It is shown that the shapes of acoustic radiation-induced static strain and displacement pulses (rectified acoustic pulses) are defined locally by the energy density of the generating waveform. Dispersive properties are introduced analytically by assuming that the rectified pulses are functionally dependent on a phase factor that includes both dispersive and nonlinear terms. The dispersion causes an evolutionary change in the shape of the energy density profile that leads to the generation of solitons experimentally observed in fused silica

Cantrell, John H.

NASA Technical Reports Server

 
 
 
 
 
 
 
 
ACOUSTIC RECTIFICATION IN DISPERSIVE MEDIA 
 
 
  John H. Cantrell 
 
 
  NASA Langley Research Center, Mail Stop 231, Hampton, VA 23681, USA 
 
 
 
ABSTRACT. It is shown that the shapes of acoustic radiation-induced static strain and    
displacement pulses (rectified acoustic pulses) are defined locally by the energy density of the 
generating waveform.  Dispersive properties are introduced analytically by assuming that the rectified 
pulses are functionally dependent on a phase factor that includes both dispersive and nonlinear terms.  
The dispersion causes an evolutionary change in the shape of the energy density profile that leads to 
the generation of solitons experimentally observed in fused silica.  
 
Keywords: Radiation-Induced Static Pulses, Dispersive Media, Solitons  
PACS: 62.65.+k, 43.25.Dc, 43.25.Qp, 43.25.Rq 
 
 
INTRODUCTION 
 
 The propagation of a sinusoidal acoustic wave through a nonlinear medium is 
known to generate not only harmonics of the original waveform but also a static strain 
(acoustic rectification or “dc shift”) associated with the propagating wave.  Using an 
approach based on the virial theorem, Cantrell [1,2] showed that the static strain 
generated by an acoustic toneburst (gated sinusoidal wave) is dependent on the energy 
density of the waveform.  He considered that the energy dependence is local and 
predicted that the derived static displacement pulse would propagate as a right-triangular 
pulse as a direct result of the locality. Experimental confirmation of the local energy 
dependence of static displacement pulses was reported by Yost and Cantrell [3] and 
Cantrell et al. [4] for monocrystals of silicon for wave propagation along the pure mode 
propagation directions. Yost and Cantrell [3] also reported results for fused silica – a 
dispersive material with a negative nonlinearity parameter.  The pronounced spreading 
and variations in amplitude of the displacement pulse profile observed in fused silica, 
however, were not predicted by the model.  We develop here a model of static pulses in 
dispersive materials that does predict the spreading and amplitude variation of the 
displacement pulse profile as the consequence of the generation of solitary waveforms 
(solitons) of the acoustic energy density.    
 
https://ntrs.nasa.gov/search.jsp?R=20090009958 2019-08-30T06:23:57+00:00Z
ACOUSTIC RECTIFICATION IN NONDISPERIVE MEDIA 
 
 We consider the propagation of an elastic wave in a lossless, non-dispersive, 
semi-infinite medium of arbitrary crystalline symmetry.  The nonlinear equation of 
motion along an arbitrary propagation direction may be transformed into the form4 
)/)](/(1[)/( 22222 auauctu ∂∂∂∂−=∂∂ εε β , where t is time, “a” is the coordinate along 
the wave propagation direction, u is the particle displacement, cε is the wave phase 
velocity where ε is a mode index associated with given directions of wave propagation 
and polarization, and βε is the modal nonlinearity parameter.  The Earnshaw particle 
velocity ∂uε/∂t solution to the nonlinear wave equation for each mode ε is [5]  
 
 )sin(/ φω εεε +−=∂∂ aktAtu  ,      )/)(2/( tucak ∂∂= εεεε βφ        (1) 
where Aε is the amplitude of the waveform, ω is the angular frequency, kε = ω/cε, and φ  
is a phase modulation factor, containing the nonlinearity parameter, that leads to wave 
distortion.  For periodic waveforms of constant amplitude the acoustic strain rectification, 
or radiation-induced static strain ‹∂uε/∂a›, is conveniently obtained by time-averaging the 
displacement gradient ∂uε/∂a (the angular brackets denote time-averaging).  The strain 
rectification (static strain) is related to the time-averaged particle velocity ‹∂uε/∂t›, hence 
time-averaged energy density ‹E›, as [1,2] 
 
      ∂uε /∂a = (βε /4cε2) (∂uε /∂t)2 = (βε /4με ) E      (2) 
where με is the modal second-order Huang coefficient.  The last equality results from the 
relations E = ρ0(∂uε/∂t)2, where ρ0 is the mass density of the propagation medium, and cε2 
= με/ρ0. 
 It is clear from Eq.(2) that the static strain is defined by the time-averaged energy 
density <E> of the propagating wave.  The constants βε and με serve as scaling factors for 
the static strain.  Since the energy density for a toneburst in a non-dispersive, lossless 
medium propagates as a stationary waveform, the static strain must be local and therefore 
exists only in that region of space for which <E> is non-zero.  Thus, from Eq.(1) the 
time-averaged energy density <E> = (1/2)ρ0Aε2 and the shape of the static strain pulse 
generated by a toneburst must be rectangular with height proportional to Aε2.  
Accordingly, the static displacement is obtained as the spatial integral of the rectangular 
static strain pulse only over the non-zero values of <E>, or energy per unit mass Aε2, 
commensurate with the stationary wave nature of the propagating energy density (or 
energy per unit mass). The locality both of the static strain and displacement pulses 
means that, in contrast to harmonic generation, the static displacement does not increase 
linearly with total distance of wave propagation, but rather increases linearly only over 
the spatial extent of the non-zero energy density or energy per unit mass.  Hence, for a 
toneburst of length L the static displacement pulse propagates in the shape of a right-
triangular pulse with the peak value ‹uε› obtained from Eqs.(1) and (2) as 
   
  
FIGURE 1.  Rectified acoustic pulses generated by a 2μs toneburst: (a) Static strain pulse in non-
dispersive media.  (b) Static displacement pulse in non-dispersive media.  (c) Static displacement pulse 
profile (soliton plus oscillatory tail) in dispersive media. 
 
 
                                                LAku 22)8/1( εεεε β=    .    (3)  
  Fig.1a shows a typical rectangular static strain pulse generated by a 2μs toneburst 
as given by Eq.(2).  Fig.1b shows a typical right-triangular static displacement pulse 
generated by a 2μs toneburst as given by Eq.(3).  The perspective in Fig.1 is the signal as 
seen by the receiving transducer.  Yost and Cantrell [3] obtained experimental 
confirmation of the right-triangular shape of the static displacement pulse in a silicon 
monocrystal by substituting a right-triangular pulse from a function generator into the 
detection electronics of the experimental set-up and comparing the output signal with that 
generated from the acoustic wave.  The overlap of the two output signals indicated that 
the static displacement was indeed right-triangular in shape.  Calculations of the 
nonlinearity parameters from measurements of the slopes of the static displacement 
pulses along the pure mode propagation directions in single crystal Si gave βε values that 
are in agreement with nonlinearity parameter measurements obtained independently from 
acoustic harmonic generation measurements [4].  Such agreement provides strong 
evidence of the right triangular shape of the static displacement pulses.   
 
ACOUSTIC RECTIFICATION IN DISPERSIVE MEDIA 
 
 Further evidence that the static strain and displacement pulses are defined locally 
by the energy density of the propagating wave is obtained from a consideration of the 
static strain and displacement pulses in vitreous silica – a dispersive material with a 
negative nonlinearity parameter.  The dispersion can be accounted by generalizing the 
phase factor φ  in Eq.(1) to include dispersive as well as nonlinear components.  We 
assume that rectified pulses are generated by the toneburst in accordance with Eq.(2), 
where <E> = (1/2)ρ0Aε2, to yield 22 )8/(/ DAcau β=∂∂ , where we have dropped the 
modal subscripts and have replaced the energy per unit mass A2 by AD2.  For dispersive 
media the energy per unit mass AD2 is no longer assumed to be constant but rather is 
assumed to evolve with propagation distance as AD2 = AD2[k0a-ω0t +φ(a,t)].  At a = 0, we 
assume AR2 = A2, the initial energy per unit mass of the toneburst launched into the 
material.  The parameters ω0 and k0 are associated with the toneburst and represent values 
of the parameters in the linear approximation.   
 We assume that the nonlinear dispersion relation depends on AD and the wave 
number k as 
 
       L+Γ+−′+−+== 22000002 )()2/1()(),( DD AkkckkcAk ωωω      ,     (4) 
where a Taylor series expansion with respect to k0 and AD2 = 0 is used to obtain the last 
equality in Eq.(4).  The c0, c′0 , and Γ are expansion coefficients.  We consider ω and k to 
be dependent on time and propagation distance, and impose the geometrical optics 
postulate that ω(a,t) = - (∂θ/∂t)  = ω0 – (∂φ/∂t) and k(a,t) = (∂θ/∂a) = k0 + (∂φ/∂a), where 
θ(a,t) = k0a -ω0t + φ(a,t).  Substituting the expressions for ω(a,t) and k(a,t) into Eq.(4) 
and adding a spatial second derivative term to account for dispersion [6], we obtain 
 
 0)/)(2/()/()2/1()/()/( 220
22
00 =∂∂′+Γ+∂∂′+∂∂+∂∂ aAAcAacact DDDφφφ  . (5) 
Transforming to a set of coordinates given by tca 0−=ξ and tc0′=τ , we re-write Eq.(5) 
in the form 
 
       0)/)(2/1()/()/()2/1()/( 2220
2
0 =∂∂+′Γ+∂∂′+∂∂ ξξφτφ DDD AAAcc   . (6) 
 Since we are interested in obtaining the propagation properties of the energy per 
unit mass AD2 = E/ρ0, it is necessary to invoke an second equaton involving both AD2 
and )/( ξφ ∂∂ .  We thus invoke the conservation of energy expression 
 
             ∂AD2 /∂t + ∂(cg AD2 ) /∂a = 0    (7) 
where from Eq.(4) cg = ∂ω/∂k = c0 + (∂φ/∂a)  is the energy propagation velocity (group 
velocity) in the linear approximation.  Again transforming to the set of coordinates given 
by tca 0−=ξ and tc0′=τ , we re-write Eq.(7) in the form 
 
        0)]/()[/(/ 22 =∂∂∂∂+∂∂ ξφξτ DD AA   .   (8) 
 
Assuming that both AD2 and )/( ξφ ∂∂  have stationary solutions to Eqs.(6) and (8) of the 
form )()( zfVff =−= τξ  where )( τξ Vz −= , we obtain from Eq.(8) 
 
          (∂ /∂z)[VAD2 − (∂φ /∂ξ)AD2 ] = 0 .    (9) 
 
We integrate Eq.(9) and set the integration constant to VA2 where A2 is the energy per 
unit mass of the toneburst at a = 0.  Solving the resulting expression for )/( ξφ ∂∂ , we 
obtain 
     )1(/ 2
2
DA
AV +=∂∂ ξφ .     (10) 
 
 Following Karpman [7], we now assume that )(1
2
0 τξφτμφ V−+−= , substitute 
the expression into Eq.(6), and use Eq.(10) to obtain 
 
         0)/)(2/1()/()1(
2
1 222
0
44
0
22
0 =∂∂+′Γ+−−− − ξμ DDDD AAAcAAV  .  (11) 
 
The stationary solution to Eq.(11) has the form of a solitary wave (soliton) given by [7]  
 
   )(/sec 0
222
0
2 τξ VAchAAA ssD −′Γ+=    (12) 
 
where  
    2/1220
2/1
0 )()/( sAAcV +′Γ=     (13) 
 
and 2sA  is the amplitude of the soliton. 
 Strictly, Eq.(12) is the general stationary solution for a perturbation imposed on a 
continuous carrier wave of amplitude A0 and is valid for an arbitrary amplitude of 
perturbation [7].  If we assume that the perturbation (the toneburst) is large compared to 
the carrier amplitude, then in the limit where the carrier amplitude becomes arbitrarily 
small we obtain 
   
          )(/sec 0
222 τξ VAchAA ssD −′Γ=    (14) 
 
where sAcV
2/1
0)/( ′Γ= .  The number of solitons generated, each having different 
amplitudes 2sA , depends on the length and amplitude of the initial perturbation.  For a 
toneburst of amplitude A the initial rectification is a rectangular static pulse of 
amplitude 2A .  The complete solution also includes an oscillatory tail wave packet similar 
in form to that generated by the Airy function.  The static strain <∂u/∂a> is obtained 
directly from Eqs.(2) and (14) as <∂u/∂a> = <∂u/∂ξ> = (β/4μ)<E> where <E> = 
(1/2)ρ0AD2. 
 Fig.1c shows the static displacement pulse generated by a 2μs toneburst obtained 
by integrating Eq.(14) and the oscillatory tail with respect to ξ.  The soliton is detected 
first, since it has the greater velocity, and is followed by the slower oscillatory tail.  It is 
important to note that the rectified acoustic pulses are generated by the toneburst at a = 0.  
The dispersive and nonlinear properties of the rectified pulses are governed by the values 
of the expansion coefficients in the dispersion relation, Eq.(4), evaluated at the values of 
k0 and ω0 associated with the toneburst.  Thus, it should not be expected that launching a 
pulse in the material directly from a pulse generator would necessarily generate solitons, 
since the values of k0 and ω0 used to evaluate the coefficients in Eq.(4) from such a pulse 
would not be the same as that of the rectified pulses generated by the toneburst.    
 
MEASUREMENT OF THE ACOUSTIC NONLINEARITY PARAMETER IN 
DISPERSIVE MEDIA 
 
Fig.1c shows the calculated static displacement profile, as observed by the 
receiving wave detector, resulting from a single soliton with an oscillatory tail generated 
by a 2μs tone-burst in a dispersive medium with β < 0.  Note that since β < 0, the 
displacement profile is inverted relative to that of Figs.1a and 1b where β > 0.  The pulse 
profile of Fig.1c is also considerably longer than the toneburst duration and shows large 
variations in displacement amplitude.  Both the spreading and the variations in amplitude 
of the static displacement pulse have been observed experimentally [3].  This observation 
provides further evidence that the static strain and displacement pulses are defined locally 
by the energy density of the waveform.   
We consider the implications of the static pulse profile to the measurement of the 
acoustic nonlinearity parameter in dispersive media.  The initial portion of the static 
displacement pulse in Fig.1c results from the soliton contribution to the profile.  The 
slope of the pulse profile following the first peak is given by <∂u/∂a> = (β/4μ)<E>.  At 
the half-peak height of that slope, where τξ V= , the average energy density is evaluated 
from Eq.(9) to be <E> = (1/2)ρ0A s2.  Generally, As2 is not equal to the square of the 
amplitude of the tone-burst.  However, for a single soliton generated by a toneburst of 
appropriate length and amplitude it can be approximately so.  If we assume that As
2 is 
very nearly equal to the square of the tone-burst amplitude, then measurement of the 
slope at τξ V=  allows a direct, although approximate, calculation of β.  Yost and 
Cantrell [3] performed such a calculation of β for Suprasil W1 vitreous silica, fortuitously 
at τξ V= , without benefit of the present analysis.  Their calculated value of -12.7 for 
β  is in quite good agreement with the value -11.6 obtained independently from acoustic 
harmonic generation measurements for the material.  The general agreement once again 
affirms the role of the local energy density in the generation of static pulses.  
 
CONCLUSION 
 
 The analytical model previously developed for acoustic rectification in non-
dispersive media [1,2] has been extended to the case of dispersive media.  The extension 
is obtained by assuming that the energy per unit mass AD2 of the rectified waves is no 
longer constant, as in the case of non-dispersive media, but now depends on phase factors 
similar to that obtained for the Earnshaw particle velocity solution to the nonlinear wave 
equation.  We assume that at a = 0 (the spatial position at which the toneburst is launched 
into the material) AD2 = A2 (the energy per unit mass of the toneburst at a = 0).  Assuming 
that the dispersion relation ω(k, AD2) is a nonlinear function both of the wave number k 
and the rectified wave amplitude AD, we expand ω(k, AD2) in a power series and use the 
geometrical optics postulate to obtain a Boussinesq type of equation.  We add a 
dispersive term to the equation and solve for the energy per unit mass waveform after 
invoking the energy conservation equation. 
 The solution predicts the generation of an energy per unit mass solitary waveform 
(soliton).  According to the analytical model the static strain pulse profile is defined 
locally by the energy density of the toneburst and thus has the shape of the energy 
density, hence energy per unit mass, profile.  The static displacement pulse profile is 
obtained by integrating over the spatial extent of the static strain profile to yield a profile 
that has been observed experimentally.  The agreement between the pulse profile 
observed experimentally and the profile predicted by the model lends validity to the 
assertion that the acoustic radiation-induced static pulses are dependent locally on the 
energy density of the propagating waveform.         
 
REFERENCES 
 
1.  J. H. Cantrell, Phys. Rev. B 30, 3214-3220 (1984). 
2.  J. H. Cantrell, J. Phys. A 26, L673-L677 (1993). 
3.  W. T. Yost and J. H. Cantrell, Phys. Rev. B 30, 3221-3227 (1984). 
4.  J. H. Cantrell, W. T. Yost, and P. Li, Phys. Rev. B 35, 9780-9782 (1987).   
5.  S. Earnshaw, Philos. Trans. R. Soc. London 150, 133 (1860). 
6.  T.Taniuti and N. Yajima, J. Math. Phys. 10, 1369-1372 (1969). 
7.  V. I. Karpman, Nonlinear Waves in Dispervive Media, Pergamon, Oxford,   
    1974. 
 


Acoustic Rectification in Dispersive Media

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