Acoustic waves propagation from underground wave guide
S.A.Kostarev*, S.A.Makhortykht
*Laborato~
of Vibration andA coustics, Tunnel Association, 21, Sadovaya-Spasskaya
Moscow, Russia 107217
flnstitute oJMathematical Problems in Bio[o@, Russian Academy of Sciences,
Pushchino Moscow reg., Russia 142292
Str.,
Abstract: An enough simple model of underground acoustic wave guide is proposed. A speeial structure of
nonhomogeneity of subsurface layer in the soil may lead to the capture of acoustic waves within a channel. bl this
mse a sound signal will propagate with a sufficiently smrdi decay in comparison with homogeneous media (due to
the lack of discrepancy of the wave tiont). b this w vibration is observed on much more distanws frc]m the
source. A maximal value of acoustic level is reached on the axis of a wave guide, k practiw this effect may be
significant for the ease of vibration generation for buildings on the piles. Under these conditions enhaneed values
of vibration may take place even when the vibration of the ground surface is negligible.
A MODEL OF UNDERGROUND WAW GU~E
For friable, powdery, plastic and water-saturated wils we sM1 use an approximation of the liquid media
c, << cl.
(1)
show that
Here c, and c1 are velocities of transverse and longitude waves in the ground. M~surements
inequation (1) is vtiid for the broad class of soils in the city renditions.
We assume that the sound speed in soil depends on the vertical coordinate. Mong with the gened
approach we sM1 consider typical city conditions for an upper soil part with a s~Klc wave guide sound
veIoeity distribution (three layers model)
~+z%-cl
h’
-~<z<o
(2)
Here c is longitude wave speed, z is depth in the soil, h is a proper layer’s boundary depth.
Parameters values are as follows
h’=lOm, hl=.2m,
hm=’ 4 m, co= 700 tis, c, =’ 900 @s,
Cml” =
’420 dS, CIO “ 91O dS.
We seek an expression for an acoustic field in the following form
p = Aq (z) exp[i(~ x - @ t)],
here A is an amplitude,
~ =&l + i~2 is a complex wave number (&2> O), @is an an~ar
(3)
wave frqueney,
i is
unit (i2 =- 1). In this case we did not fix dependence c@), so that deseribed method is applicable
for arbitrary sound speed distribution. Function q(z) is unknown.
An acoustic pressure ~z) in soil obeys the following equation (here we disregarded density \ariations
with the depth ~z))
an imaginary
(4)
C+= o /&1 is an eigenvatue of the phase veloeity in wave guide, We have for frquency 31.5 fi
MS and for frequeney 125 W C4= 779.1 ~s.
2299
C$= @/ El
ANALYSIS ~SULTS
Caldations were carried out for three basic frequency bands 31.5, 63 and 125 Hz. A distribution of
vibration magnitudes within this region is descri~ by the static equations and is determined by the lwal
strata positions and source type. For us~ soft soil an approximation of incompressible fluid is relevant.
In calculations for realistic cases we must tie into account an effwt of energy dissipation in the media
by means of the term exp(-dw) in the expressions for computed values (K is x<omponent of wave vector and
r is the distance from source to considered point).
The typical computed distribution of round wave amplitude is shown in Figure 1~). Near the surfa~
value MO) is stil and mtimrd value is rmched at some depths h) ad hz. An acoustic signal is capturd by
the wave guide and its d-y is connected ody with dissipation,
For the frequencies greater than 31.5 W we have a capture of acoustic wave by the channel. This wave
is the most important component of the total field produced by underground railways. Its dependence on the
depth is presented in Figure 1 ~= 31.5 and 125 ~), the first line is the amustic pressure p and the second
line is the value of —
~ (it is proportional to z-mportent
of vibration acceleration). For the frequency 16 Hz
an effect of signal capture is missing and we can use for approtitnate calctiations a model of homogeneous
media.
Horizontal and vertical components of vibration acceleration levels were calculated in 100 poi }\ts at the
various depths from the stiace to the depth 10 m in the soil. The distance from the source (underground
tunnel in this case) is 30 m.
values is presented in Table 1. Measurerncllts were
A comparing of these restits with experiment
tied
out in the city conditions,
T~LE
1. Calculations and measurements of vibration acceleration (3 1.5-63 Hz),
Frequency @)
h acceleration x- component
An acceleration z- componen(
level (dB)
31.5
63
level (dB)
measurements
23
26
calculations
22
24
dculations
20
30
I
measuremtll 1s
23
26
I
I
~
3“
0.4
6
0.3
4
0.2
&2
so
0.1
-2
0
(
—
-4
-2
-0,1
l—w
Z (m)
FIGW
!—P I I
L
Z (m)
—w
—P
—
“-d
b)
1. Calculation results for model (4) dependence of the pressureP and vibration acceleration Won the depth
(in fictitious units) a)f=31.5 Hz, f=125 Hz,
2300