Notes on ANISOTROPY IN CRYSTALLINE SILICON
 This talk will focus on surface acoustic waves (SAW)
in crystalline silicon. Crystalline silicon is a cubic
crystal (see ``Diamond Cubic Structure'' diagram).
Cubic crystals have the stressstrain relation shown in the
slide. Usually the strains are sufficient small that the
linear relation s_{ij} = c_{ijkl}e_{kl} is valid.
However, the strains considered here are
large enough that the nonlinear terms contribute
significantly and, in fact, give rise to waveform distortion and
shock formation.

Diamond Cubic Structure:
Si has a diamond lattice which can also be considered to
be two fcc lattices, one displaced relative to the other
by (1/4, 1/4, 1/4).
Note also that every atom has four nearest
neighbors. The lattice spacing for Si is 0.543 nm.
 The material properties of the crystal are expressed via
the elastic constants of the material. In particular,
3 SOE and 6 TOE constants are necessary to specify a cubic
crystal. For all the simulations presented in this talk,
the data for the Si elastic constants was taken from
the paper by H. J. McSkimin and P. Andreatch, Jr.,
J. Appl. Phys. 35, 33123319 (1964).
 Isotropic Material: In contrast, an isotropic material
has only 2 SOE (shear and bulk moduli) and 3 TOE
constants. The number of constants is less because the
symmetry is higher.
 Because these systems are anisotropic, the wave propagation
is different depending on how the crystal is cut and the
direction that the wave is traveling.
 The surfaces of cut crystals have traditionally
been described using a crystallographic convention called
Miller indices. Miller indices are defined by finding
three noncollinear atoms on the surface that intersect the
crystal axes and then applying the following method:
 Find the intercepts of the three basis axes in terms of
the lattice constants.
 Take the reciprocals of these numbers and reduce to the
smallest three integers having the same ratio. The
result is enclosed in parentheses (hkl).
[from C. Kittel, Introduction to Solid State
Physics, 2nd ed. (John Wiley & Sons, New York, 1965), p. 34]
Note that if the Miller indices are interpreted as a vector
components, the resulting vector is normal to the surface of the
cut.
 Directions are specified in a different way:
The indices of a direction in a crystal are expressed as the
set of the smallest integers which have the same ratios as the
components of a vector in the desired direction referred to the
axis vectors. The integers are written in square brackets,
[uvw].
The x axis is the [100] direction;
the y axis is
the [0¯10] direction. A full set of equivalent
directions is denoted this way: áuvwñ.
[from C. Kittel, Introduction to Solid State
Physics, 2nd ed. (John Wiley & Sons, New York, 1965), p. 34]
This presentation will use both of these notations frequently.
 Because it is surface phenomena that are being studied,
it is necessary to specify how the surface is oriented with
respect to the crystalline axes and, in addition, the direction
in which the wave is travelling.
The crystal cut chosen in the experiment is the (001) plane
(see diagram). For the experiment and simulations shown in
this talk, the pulses propagated in directions measured relative
to the á100ñ direction.
Notes on LINEAR SAW DIAGRAM
 This figure shows the particle motion of a typical
surface acoustic wave in an anisotropic medium.
Consider the case of a surface acoustic wave with an
initially sinusoidal velocity waveform in an isotropic
and anisotropic material. Assume that the xaxis
is in the propagation direction and that the zaxis is
normal to the surface cut.
 In order to be a surface acoustic wave, the wave must
satisfy the stressfree boundary conditions at the surface
and decay into the solid with an exponential envelope.
The depth dependence of the vertical displacement u_{z} and
horizontal displacement u_{x} is plotted against the depth
in wavelengths for the linear SAW propagating in Si in the
á001ñ direction. The amplitudes oscillate and
decay thereby giving rise to alternating regions of retrograde
and prograde elliptical motion. As can be clearly seen,
most of the motion is confined to within a wavelength of the surface.
 Isotropic material: In contrast, the amplitude of
a surface acoustic wave in an isotropic material
(usually called a Rayleigh wave) decays away purely
exponentially into the solid.
 Because the surface is typically the most experimentally accessible
part of the medium, it is the location where SAWs are examined.
The particle trajectories at the surface are shown in top and
side views. The particle motion is always confined to a plane.
For the cases shown here, this plane is generally rotated by some angle
f out of the xzplane.
 Isotropic material: In contrast, the particle trajectory
of a Rayleigh wave is always confined to the plane perpendicular
to the surface that contains the direction of propagation,
shown in the figure as the dashed rectangle.
 The particle displacement has a transverse component
for most propagation directions in the (001) plane. However,
because f is small for most propagation directions, only vertical
and longitudinal waveforms will be shown. The transverse
component is related to the other components by a relation
that can be derived from linear theory.
Notes on NONLINEAR THEORY
 Briefly, the approach used here involves calculating
the Hamiltonian energy function through cubic order in
the wave variables, choosing appropriate generalized
coordinates, applying the equations of motion in canonical
form, and deriving evolution equations for the slowly varying
amplitudes in a suitable retarded time frame. The approach was
outlined in M. F. Hamilton, Yu. A. Il'inskii,
and E. A. Zabolotskaya, ``Nonlinear surface wave propagation
in crystals,'' Nonlinear Acoustics in Perspective,
R. J. Wei, ed. (Nanjing University Press, Nanjing, China, 1996),
pp. 6469.
 Note that computing the Hamiltonian the quadratic
order would only give rise to linear terms in the model
equations. Thus, the potential
energy terms to at least cubic order in the strain must
be included to model nonlinear effects.
 Note also that this method is very general.
It is applicable to any elastic
material for which the SOE and TOE constants are known
and to any cut and direction in such a material.
 Assumptions:
 It is assumed that the nonlinear solution is close
to the linear solution; in particular the depth
dependence of each frequency is the same as in the
linear solution.
 It is assumed that the wave fronts are planar.
 It is assumed that the wave is progressive, i.e.,
travels only in one direction. (It should be
possible to extend the theory to include compound waves;
only the results will be more complicated.)
 The components of the velocity in the solid are assumed to take the
form shown in the slide. Here v_{j} is the jth component of
velocity, k is the characteristic wavenumber, and w is
the characteristic angular frequency of the signal.
Because surface acoustic waves are nondispersive, i.e.,
their wave speed is not frequency dependent, w/k = c
where c is the SAW speed in the direction of propagation.
 The coordinate system for the solution is always chosen
such that the
the zaxis is perpendicular to the surface of the solid and
the xaxis is in the direction of the propagation of the wave.
Because the elastic constants are typically given with respect
to the crystalline axes, the elastic constants must always first be
transformed into the aforementioned coordinate system before
substitution into the model equations described in the slide.
 The functions u_{nj} describe the depth dependence of the
nth harmonic of the jth component. The values of
l_{3}^{(s)} and q_{j}^{(s)} that determine these functions
are found by solving the linear problem. This is the result
of Assumption 1 above.
 Note that on the surface the expressions for the waveforms
simplify to
v_{j}(x,z,t) = 
¥ å
n = ¥

v_{n}(x) 
3 å
s = 1

b^{(s)}_{j} e^{int} [ v^{*}_{n} = v_{n}] 

where t = kxwt is the retarded time
and the b_{j}^{(s)} are constants determined from the linear
problem.
 The coupled, nonlinear spectral evolution equations
that result from this approach are shown above.
Here v_{n} is the complex amplitude of the nth harmonic,
a_{n} is the attenuation coefficient for the nth harmonic,
w is the characteristic angular frequency,
r is the density of the material, c is the SAW speed
for the direction of propagation, and S_{lm} is the
nonlinearity matrix.
 In practice, these equations are first converted to a
nondimensional form before they are solved. Let v_{0} be the
characteristic velocity amplitude of the signal.
If V = v/v_{0} and X = x/x_{0} where
x_{0} = 
rc^{4}
4S_{11}wv_{0}

, 

then the evolution equations take the form

dV_{n}
dX

+A_{n} V_{n} = 
n^{2}
8S_{11}


å
l+m = n


lm
lm

S_{lm}v_{l}v_{m} 

where A_{n} = a_{n} x_{0}.
 The ad hoc attenuation term a_{n} = n^{2}a_{1}
is added to the lefthand side for purposes of numerical
stability when solving the equations. It assumes that
the attenuation coefficient for any frequency component
is proportional to the square of that frequency as has been
observed in quartz [E. Salzmann,
T. Plieninger, and K. Dransfeld, ``Attenuation of elastic
surface waves in quartz at frequencies of 316 MHz and
1047 MHz,'' Appl. Phys. Lett. 13, 1415 (1968)].
For all the cases
shown here the dimensionless value of A_{1} = 0.025.
This attenuation is sufficiently weak that its main effect
is to stabilize the portion of the waveform in the neighborhood
of the shock without significantly the remainder of the
waveform. Note that the dimensionless value of A_{1} here
is the analog of the Goldberg number G for nonlinear
acoustic waves in fluids.
 Physically, the nonlinearity coefficients S_{lm}
represent the strength of the coupling between different
harmonics in the wave. They are given by a complicated
analytical expression which can be determined completely
by knowing the SOE and TOE constants of the material
(see the section ``Notes on Nonlinearity Matrix'' for details).
 For the case of isotropic materials,
these equations can be shown to reduce to the evolution
equations previously derived by Zabolotskaya
[E. A. Zabolotskaya, ``Nonlinear propagation of plane and
circular waves in isotropic solids,''
J. Acoust. Soc. Am. 91, 25692575 (1992)].
 While Hamilton's equations describe the evolution of a system in
time, the evolution equations listed in the slide
evolve in space, not time. Informally speaking, the
transformation between the two is done by moving into
retarded time frame and thereby replacing
¶/¶t with c ¶/¶x.
It is possible to demonstrate formally that this is the
proper transformation and that it is not an approximation
[E. Yu. Knight, M. F. Hamilton, Yu. A. Il'inskii, and
E. A. Zabolotskaya, ``General theory for the spectral evolution
of nonlinear Rayleigh waves,'' J. Acoust. Soc. Am.,
102, 14021417 (1997)].
 These equations may be solved as follows. By first
solving the linear problem for the eigenvalues, eigenvectors,
and smallsignal wave speed, the nonlinearity matrix can be
constructed. Once the nonlinearity matrix is determined, the
model equations can be integrated. The spectral evolution
equations were solved numerically using the spectral
``source'' condition corresponding to an initially sinusoidal
wave. A fourthorder RungeKutta routine was used to integrate
the system. The waveform expansions used had 200 harmonics.
 In theory, there are an infinite number of
equations to integrate. For purposes of computation, the
velocity waveform expansions were truncated such that only terms
with n = 200 to n = 200 were included in the sum.
However, because the velocity waveforms must be
realvalued, v_{n} = v^{*}_{n}. Therefore only 200 spectral
amplitudes must be determined and, correspondingly,
only 200 equations must be integrated.
 To minimize numerical aliasing effects, only the first
150 harmonics were used to reconstruct the waveforms shown later
in the talk.
Notes on COMPARISON WITH EXPERIMENT
 The theory described above has been compared with
experiment. These results were presented previously
at the Seattle ASA/ICA meeting during June 1998
[R. E. Kumon, M. F. Hamilton, Yu. A. Il'inskii,
E. A. Zabolotskaya, P. Hess, A. Lomonosov, and
V. G. Mikhalevich, ``Pulsed nonlinear surface acoustic
waves in crystals,'' in Proceedings of the 16th
International Congress on Acoustics and 135th Meeting of
the Acoustical Society of America, edited by P .K Kuhl
and L. A. Crum (Acoustical Society of America, Woodbury,
New York, 1998), Vol. 3, pp. 15571558] and hence will
only be described briefly here.
 The experimental approach used here generates SAW
via photoelastic
laser excitation. This method was described previously
by A. Lomonosov and P. Hess, ``Laser excitation and
propagation of nonlinear surface acoustic wave pulses,''
Nonlinear Acoustics in Perspective,
R. J. Wei, ed., (Nanjing University Press, Nanjing, China,
1996), pp. 106111. The waves were generated in the
(111) plane in á112ñ direction.
 The SAW pulse was
generated by a Nd:YAG laser that was focused with a cylindrical
lens into a thin strip 6 mm by 50 mm on the surface of
crystal. To detect the resulting SAW pulse, optical probe beams
were employed. This can be done because probe beam deflection
is proportional to the vertical velocity component v_{z} at the
surface. The probe beam deflections were detected by split
photodiodes with a bandwidth of 500 MHz. The probe beams
irradiated spots approximately 4 mm in diameter on the
surface at distances 5 mm and 21 mm from the excitation region.
 As can be seen in the figures, the surface wave pulses had durations of
2540 ns and peaktopeak velocity changes of 4060 m/s.
These values were typical for the pulses generated by this experiment.
 To compare the experimental data to theory, the spectral
amplitudes from the ``source'' data at x=5 mm were propagated
using the model equations shown on the previous slide. The
dashed line shows the theoretical result of this propagation.
The waveforms match closely.
Notes on NONLINEARITY MATRIX
 With confidence that the nonlinearity was being described
correctly based on the experimental data, the properties of
the nonlinearity matrix were investigated in more detail.
 The nonlinearity matrix elements are a complicated function
of many quantities. Here the d'_{ijklmn} values are derived
from the secondorder elastic constants c_{ijkl} and
thirdorder elastic constants d_{ijklmn} as shown in the slide
while the values of l_{3}^{(s)} and q_{j}^{(s)} are
parameters of the depth dependence functions u_{nj} derived by solving
the linear problem.
 Note that the elastic constants c_{ijkl} and
d_{ijklmn} are the elastic constants in a coordinate system in
which the xaxis is parallel to the direction of the wave
propagation.
 In general the nonlinearity matrix elements
are complex. However, due to the symmetry of this particular
case all of the matrix elements are real.
 Physically, the nonlinearity matrix elements can be interpreted
in two ways. First, as will be shown on the next slide,
they can be related to an effective coefficient of nonlinearity.
Secondly, they can be interpreted as describing the strength of the
energy coupling between various harmonics of the wave.
In particular, the S_{mn} matrix element describes how
energy is transferred from the mth and nth harmonics to
the (m+n)th harmonic.
 The figure shows the matrix elements S_{11}, S_{12}, and
S_{13} plotted as a function of angle from the
á100ñ direction for SAW in Si on the (001) plane.
Other matrix elements follow a similar pattern including the same
location of zero crossings, but the curves are not
multiples of one another.
 As will be shown in subsequent slides, the nonlinear evolution
of the wave can be classified into three regions as indicated in
the figure. The nonlinearity matrix elements start negative in
Region I, pass through zero, become positive in Region II, pass
through zero again, and then become negative once again in
Region III. The strongest nonlinearity occurs in the
05^{°} and 2530^{°} regions. On the other extreme,
the waveform evolution is predicted to be linear (to third order
in the wave variables) around 21^{°} and 32^{°}. While the
matrix elements go to zero near 45^{°}, the wave also
degenerates into an exceptional bulk shear wave in that
direction and the assumptions of the theory (namely that the
amplitude of the wave decays away exponentially into the solid)
are no longer satisfied.
 As can be seen in the graph, the nonlinearity
matrix elements not only change sign twice over the interval
0^{°} < q < 45^{°} but the magnitude changes
substantially, nearly tripling in Region II. Thus the sensitivity
to changes in direction is much greater for the nonlinearity
matrix elements than in the case of the linear SAW speed.
Notes on SHOCK FORMATION DISTANCE
 While the nonlinearity matrix may be a unfamiliar concept,
it may be related to a quantity that is more familiar: the
shock formation distance.
 For an initially sinusoidal wave, the shock formation distance
can be estimated by the expression shown on the slide where
b_{x} is the coefficient of nonlinearity, e_{x}
is the acoustic Mach number or, equivalently, peak strain,
and k is the wavenumber. The coefficient of nonlinearity
contains the nonlinearity matrix element S_{11}, the density
r, and the SAW sound speed c.
 In general the nonlinearity matrix elements are complex.
Hence the coefficient of nonlinearity may also be complex.
The interpretation of a complexvalued coefficient of
nonlinearity is not clear and is a topic of current research.
 Hence the shock formation distance is inversely proportional
to the magnitude of the nonlinearity matrix element S_{11}.
The figure shown plots this estimate of the shock formation
distance on the (001) plane of silicon as a function of
direction for the case of w/2p=50 MHz and v_{x0} = 36 m/s
(e_{x} » 0.007).
 Where the nonlinearity matrix element S_{11} approaches zero,
the waveform evolution is linear and, correspondingly,
the shock formation distance approaches infinity. Where the
nonlinearity matrix element S_{11} is maximized, the shock
formation distance is minimized.
 The importance of the relationship between the nonlinearity
matrix elements and the shock formation distance is that it
allows for the prediction of regions where experiments would be
most likely to measure significant nonlinear waveform
distortion.
Notes on SIMULATIONS WITH SINUSOIDS
 The nature of the waveform distortion can be characterized
by looking at the nonlinearity matrix elements. In the center
of the figure is a reproduction of the plot of the S_{11}
matrix element as a function of propagation direction.
Waveforms are examined for two directions, 10^{°} and 26^{°}
degrees, that exhibit distortion characteristic of propagation
in their respective regions.
 First look at the 26^{°} direction in Region II.
The waveforms shown in
the top two graphs correspond to an initially sinusoidal wave
propagating along the surface in this direction. The graphs
show snapshots of the longitudinal and vertical velocity
waveforms in the retarded time frame (moving along with the
wave at the linear wave speed) at the various distances shown.
Here the distance is scaled such that X=1 corresponds to
the estimated shock formation distance of 10 mm.
 For both cases shown, there is a small transverse component.
However it is just related to the other components by a
transformation described by linear theory.
 Note that S_{11} is positive. Hence the coefficient of
nonlinearity will also be positive and the waveform should
distort like a fluid does with the peaks advancing and
the troughs receding.
 However, unlike a fluid, cusps form in the
longitudinal velocity waveform while a peak forms in the
vertical velocity waveform in the shock front region.
This distortion is characteristic of nonlinear surface acoustic waves
even in isotropic media. This occurs because the generation
of higher harmonics causes more of the energy of the wave to
be concentrated at the surface. Recall that the energy of a
sinusoidal wave is concentrated within approximately one
wavelength of the surface (see Notes on LINEAR SAW DIAGRAM).
 Now look at the 10^{°} direction in Region I. Here S_{11} is
lower in magnitude and negative. The reduced magnitude of the
nonlinearity results in an increase of the shock formation
distance to 23 mm. Because S_{11} is negative, the
coefficient of nonlinearity is also negative and the waveform
will distort in a fashion opposite of that of a fluid with the
peaks receding and the troughs advancing. This is clearly seen
in the waveforms shown.
 The waveforms in Region III distort in a way similar to
those in Region I. However, there the nonlinearity is much
weaker. This is due to the fact that in this region the
wave is gradually degenerating into a bulk wave as the
45^{°} direction is approached (see Notes on Supplement:
RESULTS FROM LINEAR THEORY and Notes on Supplement:
WAVEFORMS: REGION III) As this occurs, the
energy of the wave penetrates deeper and deeper into the solid and is
correspondingly weaker at the surface. This reduced amplitude
then makes the nonlinear effects weaker.
Notes on CONCLUSION
 The nonlinear wave propagation of surface acoustic waves in
crystalline silicon was investigated on the (001) plane over
all directions. Due to the symmetries of the crystal in this
plane, only the range 0^{°} < q < 45^{°} from the
á100ñ direction need be studied.
This is the first systematic theoretical
study of nonlinear SAW of its kind.
 It was found that the behavior of the nonlinearity matrix
elements allows the waveform distortion to be classified into
into three distinct regions:
Region  Angular range  Waveform Behavior

I  0^{°} < q < 21^{°}  Steepens ``backward''

II  21^{°} < q < 32^{°}  Steepens ``forward''

III  32^{°} < q < 45^{°}  Steepens ``backward'' 
 The theory predicted that SAW propagation in two particular
directions (q » 21^{°} and
q » 32^{°} from the á100ñ direction)
will be linear to within the accuracy of the theory
even for finite amplitude SAW.
 Nonlinearity effects vary more than linear SAW speed
as a function of direction (see the Notes on Supplement:
RESULTS FROM LINEAR THEORY).
Notes on Supplement: NONLINEARITY MATRIX & LINEAR THEORY
 This procedure will be more fully described in the forthcoming paper
[M. F. Hamilton, Yu. A. Il'inskii, and E. A. Zabolotskaya,
``Nonlinear surface acoustic waves in crystals,'' J. Acoust. Soc. Am.
(in review)].
 The elastic constants with the tilde are transformed into
the coordinate system in which the xaxis is the
direction of propagation and the zaxis is perpendicular to
the surface of the solid.
Notes on Supplement: RESULTS FROM LINEAR THEORY
 Nonpiezoelectric crystals have three bulk acoustic modes
(one longitudinal or quasilongitudinal and two shear or
quasishear) and one surface acoustic wave mode. The wave
speed for each mode is generally a nonconstant
function of the direction of propagation.
 The acoustic modes in a crystal and isotropic solid differ
in two ways: (1) for an isotropic solid the wave speeds for all
modes are independent of the direction of propagation and (2)
for an isotropic solid the two shear modes are always degenerate
in wave speed and have polarizations that are mutually
perpendicular.
 While some of the individual particle velocity
polarizations of the bulk acoustic modes may not be parallel or
perpendicular to the direction of propagation, the polarizations
of the three modes are always mutually perpendicular
[B. A. Auld, Acoustic
Fields and Waves in Solids, 1st ed. (John Wiley & Sons, New
York, 1973), pp. 219220.].
 In this case, silicon has one quasilongitudinal, one pure shear
([001]polarized), and one quasishear mode. Because the pure shear
mode has constant wave speed for all directions, it was used
as a reference velocity. Here the wave speed of the acoustic
modes relative to the pure shear wave speed are plotted as a
function of the angle of propagation q from the
á100ñ direction. Because of the symmetry of the
crystal for this particular cut and direction, the wave speeds
are periodic every 90^{°} and symmetric about q = 45.
 At q = 0, the bulk acoustic waves are similar to those in
an isotropic solid with a pure longitudinal mode and two
degenerate pure shear modes.
 For reference, the bulk shear wave speed for Si
at q = 0 is 5829 m/s while the bulk longitudinal wave speed
is 8413 m/s (based upon the McSkimin and Andreatch data).
 The surface acoustic wave (SAW) is shown as the dashed line.
It is always less than the lowest bulk wave speed.
For reference, the SAW speed at 0^{°} is 4902 m/s.
As q® 0, the SAW speed approaches the speed of
the quasishear bulk mode. At q = 45^{°}, the modes
become degenerate thereby allowing for a bulk shear wave that
satisfies the surface boundary conditions. This wave is usually
called an exceptional bulk wave or ``surfaceskimming'' bulk
wave.
 Note that the variation in the wave speed from 0^{°}
to 45^{°} is relatively small for all the modes:
Mode  Variation

Quasilongitudinal  +5%

Pure shear  0%

Quasishear  20%

SAW  8% 
In contrast, it will be shown subsequently
that the nonlinear effects change more than this linear effect
as a function of direction.
Notes on Supplement: WAVEFORMS: REGION I
 These plots show the waveform distortion due to nonlinear
processes for Si in the (001) plane in the direction 10^{°}
from á100ñ for the waveforms at the surface.
It is a typical example of surface waveforms in the region
0^{°} < q < 21^{°}. For convenience, the
plot of the nonlinearity matrix element S_{11} is reproduced
in the upper right graph. The thick solid line marks the
location of 10^{°} on the graph.
 The graphs in the left column show the x, y, and
zcomponents of the velocity. In each waveform, the
nondimensional velocity is plotted versus a nondimensional
retarded time. The initial waveform is a single frequency,
continuous wave. The velocity V is scaled such that the imaginary
part of the Fourier amplitude of the fundamental is unity, and
the retarded time t is scaled by the period of the
initial sinusoidal signal. The propagation distance X is scaled
such that X = 1 corresponds to approximately one shock formation
distance (see the slide on Shock Formation Distance). The
estimate of the shock formation distance ¯x = 23 mm.
 Because the nonlinearity matrix elements are different in each
direction, the shock formation distance is also different in
each direction. Hence it is not possible to directly compare
the waveforms at the same nondimensional distance X for
waveforms propagating in different directions.
 The waveforms clearly exhibit significant distortion and shock
formation. The longitudinal velocity component V_{x} has the
cusps near the shock front while the vertical velocity component
V_{z} has a sharp peak. Both of these effects are similar to
those observed in SAW in isotropic media
[E. A. Zabolotskaya, ``Nonlinear propagation of plane and
circular waves in isotropic solids,''
J. Acoust. Soc. Am. 91, 25692575 (1992)].
 Due to the symmetry of this cut, the waveform distortion
is symmetric. However, this is not generally true for
any cut and direction (e.g., Si in the (111) plane in the
á11¯2ñ direction).
 Because the nonlinearity matrix element S_{11} is negative,
the coefficient of nonlinearity b is also negative.
This causes the longitudinal velocity waveform to steepen
``backward'' from the way a sinusoid would steepen in a fluid
with nonlinear distortion modelled by the Burgers equation.
Here the ``peaks'' of the waveform move slower than the SAW
speed while ``troughs'' move faster.
 Unlike an isotropic medium, here the wave has a transverse
velocity component V_{y}. This effect can be more clearly
visualized by looking at a top view of the particle trajectory
(projected into the (001) plane) as shown in the figure.
Looking from above the (001) plane, the
plane of the particle motion is rotated clockwise out of the
sagittal plane by approximately f = 2^{°}.
 The overall motion of the wave can be seen in the side view of
the particle trajectory (projected into the (100) plane). The
motion is retrograde at the surface. It begins as an ellipsoid
with eccentricity e = 0.590 and distorts to more asymmetric
``egglike'' shape due to the nonlinear effects. The decrease
in the in the path length visible in the graph is due to energy
loss at the shock front.
 Each waveform was generated from a numerical calculation
using 200 harmonics. However, in order to minimize the
effects of numerical aliasing due to the finite spectral
width taken only the first 150 harmonics were used to
reconstruct the time waveform from the harmonic amplitudes
(the remaining 50 harmonics set to zero before the inverse
Fourier transform was taken).
Notes on Supplement: WAVEFORMS: REGION II
 These plots show the waveform distortion due to nonlinear
processes for Si in the (001) plane in the direction 26^{°}
from á100ñ for waveforms at the surface.
It is a typical example of surface waveforms in the region
21^{°} < q < 32^{°}. For convenience, the
plot of the nonlinearity matrix element S_{11} is reproduced
in the upper right graph. The thick solid line marks the
location of 26^{°} on the graph.
 The graphs in the left column show the x, y, and
zcomponents of the velocity. In each waveform, the
nondimensional velocity is plotted versus a nondimensional
retarded time. Note that for purposes of plotting the
waveform in a clearer way, the phase of the wave has been shifted by
p from the 10^{°} case.
The initial waveform is a single frequency,
continuous wave. The velocity V is scaled such that the imaginary
part of the Fourier amplitude of the fundamental is unity, and
the retarded time t is scaled by the period of the
initial sinusoidal signal. The propagation distance X is scaled
such that X = 1 corresponds to approximately one shock formation
distance (see the slide on Shock Formation Distance). The
estimate of the shock formation distance ¯x= 10 mm.
 Because the nonlinearity matrix elements are different in each
direction, the shock formation distance is also different in
each direction. Hence it is not possible to directly compare
the waveforms at the same nondimensional distance X for
waveforms propagating in different directions.
 The waveforms clearly exhibit significant distortion and shock
formation. The longitudinal velocity component V_{x} has the
cusps near the shock front while the vertical velocity component
V_{z} has a sharp peak. Both of these effects are similar to
those observed in SAW in isotropic media
[E. A. Zabolotskaya, ``Nonlinear propagation of plane and
circular waves in isotropic solids,''
J. Acoust. Soc. Am. 91, 25692575 (1992)].
 Due to the symmetry of this cut, the waveform distortion
is symmetric. However, this is not generally true for
any cut and direction (e.g., Si in the (111) plane in the
á11¯2ñ direction).
 Because the nonlinearity matrix element S_{11} is positive,
the coefficient of nonlinearity b is also positive.
This causes the longitudinal velocity waveform to steepen
``forward'' in the same way that a sinusoid would steepen in a fluid
with nonlinear distortion modelled by the Burgers equation.
Here the ``peaks'' of the waveform move faster than the SAW
speed while ``troughs'' move slower.
 Unlike an isotropic medium, here the wave has a transverse
velocity component V_{y}. This displacement is significantly
greater than in the 10^{°} case. This effect can be more clearly
visualized by looking at a top view of the particle trajectory
(projected into the (001) plane) as shown in the figure.
Looking from above the (001) plane, the
plane of the particle motion is rotated counterclockwise out of the
sagittal plane by approximately f = 13^{°}.
 The overall motion of the wave can be seen in the side view of
the particle trajectory (projected into the (100) plane). The
motion is retrograde at the surface. It begins as an ellipsoid
with eccentricity e = 0.659 and distorts to more asymmetric
``egglike'' shape due to the nonlinear effects. The decrease
in the in the path length visible in the graph is due to energy
loss at the shock front. Note that the small end of the ``egg''
shape of the X = 2 trajectory is on the opposite side to the
similar trajectory in the 10^{°} case.
 Because the particle motion is rotated out of the
sagittal plane, the side view as a projection is distorted
so that the elliptical motion looks narrower in the
horizontal direction.
 While the dimensionless value of the attenuation
A_{1} = a_{1}[`(x)] = 0.025 is held constant in all
the cases shown, the physical attenuation coefficient
a_{n} is changing as the shock formation distance
changes. Hence for this case the physical attenuation
coefficient is approximately twice as high as in the previous case
because the shock formation distance is approximately
half as much.
 Each waveform was generated from a numerical calculation
using 200 harmonics. However, in order to minimize the
effects of numerical aliasing due to the finite spectral
width taken only the first 150 harmonics were used to
reconstruct the time waveform from the harmonic amplitudes
(the remaining 50 harmonics set to zero before the inverse
Fourier transform was taken).
Notes on Supplement: WAVEFORMS: REGION III
 These plots show the waveform distortion due to nonlinear
processes for Si in the (001) plane in the direction 35^{°}
from á100ñ for waveforms at the surface.
It is a typical example of surface waveforms in the region
32^{°} < q < 45^{°}. For convenience, the
plot of the nonlinearity matrix element S_{11} is reproduced
in the upper right graph. The thick solid line marks the
location of 35^{°} on the graph.
 The graphs in the left column show the x, y, and
zcomponents of the velocity. In each waveform, the
nondimensional velocity is plotted versus a nondimensional
retarded time. Note that for purposes of plotting the
waveform in a clearer way, the phase of the wave has been shifted
back by p so that it is the same as in the 10^{°} case.
The initial waveform is a single frequency,
continuous wave. The velocity V is scaled such that the imaginary
part of the Fourier amplitude of the fundamental is unity, and
the retarded time t is scaled by the period of the
initial sinusoidal signal. The propagation distance X is scaled
such that X = 1 corresponds to approximately one shock formation
distance (see the slide on Shock Formation Distance). The
estimate of the shock formation distance ¯x = 110 mm.
 Because the nonlinearity matrix elements are different in each
direction, the shock formation distance is also different in
each direction. Hence it is not possible to directly compare
the waveforms at the same nondimensional distance X for
waveforms propagating in different directions.
 The waveforms clearly exhibit significant distortion and shock
formation. This is true even though the wave is almost degenerate
with the quasishear mode.
 Due to the symmetry of this cut, the waveform distortion
is symmetric. However, this is not generally true for
any cut and direction (e.g., Si in the (111) plane in the
á11¯2ñ direction).
 Because the nonlinearity matrix element S_{11} is negative,
the coefficient of nonlinearity b is also negative.
This causes the longitudinal velocity waveform to steepen
``backward'' from the way a sinusoid would steepen in a fluid
with nonlinear distortion modelled by the Burgers equation.
Here the ``peaks'' of the waveform move slower than the SAW
speed while ``troughs'' move faster.
 The top view of the particle trajectory
(projected into the (001) plane) is shown in the figure.
Looking from above the (001) plane, the
plane of the particle motion is rotated clockwise out of the
sagittal plane by approximately f = 83^{°}.
 The overall motion of the wave can be seen in the front view of
the particle trajectory (projected into the (010) plane). The
motion is retrograde at the surface. It begins as an ellipsoid
with eccentricity e = 0.956 and distorts to more asymmetric
``egglike'' shape due to the nonlinear effects.
 While the dimensionless value of the attenuation
A_{1} = a_{1}[`(x)] = 0.025 is held constant in all
the cases shown, the physical attenuation coefficient
a_{n} is changing as the shock formation distance
changes. Hence for this case the physical attenuation
coefficient is approximately eleven times less than in the
10^{°} case because the shock formation distance is
approximately eleven times more.
 Each waveform was generated from a numerical calculation
using 200 harmonics. However, in order to minimize the
effects of numerical aliasing due to the finite spectral
width taken only the first 150 harmonics were used to
reconstruct the time waveform from the harmonic amplitudes.
(the remaining 50 harmonics set to zero before the inverse
Fourier transform was taken).