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a standing sound wave in a tube with an open end has an antinode bit outside the end point .
such a small distance is called the open end correction ( hereafter abbreviated as oec ) .
the end correction @xmath0 was calculated for an infinite flanged circular tube by rayleigh @xcite , and after that , h. levine and j. schwinger calculated the unflanged end correction as a function of wavelength @xcite .
oec for an infinite flanged circular tube as a function of wavelength was determined by y. nomura et al . by the radiation impedance method @xcite .
the recent application of oec was given by m. s. howe @xcite ( and related references therein ) .
the idea of obtaining oec as discussed by rayleigh comes from the fluid mechanical consideration @xcite , @xcite .
let us suppose a half infinite space with a connected circular tube filled with nonrotational and noncompressive fluid , and assume a piston moving at a speed of @xmath1 in the tube as seen in fig .
then , the total kinetic energy of the fluid is given by the definition of oec .
@xmath2    where @xmath3 is the fluid mass density , @xmath4 is the velocity potential , @xmath5 is the cross - sectional area of the tube , and @xmath0 is the oec . the integration region is taken toward the piston inner circular tube and half infinite 3d space .
the region with a nonzero velocity is not only in the tube but is also spread around a branching bay with a speed lower than @xmath1 .
such a region contributes to oec .    in this study
, we calculate oec quite differently . as shown in fig .
2 , we place a circular well at @xmath6 with a radius @xmath7 and a depth @xmath8 , and consider the particle diffusion of density @xmath9 .
we put a particle bath with a density @xmath10 at the bottom of the well ; we thus have @xmath11 at infinity . by using the above boundary conditions
, we approximately solve the static diffusion equation under the condition that one - dimensional diffusion is occurring in the well . by using a solution for a diffusion field , we define oec and show that its value is @xmath12 , consistent with that given by rayleigh and nomura et al .
, when the tube radius is small enough compared with the wavelength @xmath13 @xcite .
in the well , we approximate the diffusion process by using a one - dimensional diffusion equation ( see fig .
this is very simple , but is essential for our further discussion . at the top of the well , the space is widely opening and it is therefore possible to suppose that @xmath14
. then , the constant diffusion flow comes up from the bottom . @xmath15
our purpose is to correct this flow formula .
+ we take @xmath16 and then obtain the diffusion field and constant diffusion flow .
@xmath17 @xmath18 is the number of particles supplied per unit time from the well to the open region while leaving @xmath19 undetermined .
then , we solve the diffusion equation in open space with the boundary conditions    @xmath20    @xmath21    the additional condition @xmath22 is given below using the solution of the diffusion equation in an open region .
then , two diffusion fields in the inner and outer wells are connected , and the diffusion equation is completely solved .
we consider the diffusion equation @xmath23 in the open region @xmath24 .
the boundary conditions are ( [ eq : bc1 ] ) and ( [ eq : bc2 ] ) .    .
the particle server is at the bottom .
particles diffuse from the bottom of the well into the outer region.,width=226 ]    by supposing an axial symmetry , we obtain the following static diffusion equation for the diffusion field @xmath25 :    @xmath26    then , we have a solution in the form    @xmath27    on the plane @xmath28 , the diffusion flow in the @xmath29 direction is given by    @xmath30    @xmath31 should be selected to obtain the boundary condition ( [ eq : bc2 ] ) .
the following formula solves the problem .
@xmath32    from ( [ eq : bc2 ] ) , ( [ eq : flow ] ) , and ( [ eq : formula ] ) , we determine the function @xmath31 given as @xmath33    then , we obtain the following solution for @xmath25 :    @xmath34    next , we must determine @xmath35 . from ( [ eq : density ] ) ,
we obtain @xmath36    . at @xmath37
, we have a nonzero density .
this shows that particles diffused from the well and distributed around it .
the square line shows the mean @xmath38.,width=264 ]    the function @xmath39 is expressed by the following hypergeometric function : @xmath40 the form of this function is given in fig .
the smoothed curve is the function @xmath39 . to obtain a consistent solution with a diffusion field in the well , the matching condition ( [ eq : bc4 ] )
is required at @xmath41 . however , as shown in figure 3 , the distribution at @xmath42 is not constant .
this shows that our solution ( [ eq : density ] ) is not consistent with the inner - well solution , since our approximated solution ( [ eq : innerwell ] ) counts the diffusion out in the @xmath43-direction . to further achieve what in this approximation
, we approximate the form of @xmath39 by its mean in the region @xmath42 .
@xmath44    this mean is shown by a solid square line in fig .
3 . then , an additional boundary condition ( [ eq : bc4 ] ) leads to @xmath45    from ( [ eq : innerwell ] ) and ( [ eq : boundary1 ] ) , we obtain    @xmath46    this result should be compared with the second equation of ( [ eq : innerwell_1 ] ) ; it shows that the effective depth of the well is slightly larger than @xmath8 and that oec is defined by    @xmath47    this is the same oec for sound in a long - wavelength limit @xcite .
the point @xmath48 upwards of the well is the place where the density @xmath25 vanishes effectively .
note that , from eq .
( [ eq : boundary1 ] ) oec is defined by @xmath49 where @xmath50
why do we have oec for sound in a long - wavelength limit by our method ?
we discuss this point and show that the definition of oec by the radiation impedance method is quite similar to that given in this paper .
first we review the radiation impedance method of obtaining oec .
let us put a disk as a sound source oscillating in the @xmath51 direction and having a radius @xmath7 on the surface @xmath28 .
the sound emitted from the source satisfies the wave equation ,    @xmath52    where @xmath53 is the pressure field and @xmath54 is the speed of sound .
the linearized euler equation is necessary to consider the speed of sound source : @xmath55 where @xmath3 is the mass density .    from the solution of eq .
( [ eq : wave ] ) and ( [ eq : euler ] ) , we obtain the fluid velocity @xmath56 .
the boundary conditions for @xmath53 at the surface @xmath28 are as follows :    @xmath57    where the time dependence is supposed to be @xmath58 .    from the emitted sound @xmath59
, we obtain the force of reaction @xmath60 to the sound source as    @xmath61    then , the radiation impedance is defined by @xmath62 from the parallelism to the mechanical impedance , the additional inertial mass due to the radiation is defined using the maginary part of radiation impedance .
@xmath63    this additional mass is supplied by oec as    @xmath64    so , we obtain oec by using the formula    @xmath65    where @xmath66    now , let us check the similarity of this oec in a diffusion process
. the long - wavelength limit is given as    @xmath67    then eq . ( [ eq : wave ] ) leads to the laplace equation . the linearized euler equation ( [ eq : euler ] )
should be compared with the equation of a diffusion flow .
@xmath68    therefore , we have the correspondences ; @xmath69 and @xmath70 .
the boundary condition ( [ eq : bc5 ] ) corresponds to @xmath71 which is the same as ( [ eq : bc2 ] ) .    the definition of oec as eq .
( [ eq : oec4 ] ) corresponds to @xmath72 this is the same as the definition of oec for the diffusion equation ( [ eq : oec3 ] ) .
therefore , the equations , boundary conditions , and definitions of oec are the same in the long - wavelength limit of these two methods .
this is the reason why we can calculate oec of sound waves simply using a diffusion equation .
the difficulty in obtaining oec is quite simplified in the long - wavelength limit by considering the diffusion equation .
the authors would like to thanks prof.yokoyama in gakushuin university for the helpful suggestion .
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in the physics lessons on waves and resonance phenomena in high school and college , we usually consider sound waves in a tube with open or closed ends @xcite . 
 however , it is well known that we need a tube with open end correction @xmath0 . the correction for a flanged circular tube 
 was first given by rayleigh @xcite and experimentally checked by several authors @xcite . in this paper 
 , we show the different methods of obtaining the end correction for a circular tube by a diffusion process .