(1-7) Asymptotic and exponential approximation for the
Rubinowicz line integral
By Y. Sakurai, K. Takahashi (Yamaha) and H. Morimoto
The asymptotic expansion of the
Kirchhoff-Rubinowicz line integral for the boundary diffraction wave from an
edge of infinite length was obtained by the stationary phase method. The
integral was approximated by two exponential terms in the range where the
method was not applicable and a numerical solution was found for the integral.
From the behavior of the boundary
diffraction wave traveling along the edge, the contribution of the two ends of
an edge having limited length was easily estimated, i.e. the wave loses half
its amplitude on meeting the nearest end and vanishes at the other.
The double reflection of a
boundary diffraction wave at two limited edges in an enclosure was calculated
by an extension of Keller's geometrical acoustical treatment and the method was
compared with the numerical double line integral where each element of the
first boundary diffraction wave was treated as a directional source.
In the sound field of
an enclosure, there is no diffraction into the rear surface of each boundary
and its rear velocity potential can be treated as zero. The surface integral of
each term in a successive substitution of the boundary integral equation1)
by T.Terai corresponds to a multiple reflection and is transformed into a line
integral by the Rubinowicz transformation, along the edges of each of the plane
surfaces that compose the enclosure. The line integral along an edge is called
a boundary diffraction wave, which has been called a boundary wave and is
changed to distinguish from the reflection at a boundary in an enclosure, and
has been calculated numerically by dividing the edge into the small segments
where the integrand of the line integral can be regarded as constant and then
summing the small segment contributions2). However, when the number
of plane surfaces in an enclosure increases, the numerical calculation of the
boundary diffraction waves is enormously time consuming not only if we consider
multiple reflections but also for the first reflections alone.
For the line integral
of an edge with infinite length, its approximation was first obtained as an
asymptotic expansion using the stationary phase method3) and the
limit of its application was investigated. Here, the numerical calculation was
simulated by an approximation using two exponential terms.
The behavior of the
travelling wave along the edge was investigated and the contribution at the
ends of an edge was explained. We show that the boundary diffraction wave from
an edge with limited dimension is easily estimated. The double boundary
diffraction wave reflection was calculated by the convolution of the effects of
two edges of limited dimension applying the expanded treatment of Keller's
geometrical acoustics4). This was compared with the result obtained
by the numerical double line integral5) to find a good
1) Calculation of a boundary diffraction wave from an
edge of infinite length
1-1) Asymptotic expansion of the stationary phase
Each term of the
successive substitution in the boundary integral equation corresponds to the
reflection of a surface whose velocity potential in its rear is treated as zero2).
Such a reflection, H (P,ω), is given
by the Rubinowicz line integral (see Fig.1),
Fig.1 Boundary diffraction wave along an edge
of infinite length.
where k is the
wave number, r and rs are vectors from the
receiving point and the point source to a line element dg, respectively. ε
takes 1/2 when the angle between r
and rs is π or one of
them is zero. When a specular reflection point is on the surface, ε takes one.
It takes zero when the specular reflection point is outside the plate. The
impulse response, h(P,t), is found by
the inverse Fourier transform of Eq.(1),
where δ(t) is the Delta function and C is sound velocity.
Eq.(1) can be
rewritten as follows,
where n and ns are inward normal on the planes which are composed by
the edge, and the receiving point and the source, respectively.
When the integrand of
Eq.(3), f(g), changes smoothly, the
stationary phase method which we shall refer to as S.P.M. in this paper is
applicable. Using the formula3) in Eq.(4), the contribution, Hs(P,ω), from the boundary is
obtained as in Eq.(5),
go is the point where the distance between the point
source and the receiving point through the edge is shortest.
By taking an
inverse Fourier transform, the impulse response, hs(P,t), is obtained as,
phase point (S.P. point) go on the edge that connects the point
source and the receiving point via the shortest distance (ro+rso) was estimated by numerical
1-2) Approximation by exponential terms
When the denominator
of f(g) in Eq.(3) becomes small, it
does not change smoothly along the edge. In practice, if the denominator is in
the range from 0.0 to 0.5, Eq.(4) can not be applied. When the denominator
becomes zero because of the angle term or r
and/or rs, , the line
integral jumps to 1/27).
However, it is
interesting that even in the inapplicable range, the peak value at (ro+rso)/C in the
time domain as determined by numerical calculation of Eq.(2) takes the same as
that by the S.P.M.. This means that even if the function f(g) changes abruptly around the S.P. point, the integration
around the point is precisely estimated by the S.P.M.. It should be noticed
that when the discrete numerical calculation, such as a discrete sampling for
FFT processing, is referred to, Eq.(6) must be integrated in the sampled time
interval where the comparison is being made.
Therefore, the ratio
of Eq.(6) to Eq.(2) was calculated in the time domain, and this is shown in
Fig.2. The ratio is always unity at the time t = (ro+rso)/C and then
Possibility of the approximation by exponential functions
integrand f(g) versus g at each curve
is drawn in the figure. The horizontal line of unity shows when the calculation
by S.P.M. in Eq.(6) is the same as that by Eq.(2).
gradually. We have attempted to simulate this decreasing function, which is a
correction function, E(t), for Eq.(6) in order to obtain Eq.(2), by the
summation of two exponential terms. E(t) is multiplied to hs(P,t) to have
We note that Eq.(2) has parameters cos (r, rs) and r
rs which change the integral and since it can
be shown that they change the integral independently, E(t) was thought to be
separable into two functions E1(t) and E2(t), for cos (r,rs) and r
rs, respectively. Namely,
and E2(t) were simulated by two exponential terms as follows; for
the case cos(r, rs) <
for the case
coefficients were found by a comparison with the numerical calculation of
Eq.(2) using the least squares and the resultant expressions are given in
Eqs.(10) to (15),
Fig.3 shows how
they change depending on each parameter.
Fig.3 Six factors
at the simulation using exponential functions for the terms depending on cos (r, rs) and r
i) Change of A1, B1
and C1 to cos(r, rs).
ii) Change of A2, B2
and C2 to
When cos(r, rs) approaches -1, A1, B1 and C1
change rapidly. When cos(r, rs)
is -1, the correction function is zero, and the line integral along the edge
becomes zero and ε takes 1/2 in Eq.(2). For the square root of the parameter r
rs in (ii), B2 and C2
become small when it is non-zero. When it is zero, ε in Eq.(2) takes 1/2.
In order to see how
well the simulation using the correction function E(t) having six coefficients,
compares with a precise result from Eq.(2), a number of examples were
calculated and compared as shown in Fig.4.
of the line integral in Eq.(2) by the exponential functions in Eqs.(6) - (15);
a solid line is calculated by Eq.(2) and a dotted one by Eqs.(6)-(15).
2. The boundary diffraction wave from an edge of limited
contribution in Eq.(2) from the edge is given at the S.P. point go at the time (ro+rso)/C, i.e.
via the shortest path between source and receiver. After that moment, the two
parts of the edge either side of go
produce Delta functions received at the receiver with the same amplitude as
that received at (r+rs)/C.
Namely, the integrand
in Eq.(2) is symmetrical regarding (r+rs)
after the S.P. point for any geometrical condition of point source, receiving
position and edge location, and its integral gives the same amplitude at the
same time from each side.
When plane wave
incidence is applicable for Eq.(1), e.g. in the far field, the
diffraction, ha(P,t), along the edge from g1 to g2 in Fig.1 is
approximated as follows8),
where U(t) is the
unit step function, and
Where C is a sound
velocity. From the equation, it is evident that we have double the amplitude of
the unit step function at the moment t=(ro+rso)/C
and then loses as each end is reached.
When the boundary
diffraction wave along a limited edge is numerically calculated from Eq.(2),
the same behavior is evident i.e. it loses half its amplitude at the near end
and half at the other end thus the two techniques of calculation agree on the
contribution of the ends of an edge. When a wave front arrives at one end, the
contribution from that side vanishes leaving only one half of the amplitude of
the boundary wave along the infinite edge while the wave travels to the other
end, and no contribution occurs after that. This is illustrated in Fig.5.
diffraction wave from an edge of limited length
i) In the case when a S.P. point is on
ii) In the case when a S.P. point is
outside the edge.
3. Double reflection of a boundary diffraction wave
reflection of a boundary diffraction wave for the situation shown in Fig.6 was
investigated, where a point source is at Ps
and a receiving point is at P.
The sound field is
identical with that obtained with the source at the image position Ps' and the receiving point
at the image position P'. Because
each small line element of an edge can be treated as a point source for the
boundary diffraction wave, the second reflection of the boundary diffraction
wave can be calculated as boundary diffraction at the
second plate edge
of the source which has the directivity given by the effect of the first boundary
edge. It is expressed by the following double line integral5),
reflection of a boundary diffraction wave at two edges
i) The double line integral method
treating the edge element as the
ii ) "Convenient" method
based on an expansion of geometrical acoustic
calculation time for this integral is long.
However, it is
possible to perform a more convenient calculation using the Keller's
geometrical acoustic treatment4). First, the shortest path A-B-C
which connects the source and receiving positions crossing at g1O and
g2O on each edge respectively is found. The point P" is on the
extension of the line
B and has the same distance from P'
to g2O. The boundary
diffraction wave over the first edge at the point P" from Ps' is
calculated, and the decrease due to distance and phase shift between
eliminated. The result we have called the diffraction factor q1(P", t).
factor is applied to the strength of the source for the second edge,
Ps", which is on the extension of
the line B and at the same distance from Ps
to g1O, the boundary
diffraction wave is the double reflection hb(P,
t) over the two edges. When the decrease with distance is expressed by
d(P, t) and the diffraction factor of
the second edge by q2(P,t)
, hb(P, t) is given by the
where * indicates
convolution, and q1(P",
t) and q2(P, t) are
calculated by the method in the section 2.
explanation for the convolution in Eq.(18) is shown in Fig.7.
of the diffraction factors at two edges based on the "convenient"
i) In the case where the crossing
points on the two edges are within their lengths.
ii) In the case where one of them is
outside the edge but on its extension.
iii) In the case where the crossing points
are outside both edges.
(i) shows the case where the crossing points on the two edges are within
their lengths, (ii) shows the case where one of them is outside the edge but on
its extension, and (iii) is the case where the crossing points are outside both
edges. The shortest path of course must be expressible analytically, but it was
estimated numerically for this work.
The impulse response
of the boundary diffraction wave from each edge of infinite length is
calculated by the asymptotic expansion of Eq.(6) or the exponential
approximation with both Eqs.(6) and (7). Following the method in section 2, the
impulse response for each edge of limited dimension was calculated. And the
double reflection of a boundary diffraction wave between the two edges was
obtained by the convolution in Eq.(18). The double reflection thus calculated
was compared with the double line integral of Eq.(17) and this comparison is
shown in Fig.8.
reflection of boundary diffraction waves between two plates calculated by the
two different methods.
line shows that by the double line integral and a dotted line that by the
i) Between two parallel plates.
ii) Between two plates which have an
There is a fairly
reasonable correspondence between the two. The reason for the discrepancy at
low frequencies is the slight path difference at the ends, between the two
method in Eqs.(17) and (18). Boundary diffraction wave from the end of an edge
is usually smaller and the discrepancy gives only a little difference.
The calculation of
the Rubinowicz line integral for the boundary diffraction wave of an edge of
infinite length was achieved by both an asymptotic expansion and an exponential
diffraction wave along an edge of limited dimension was found to lose half of
its corresponding amplitude of the infinite edge when reaching the nearest end
and the remaining half at the other end.
A double reflection
at the edges of two plates was estimated using the boundary diffraction wave
thus calculated and applying an extension of Keller's geometrical acoustic
treatment. This method will be useful for the calculation of multiple
reflections of boundary diffraction waves in an enclosure.
The authors are
grateful to Dr. E.Walerian, Associate Professor at the Polish Academy of
Sciences, for her helpful discussions on the asymptotic expansion and to Dr.
George Dodd, Senior Lecturer at the Acoustic Research Center, Auckland
University, for his help with English translation.
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