A square wave shape is used in the Pestorius algorithm to calculate the risetime of a step shock in the atmosphere. These results agree closely with steady shock calculations. The healing distance of perturbed shocks due to finite wave effects is then investigated for quasi-steady shocks. Perturbed 100 Pa shocks require on the order of 1.0 km travel distance to return to within 10 percent of their steady shock risetime. For 30 Pa shocks, the minimum recovery distance increases to 3.0 km. It is unlikely that finite wave effects can remove the longer risetimes and irregular features introduced into the sonic boom by turbulent scattering in the planetary boundary layer

Bass, Henry

Raspet, Richard

Wu, Wenliang

Yao, Lixin

English

NASA Technical Reports Server

N92-33882.
STEADY STATE RISETIMES OF SHOCK WAVES IN THE ATMOSPHERE
Richard Raspet, Henry Bass, and Lixin Yao
Department of Physics and Astronomy
University of Mississippi
University, MS 38677
Wenliang Wu
Department of Electric al Engineering
University of Mississippi
University, MS 38677
SUMMARY
A square wave shape is used in the Pestorius algorithm to calculate the risetime of a step shock in
the atmosphere. These results agree closely with steady shock calculations. The healing distance of
perturbed shocks due to finite wave effects is then investigated for quasi-steady shocks. Perturbed 100
Pa shocks require on the order of 1.0 km travel distance to return to within 10% of their steady shock
risetime. For 30 Pa shocks the minimum recovery distance increases to 3.0 km. It is unlikely that finite
wave effects can remove the longer risetimes and irregular features introduced into the sonic boom by
turbulent scattering in the planetary boundary layer.
INTRODUCTION
In a previous paper 1 we compared the risetimes calculated by the enhanced Pestorius 2,3,4 algorithm
with risetimes calculated using the augmented Burger's 5,6,7 equation under the assumption of a steady
step shock. Good agreement was obtained if the N-wave used in the enhanced Pestorius algorithm had
a duration on the order of 100 times the risetime.
Sparrow 8 has applied his numerical method for general finite amplitude wave propagation to the
propagation of square pulses as displayed in Fig. 1. These pulse shapes are particularly useful for
performing the comparison described in Ref. 1 since the shock front is a better approximation of a steady
PR'EC._DiNG PA_ 8LANK NOT FILMED
109
https://ntrs.nasa.gov/search.jsp?R=19920024638 2020-03-17T10:00:46+00:00Z
shock than a long duration N-wave. In addition, if the calculation is performed without geometric
spreading, the shock overpressure will remain constant until the central linear position of the wave is
convected to the shock front (see Fig. 1). ta may be calculated using weak shock consideration as
pc2To
t a --
413Po
(1)
where P0 is the overpressure, T O is the duration, c is the speed of sound, p is the density of air, and 13
= 1.2.
We have also used this waveform to investigate the distance a perturbed shock must propagate
before its risetime approaches its steady state value. Plotkin and George 9 suggest that finite wave effects
might significantly shorten the risetimes after scattering has initially increased them.
CALCULATIONS
The numerical algorithm which includes vibrational relaxation absorption and dispersion 3 was
applied to the square wave form displayed in Fig. 1. The initial rise portion of the wave was modeled
by a hyperbolic tangent as illustrated in Fig. 2 for the initial wave shape. This rise shape is the solution
to the Burger's equation for a steady shock in a non-relaxing media.
4-1 I II
100'
• RH =30N
Po= I0 Pa
80- rt=4ms
_60-
_, 4o-
.
20-
0 4
110
/" 7-
!
t i
80km] 16km[ /Okm
I i
i I
t J
, //II
, ////
/ l
i i i
Time from first point
in numerical domain [ms]
140
Figure 1. Square shock propagation without
geometric spreading.
Figure 2. Development of the shock front with
distance for a square wave with a 10
Pa overpressure at 295°K and a rela-
tive humidity of 30%.
110
For all calculations the 250 ms duration waveform is embedded in 125 ms of zeroes in front of and
behind the pulse to allow for non-linear duration changes and to minimize sampling errors. 8192 points
are used to assure that the rise portion of the wave contains sufficient points to model the physical
properties accurately. We note that 8192 points were only required for 100 Pa overpressure waves but
this number of points was used in all calculations.
RISETIME OF STEADY SHOCKS
The numerical algorithm was started with a hyperbolic tangent rise portion with a best guess risetime
for three steady state overpressures (100, 30 and 10 Pa), two relative humidities (30% and 10%) and a
temperature of 295°K. These conditions were chosen for comparison with Kang and Pierce's
calculations 6,7 and with our earlier work with non-steady shocks 2 in the atmosphere. The shocks were
propagated until a steady risetime was achieved.
In some cases the wave had to be propagated extreme distances to achieve an equilibrium risetime
since the rise shape was quite different from the initial hyperbolic tangent. We will discuss this in more
detail in the next section.
Table I contains the risetimes computed from the enhanced Pestorius algorithms with results read
from Fig. 5.12 of Kang's dissertation. These values are read from a log-log scale and are only accurate
to two significant digits.
Table I. Comparison of rise times computed using a square wave in the
enhanced Pestorius algorithm with Kang's steady state calculations.
10% Relative Humidity
30% Relative Humidity
Pressure
Risetime (ms)
square wave steady state
100 Pa 1.00 1.10
30 Pa 3.90 4.70
10 Pa 11.80 13.60
100 Pa 0.45 0.34
30 Pa 1.40 1.60
10 Pa 4.10 4.90
111
The two results agree within about 20%, confirming the validity of both methods as means of
estimating the risetimes of steady shocks. This relatively good agreement is in contrast to the comparison
of the steady state risetimes with the risetime of spherically decaying relatively short duration explosion
waves calculated using the enhanced Pestorius algorithm. The risetimes of the spherically decaying short
duration waves are significantly shorter than the square wave or steady state risetimes.
HEALING DISTANCE OF PERTURBED WAVES
Square pulses were started with a range of risetimes about the equilibrium risetimes listed in Table I.
All waves had a hyperbolic tangent rise function. Figure 3 displays the results for 30% relative humidity
and 30 Pa overpressure. The pulse started with a 1.5 ms risetime, dips to a lower risetime, then
approaches a 1.4 ms risetime. The pulse started at 1.0 ms has a risetime within 0. I ms of the 1.5 ms
risetime pulse after propagating about 2.5 km, while the pulse started at 2.0 ms is within 0.1 ms after
4.0 km.
The healing distance is longer for atmospheric conditiotas with larger attenuation. The 30 Pa pulses
with a 10% relative humidity started with 4.0 ms and 3.5 ms risetimes which agree within 0.1 ms by
10 kin; pulses started at 4.5 ms and 4.0 ms are within 0.1 ms at 12 kin. The behavior of the
risetime versus distance for different starting risetimes is displayed in Fig. 4.
E
2.02"5"_t I i i i i i
1.5I_
1.0
0.5
o.o 5',o16.o lg.O26.0
Distance [kml
23.0 30.0
E
5°5"
5.0-
4.5'
r
\
I ,, I I I I
4.0 t
30_ N'--J ,
0.0 5.0 10.0 15.0 20.0 25.0 30.0
Distance lkml
Figure 3. Development of risetime for a 30 Pa
overpressure shock wave for T=295°K
and a relative humidity of 30%.
Figure 4. Development of risetime for a 30 Pa
overpressure shock wave for T=295°K
and a relative humidity of 10%.
112
Thedip in risetimedisplayedin Figs. 3 and4 occursbecausethesteadystaterise function doesnot
resemblea hyperbolictangentfunction.Figure5 displaysthe behaviorof therisetimefor thecasewhich
displayedthelargestrelativedecreasein risetime. Therisetimefor the 10%relativehumidity caseat 100
Padropsfrom 1.0ms to 0.5 ms,then risesto 1.0ms at a distanceof 5 kin. The developmentof the
rise function with distancefor this caseis shownin Fig. 6. The 1.0ms risetimeat 5.0 km is determined
principally by thefoot of thewave. This portionof thewaveis dueto velocity dispersioncausedby
molecularrelaxation. This time developmentof thewaveshapeshouldbecontrastedwith Fig. 2, where
thefinal waveformapproximatesthehyperbolictangentfunction.
_E
2.0
1.5
1.0
0,5 ¸
I I I |
\
0.0
o.o 11o 2;0 a'.o 41o s.o
Distance [km]
100-
80-
_60-
_40-
es
20-
0 [,!
110
Figure 5. Development of risetime for a 100 Pa Figure 6.
overpressure shock wave for T=295°K
and a relative humidity of 10%.
I I I
RH =10_
Po = 100 Pa
rt=l.0ms
5.0kin
| I I I
P"_ r-
1.0
km
Okm
, J/
i I
' 1'_0 130 140
Time from first point
In numerical domain [ms]
Development of the shock front with
distance for a square wave with 100
Pa overpressure at 295°K and a rela-
tive humidity of 10%.
Table IIcontains a chart of the approximate healing distance for a shock risetime perturbed by 0.5
ms to return to within 0.1 ms of a shock started with the correct risetime (but not the final rise shape).
These distances are an average of the distance for a pulse started with the equilibrium risetime plus 0.5
ms and a pulse started with the equilibrium risetime minus 0.5 ms.
Table II. Approximate healing distance for different shocks perturbed by .5 ms in risetime
RelativeHumidity Pressure Distance (km)
10%
30%
100 0.8
30 10.0
10 70.0
100 0.8
30 3.0
10 35.0
113
CONCLUSION
The enhanced Pestorius algorithm applied to square pulses agrees closely with steady shock
calculations of risetimes. The use of the square pulse is convenient for comparisons.
The square pulses were also used to examine the finite wave healing of perturbed pulses. This
rough calculation shows that if scattering in the planetary boundary layer induces a longer risetime in a
sonic boom, the longer risetime will still be present at the ground. Finite wave calculations on the
scattered waveforms predicted by Yao 1° will be performed in the near future.
Another implication of the large healing distance is that atmospheric conditions at higher altitudes
may determine quiet atmosphere risetimes at the ground.
REFERENCES
. Richard Raspet and Henry E. Bass, "Comparison of sonic boom risetime prediction techniques,"
13th Aeroacoustics Conference of the American Institute of Aeronautics and Astronautics,
Tallahassee, FL, October 1990.
2. Henry E. Bass and Richard Raspet, "Vibrational relaxation effects on the atmospheric attenuation
and risetimes of explosion waves," J. Acoust. Soc. Am. _ 1208-1210 (1978).
3. Henry E. Bass, Jean Ezell and Richard Raspet, "Effect of vibrational relaxation on risetimes of
shock waves in the atmosphere," J. Acoust. Soc. Am. _ 1514-1517 (1983).
4. Henry E. Bass, Bruce A. Layton, Lee N. Bolen, and Richard Raspet, "Propagation of medium
strength shock waves through the atmosphere." J. Acoust. Soc. Am. _ 306-310 (1987).
5. Allan D. Pierce,"Weaker sonic booms may be considerably more quiet," J. Acoust. Soc. Am. 86,
$30 (1989).
6. Jongmin Kang and Allan D. Pierce, "Propagation of nonlinear transients, such as sonic booms,
through an absorbing and relaxing atmosphere," J. Acoust. Soc. Am. _ $81 (1989).
. Jongmin Kang, "Nonlinear acoustic propagation of shock waves through the atmosphere with
molecular relaxation," a dissertation in Mechanical Engineering, The Pennsylvania State University,
May 1991.
114
8. Victor W. Sparrow, "Time domain computations in nonlinear acoustics without one way
assumptions," 3rd IMACS Symposium on Computational Acoustics, Boston, MA, 26-28 June.
9. K.J. Plotkin and A.R. George, "Propagation of weak shock waves through turbulence," J. Fluid
Mech. _ 449-467 (1972).
10. Lixin Yao, Henry E. Bass and Richard Raspet,"A statistical study of the relaxation between weather
conditions and sonic boom waveforms," J. Acoust. Soc. Am. _ S1 (1991).
115


Steady state risetimes of shock waves in the atmosphere

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