Mathematicallythisisequivalenttothefollowingcausality
where (β (x,ω)) > 0 is even in ω and (β (x,ω))
∗ ∗ condition: for each c c we have
ℜ | | ℑ | | 1 F
is odd in ω. Here Gˆ(x,ω) denotes the Fourier transform x ≥
of G(x,t) with respect to time t. The last two conditions G x,t+ | | =0 if t<0 and x =0. (5)
guaranteethatG(x,t)isreal-valued.Wefocusonthecases (cid:18) c1 (cid:19) | |6
β (x,ω)=α (ω) x . (3)
∗ ∗ Variousdissipativewavemodelsareanalysedwithrespect
| |
to causality in Kowar et al. (2012). In particular, it is
We call α = α (ω) and α = (α (ω)) the attenuation-
dispersion∗law a∗nd the attenuatℜion∗law, respectively. shown that (3) implies cF = const. if cF < ∞. See
also Kelly et al. (2008).
1.2 The wave equation
2.2 Attenuation-dispersion laws (γ (1,2])
∈
The above dissipative wave model satisfies the following
integro-differential equation In this paper we consider the attenuation-dispersion laws
αc∗(ω)= c 1α+1((−iiωτ)ω)γ−1 (6) • Gasplin(x(,7t)),denote the Green function with αp∗l defined
0 0
− Gˆ (x,ω) be defined as in (9) and
M
and p • G (x,t):= −1 Gˆ (x,t).
M M
( iω)γ • F { }
αp∗l(ω)=a1 cos−(γπ/2) +a2(−iω) (7) Theorem 1. Let M >0 and A0, A1, A3 >0 be such that
α(M)+A ω M αc(ω) A ω2+A ω (11)
0 1 2
| − |≤ ≤ | |
for ω R with γ (1,2], c (0, ) and α , τ , a > 0.
0 1 0 1
αpl =∈αpl(ω) is cal∈led the freq∈uency∞power law. holds for ω >M. For x =0 it follows that
∗ | | | |6
Let a1 = α12τc0γ−1 |cos(γπ/2)| and a2 = αc1 , (8) kGc(xk,G·)c−(xG,·cM)k(Lx2,·)kL2 ≤ s4 π2AA120 e−(cid:16)α(M4)+|x4A|A221(cid:17)|x| .
0 0
p
then it follows from Proof. ω 7→ Gˆc(x,ω) and ω 7→ GˆcM(x,ω) are square
π π integrable and thus their inverse Fourier transforms t
( iω)γ = ω γ cos γ i sin γ sgn(ω) Gc(x,t) and t Gc (x,t) are square integrable, to7→o.
− | | 2 − 2 7→ M
that n (cid:16) (cid:17) (cid:16) (cid:17) o Because of the Plancherel-Parseval equality, it follows
αc(ω)=ℜ(αc∗(ω))≈a1|πω|γ =αpl(ω), kGc(x,·)−GcM(x,·)kL2 = kGˆc(x,·)−GˆcM(x,·)kL2 .(12)
ℑ(αc∗(ω))≈−a1 tan γ 2 sgn(ω)|ω|γ −a2ω kGc(x,·)kL2 kGˆc(x,·)kL2
for τ ω γ−1 << 1. We s(cid:16)ee th(cid:17)at αpl(ω) is a good ap-
prox|im0at|ionofαc(ω)forsufficientlysm∗ all frequencies.For From (11) and
the case τ = 10−∗6µs (liquid), the small frequency range ∞
0
condition τ ω γ−1 0.1 is visualized in Table 1. exp ω2/2 dω =√2π,
| 0 | ≤ {− }
Z
Table1.Visualizationofsmallfrequencyrange −∞
condition τ ω γ−1 0.1 for τ =10−6µs. it follows
0 0
| | ≤ ∞
e−2A0(ω−M)|x|
γ range boundM Gˆc(x, ) Gˆc (x, ) 2 2e−2α(M)|x| dω
1.1 |ω|≤10−4MHz 10−4MHz k · − M · kL2 ≤ (4π x)2
1.5 |ω|≤10+4MHz 10+4MHz MZ | |
2.0 |ω|≤10+5MHz 10+5MHz e−2α(M)|x|
≤ (4π x)2A x
0
| | | |
2.3 Two results on causality and
∞ e−2(A2|ω|+A1ω2)|x|
tInheKwoawvaerfreotnatl.of(G20(x12,t)), iwtiitshpαrcovdeenfinthedatatshiens(p6e)esdatciFsfieosf kGˆc(x,·)k2L2 ≥ (4π x)2 dω
∗ −Z∞ | |
cF ≤ c0 and in Kowar (2010) it is proven that cF = c0. A22|x|
Moreover,itisshownthatG(x,t)withαpl definedasin(7) √πe 2A1
∗ =
witha2 =0hasnotaboundedwavefrontspeed.Thesame (4π x)2 2A1 x
result is true for a =0. | | | |
2
6 The theorem follows from (12) anpd the last two results.
Loosely speaking, causality restricts the growth of α(ω)
and consequently ω Gˆ(x,ω) must not decrease too fast Remark. Because ω Gˆc (x,ω) vanishes on a non-
for fixed x R3 07→. If ω Gˆ(x,ω) decreases too fast, empty interval, causality7→condMition (5) is not satisfied for
then it beha∈ves l\ik{e }the trun7→cated Green function Gc . However, Theorem 1 shows that Gc (x,t) is a good
M M
GˆM(x,ω):=Gˆ(x,ω)χ[−M,M](ω), (9) L2−approximationof Gc(x,t) if M is sufficiently large.
for which causality condition (5) cannot hold. Here Theorem 2. Let B1 :=ℜ(αp∗l−αc∗), B2 :=ℑ(αp∗l−αc∗),
1 if ω [ M,M] C := 1 2e−B1(ω)|x| cos(B (ω) x)+e−2B1(ω)|x| .
χ[−M,M](ω):= 0 othe∈rwi−se (10) − 2 | |
(cid:26) and Mδ(cid:12)(cid:12) (0,M] be such that (cid:12)(cid:12)
(cid:12)∈ (cid:12)
denotes the characteristic function of ω on [−M,M]. kGˆcMδ(x,ω)k2L2 =(1−δ)kGˆc(x,ω)k2L2 (13)
3. SMALL FREQUENCY APPROXIMATION for some 0<δ <<1. For x =0 it follows that
| |6
We now derive two theorems that permit to estimate the kGcM(x,·)−GpMl(x,·)kL2 (1 δ)D (x)+δD (x)
quality of small frequency approximations. Let Gc(x, ) L2 ≤ − 1 | | 2 | |
k · k
Gc(x,t)denotetheGreenfunctionwithαc definedas with D (x) := max C(px,ω)2 for I := [ M ,M ]
• in (6), ∗ and I :j=|R| I . ω∈Ij | | 1 − δ δ
2 1
\
Proof. As in the proof of Theorem 1, it follows that for different distances L. We see that Gc (x,t) and
M
kGcM(x,·)−GpMl(x,·)kL2 = kGˆcM(x,·)−GˆpMl(x,·)kL2 . Table 2. Visualizationof errorǫM for different
kGcM(x,·)kL2 kGˆcM(x,·)kL2 distances L (in cm).
The theorem follows from the last result, (13) and
L 10−6 10−3 10−1 10
Gˆc (x, ) Gˆpl(x, )2dω ǫM 7.62·10−8 7.35·10−5 4.46·10−4 7.13·10−5
| M · − M · |
Z ≤D1(|x|) Z |GˆcM(x,·)|2dω GpMl(x,t) aMre go1o0d0aMppHrozximanadtionsxof G1c0(−x6,·c)mas. long as
|ω|≤Mδ ≥ | |≥
In particular, this means that the wave can be predicted
+D (x) Gˆc (x, )2dω.
2 | | | M · | for
Z 10−6cm
|ω|≥Mδ t =6.67 10−6µs.
Cf. Fig. 3 from the example in Section 4 for which δ 6 ≥ c ·
0
10−4, M 10MHz,D (x) 5.6 10−13 and D (x≈) ·
1.04, i.e.δ ≈(1 δ)D (x1)|+|δ≈D (x·)=0.025. 2 | | ≈ Remark on causality. Because the wave Gc has the
− 1 | | 2 | | finite wave front speed c = c and Gpl(x, ) is a good
F 0 M ·
p 4. NUMERICAL EXAMPLE L2 approximation of Gc(x, ), it follows that Gpl has
− · M
in “some sense” the same wave front speed. In “some
We now present a numerical example. We choose the sense”meansthatGpl containspartialwavespropagating
following parameter values M
arbitrarily fast but their sum is small in the L2 sense.
γ 1.66, c 0.15cm, τ 10−6µs, −
0 0
≈ ≈ µs ≈ Remark on large frequencies. We note that the power
2c0a1 attenuation law αpl(ω) increases much faster than αc(ω)
α = 138.08MHz
1 τγ−1 cos(γπ/2) ≈ for large frequencies and that the respective phase speed
0 | | cpl(ω) has a singularity at ω 7.950959 106MHz and
(where we have used (8) and a1 ≈0.04344c1m) and at ω2 = ω1 (cf. Fig 2), res1pe≈ctively. In c·ontrast to the
a = α1 920.55 1 latter fac−t, the phase speed cc(ω) of the causal model has
2
c0 ≈ cm no singularity.
tomodelamediumsimilartocastoroil.(Fortherelaxation
timeτ0seeKinsleretal. (2000)andfora1andγ seeSzabo 5. CONCLUSIONS
(1995)).
The differences of the attenuation laws and the phase We showed that - for appropriate situations - the causal
speeds of the attenuation-dispersion laws αc and αpl are wave Gc defined by (2), (3) and (6) can be approximated
visualized in Fig. 1. From the figure we∗see tha∗t αpl by the noncausal wave GpMl defined by (9) (2), (3), (7)
∗ and (8). In words, the wave Gpl contains partial waves
is a good approximation of the attenuation-dispersion
law αc for the (practically measurable) frequency range propagating arbitrarily fast but the sum of the noncausal
[0,60]∗MHz.Inthefollowingwechooseassmallfrequency wavesissmallintheL2 sense.Thisresultfits inwiththe
−
results in Kelly et al. (2008).
bound:
1
M =100MHz<< .
τ REFERENCES
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0 0
kGcM0(x,·)kL2 and ω 7→ C(|x|,ω) for |x|7→= 1cm. Ac- iinnPBhioomtoeadciocaulstIimcsa.gtinogapIIpLeaecrtiunr:eMNaottheesminatMicaalthMemodaetliicnsg,
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300
x 105 200
2
d
1.8 e 100
e
w1.6 p
a s 0
on l11..24 ase −100
ati 1 ph−200
u
n0.8
e −300
att0.6
0.4 −400
7.945 7.946 7.947 7.948 7.949 7.95 7.951 7.952 7.953 7.954 7.955
0.2 frequency (MHz) x 106
0
0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000
frequency (MHz) Fig. 2. Numerical comparison of the attenuation laws
and phase speeds for large frequencies The dashed
x 10−3
1.102 lines correspond to model αc(ω) and the solid lines
∗
1.1 correspond to model αpl(ω).
∗
d1.098
e
e
p1.096
s
e 1.094 0.2
s
a1.092 0.18
h
p 1.09 0.16
0.14
1.088
0.12
g
1.0860 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 0.1
frequency (MHz)
0.08
0.06
Fig. 1. Numerical comparisonof the attenuation laws and
0.04
phase speeds. The dashed lines correspond to model
0.02
αc∗(ω)andthesolidlinescorrespondtomodelαp∗l(ω). 00 10 20 30 40 50 60 70 80 90 100
M (MHz)
0
1.4
1.2
1
0.8
C

0.6
0.4
0.2
0
0 100 200 300 400 500 600 700 800 900 1000
frequency (MHz)
Fig. 3. Visualization of functions M g(M ) :=
0 0
kGcM0(x,·)kL2 and ω 7→C(|x|,ω) for |x7→|=1cm.