(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 approximation.

1) Calculation of a boundary diffraction wave from an edge of infinite length
1-1) Asymptotic expansion of the stationary phase method
     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,


The stationary 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 calculation of was also numerically obtained.

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

Fig.2 Possibility of the approximation by exponential functions

The 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).

decreases 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 h(P,t).
     We note that Eq.(2) has parameters cos (r, rs) and rrs 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 rrs, respectively. Namely,


and E1(t) and E2(t) were simulated by two exponential terms as follows; for the case cos(r, rs) < -0.5,


for the case < 200(cm),

The six 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),





       (14), and


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 rrs

  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 rrs 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.

Fig.4 Simulation 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 dimension.
     The first 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.

Fig.5 Boundary diffraction wave from an edge of limited length
  i) In the case when a S.P. point is on the edge.
  ii) In the case when a S.P. point is outside the edge.

3. Double reflection of a boundary diffraction wave
      The double 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),

Fig.6 Double reflection of a boundary diffraction wave at two edges
  i) The double line integral method treating the edge element as the
    secondary source.        
  ii ) "Convenient" method based on an expansion of geometrical acoustic


Unfortunately, the 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 them is eliminated. The result we have called the diffraction factor q1(P", t). Now, this 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 following,


where * indicates convolution, and q1(P", t) and q2(P, t) are calculated by the method in the section 2.
     The schematic explanation for the convolution in Eq.(18) is shown in Fig.7.

Fig.7 Convolution of the diffraction factors at two edges based on the "convenient" method.
  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.

Fig.8 Double reflection of boundary diffraction waves between two plates calculated by the two different methods.
A solid line shows that by the double line integral and a dotted line that by the "convenient” method.
  i) Between two parallel plates.
  ii) Between two plates which have an open angle.             

     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 approximation.
     The boundary 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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