[ASJ2020a] 平面アレイと鉛直スピーカ対を用いた3次元近傍エリア再生

2fb31d9e685d0533a22475951301ec67?s=47 Takuma OKAMOTO
September 10, 2020

[ASJ2020a] 平面アレイと鉛直スピーカ対を用いた3次元近傍エリア再生

2fb31d9e685d0533a22475951301ec67?s=128

Takuma OKAMOTO

September 10, 2020
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  1. ฏ໘ΞϨΠͱԖ௚εϐʔΧରΛ༻͍ͨ ࣍ݩۙ๣ΤϦΞ࠶ੜ Ԭຊ୓ຏ ৘ใ௨৴ݚڀػߏ  (FOFSBUJOH%MPDBMJ[FETPVOE[POFVTJOHBQMBOBSBSSBZ DPNCJOFEXJUIBWFSUJDBMMPVETQFBLFSQBJS UI4FQU "4+"OOVBM.FFUJOH"VUVNO! 

  2. *OUSPEVDUJPOMPDBMJ[FETPVOE[POFHFOFSBUJPO 1SFWJPVTNFUIPETBOEQSPCMFNT 1SFWJPVTQSPQPTFENFUIPEBOEQSPCMFN50LBNPUP 8"41"" 1SPQPTFENFUIPE $PNQVUFSTJNVMBUJPO $PODMVTJPOT 0VUMJOF  (b)

    Exterior sound field synthesis + (c) Proposed method D D D0 D0 rQ (a) Monopole sound field Residual sound field for r < rQ ˇ B0 n for r > 0 − ˇ B0 n for r > rQ BOE
  3. 4PVOE" 4PVOE" 4PVOE# B C 4JMFOU[POF 4PVOE" 4JMFOU[POF D ଟνϟωϧεϐʔΧΛ༻͍ͨہॴ࠶ੜɾϚϧνεϙοτ࠶ੜ

    B ہॴ࠶ੜ ͋ΔྖҬʹͷΈ͋ΔಛఆͷԻ͕ฉ͑͜ɼଞͷ৔ॴͰ͸ฉ͑͜ͳ͍ C Ϛϧνεϙοτ࠶ੜ ͦΕͧΕͷྖҬʹͦΕͧΕผͷԻ͕ฉ͑͜Δ D ۙ๣ΤϦΞ࠶ੜ εϐʔΧΞϨΠͷۙ๣ͰͷΈԻ͕ฉ͑͜ɼ཭ΕΔͱٸܹʹݮਰ Ԡ༻ྫ ύʔιφϧΦʔσΟΦγεςϜ ඒज़ؗͳͲͷଟݴޠΨΠυ ͦͷଞόʔνϟϧϦΞϦςΟٕज़ ɹɹɹɹɹɹɹˠϔουϗϯ૷ணͳ͠ *OUSPEVDUJPO 
  4. Τόωοηϯτ೾࠶ੜʹجͮ͘ํࣜ)*UPVFUBM 8"41"" *$"441 ࣍ݩΞϨΠͰ࣍ݩۙ๣ΤϦΞ࠶ੜɿ(PPEؔΒɼԻڹࢽ ߴ͍प೾਺͸΄ͱΜͲಧ͔ͳ͍ɿ#BE ΞϨΠؒͷ࣍ݩϛεϚονʹجͮ͘ํࣜ50LBNPUP +*).41 *8"&/$ ߴ͍प೾਺΋੍ޚՄೳɿ(PPE Ԗ௚ํ޲ʹ͸Ի͕์ࣹͯ͠͠·͏ɿ#BE

    ҟͳΔΞϨΠΛ༻͍Δํࣜ50LBNPUP *$"441 +*).41 ࣍ݩۙ๣ΤϦΞ࠶ੜΛ࣮ݱɿ(PPE ΞϨΠܗঢ়͕ෳࡶˠ࣮૷ʹ͸ෆ޲͖ɿ#BE 1SFWJPVTNFUIPETBOEQSPCMFNT  z = 0 [m] [dB] -1 0 1 2 3 4 y [m] -2 -1 0 1 2 x [m] -40 -35 -30 -25 -20 -15 -10 -5 0 Y axis [m] X axis [m] z = 0 [m] [dB] ï2 ï1 0 1 2 ï2 ï1.5 ï1 ï0.5 0 0.5 1 1.5 2 ï40 ï35 ï30 ï25 ï20 ï15 ï10 ï5 0 Y axis [m] X axis [m] z = 0 [m] [dB] ï2 ï1 0 1 2 ï2 ï1.5 ï1 ï0.5 0 0.5 1 1.5 2 ï30 ï25 ï20 ï15 ï10 ï5 0 Y axis [m] X axis [m] [dB] 1 2 3 4 ï2 ï1.5 ï1 ï0.5 0 0.5 1 1.5 2 ï50 ï40 ï30 ï20 ï10 0 ߴ͍प೾਺ʹ΋ରԠͨ࣍͠ݩΞϨΠΛ༻͍ͨ࣍ݩۙ๣࠶ੜ๏ͷఏҊ
  5. ฏ໘ෳ਺ԁܗΞϨΠΛ༻͍ͨ࣍ݩۙ๣ΤϦΞ࠶ੜ50LBNPUP 8"41"" த৺ͷ఺Իݯͷ์ࣹԻ৔ΛपΓͷෳ਺ԁܗΞϨΠͰ૬ࡴ B ɿݪ఺ʹ഑ஔͨ͠ແࢦ޲ੑԻݯΛ࠶ੜ શ͕ͯ֎෦Ի৔  C ɿ B

    ͷԻ৔ͷٯҐ૬Λฏ໘ෳ਺ԁܗΞϨΠͰ࠶ੜ ֎෦Ի৔ͱ಺෦Ի৔ͷࠞ߹ܥ  D ɿΞϨΠ൒ܘͷ࠷େʹରԠ͢Δ൒ܘͷٿͷ಺෦ͰͷΈԻѹ ಺෦Ի৔ ͕࢒Δ ໰୊఺ɿෆ҆ఆͳԻѹ෼෍ܗঢ় த৺ͷ఺Իݯ͸࣍ͷٿ໘ௐ࿨εϖΫτϧͷΈˠ์ࣹ੒෼͕ΞϨΠ಺෦Ͱ΋ଧͪফ͞Εͯ͠·͏ͨΊ 1SFWJPVTQSPQPTFENFUIPEBOEQSPCMFN  (b) Exterior sound field synthesis + (c) Proposed method D D D0 D0 − ˇ B0 0 = − jk √4π for r > rQ rQ ˇ B0 0 = jk √4π for r > 0 (a) Monopole sound field Residual sound field for r < rQ ࠶ੜྖҬʹ͓͚ΔԻѹ෼෍ܗঢ়ͷ҆ఆԽ͕՝୊
  6. ฏ໘ෳ਺ԁܗΞϨΠΛ༻͍ͨ࣍ݩۙ๣ΤϦΞ࠶ੜʹ͓͚ΔԖ௚εϐʔΧରͷಋೖ Ԗ௚εϐʔΧରͷ์ࣹԻ৔ΛपΓͷෳ਺ԁܗΞϨΠͰ૬ࡴ Ԗ௚εϐʔΧରಋೖͷϙΠϯτ ߴ࣍ͷٿ໘ௐ࿨εϖΫτϧ΋ؚ·ΕΔˠΞϨΠ಺෦ͷ์ࣹԻ৔͕ଧͪফ͠ਏ͘ͳΔˠ࠶ੜྖҬͷ҆ఆԽ ์ࣹԻ৔͸ґવͱͯ͠ɹɹɹɹ͕ۮ਺੒෼ͷΈˠฏ໘ΞϨΠͰͷ૬ࡴ͕Մೳ ࣠ର৅ͷԻ৔ˠ֤ԁܗΞϨΠ͸࣍ͷ࣍ݩԁ౵ௐ࿨εϖΫτϧͷΈˠಉҰΞϨΠͰͷ৴߸͸શͯ౳͍͠ ˠͭͷԁܗΞϨΠΛ༻͍ͨ৔߹ɿ ֤ԁܗΞϨΠ  Ԗ௚ର

    νϟωϧͷ৴߸ %"ม׵ث Ͱ੍ޚՄೳ 1SPQPTFENFUIPE  (b) Exterior sound field synthesis + (c) Proposed method D D D0 D0 rQ (a) Monopole sound field Residual sound field for r < rQ ˇ B0 n for r > 0 − ˇ B0 n for r > rQ |m| + n <latexit sha1_base64="9ePhyBN+VlciO3WdBtf6oHgMzME=">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</latexit> z <latexit sha1_base64="2nAWqti7tIwnqDQtKHSZtxGcVsA=">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</latexit>
  7. ݪཧɿෳ਺ԁܗεϐʔΧΞϨΠΛ༻͍ͨ࣍ݩԻ৔੍ޚ ֎෦Ի৔ͷٿ໘ௐ࿨ల։ ෳ਺ԁܗΞϨΠʹΑΔԻ৔ ෳ਺ԁܗΞϨΠͷۦಈ৴߸ɿٿ໘ௐ࿨εϖΫτϧͷۮ਺੒෼ͷΈ ैདྷ๏ɿݪ఺ͷ఺ԻݯʹΑΔԻ৔Λ૬ࡴ͢Δํࣜ ݪ఺ʹ഑ஔͨ͠఺ԻݯʹΑΔԻ৔ ෳ਺ԁܗΞϨΠͷۦಈ৴߸ 1SFWJPVTGPSNVMBUJPO  S(r,

    ✓, ) = 1 X n=0 n X m= n ˇ Bm n hn(kr)Y m n (✓, ) <latexit sha1_base64="CGN8lKd9T9sqiB0gRW/T9PgOq6Y=">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</latexit> ˇ Bm n + 2⇡jk Q X q=1 ˚ Dm,qjn(krq)P|m| n (0) = 0 <latexit sha1_base64="TiKjBu+6sjWVr0l4xLtD9UbCnQY=">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</latexit> ˚ D even m = 1 2⇡jk ⇣ UT |m| U|m| ⌘ 1 UT |m| ˇ Beven m <latexit sha1_base64="YKPL9ethyxwVPaHP8G5o8gGuD6I=">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</latexit> S0(r, ✓, ) = D0G(r, r0 = 0) = D0 0 X n=0 n X m= n jk p 4⇡ hn(kr)Y m n (✓, ) = D0 ejkr 4⇡r <latexit sha1_base64="WnOgS4xieexwhNZ+QB4yrAVJRDw=">AAADNnichVHPaxNBFH67/qrxR1O9CF4W00oCMcy2gYoQKCjosW1MW8kmYXczScbszq6zk2gd5h/wH/DgSUFE/Ae8e/HqwUMv3kTEi1DBiwffbhZFi/UtO/O9773vvTczXhywRBKyZ5hHjh47fmLuZOHU6TNn54sL57aSaCJ82vKjIBI7npvQgHHakkwGdCcW1A29gG574+tpfHtKRcIiflvuxrQTukPOBsx3JVK94juH0/t+FIYu7yvHY7ptd5QTetED9KKgH7pyZC2W7EWtC82eIrosqo4cUelWLScesUrjRsreLKdiS+iqSN0GqViFWcRJJmFP8QbR3V9e2LjC0eXaGQjXV3fHWjnJPSFV3YkZNhqhQJfHonInBV0V6nLeM2tpNaxZ6UxNu1hAaJ2JcYJesURqJDPrILBzUILc1qPiC3CgDxH4MIEQKHCQiANwIcGvDTYQiJHrgEJOIGJZnIKGAmonmEUxw0V2jOsQvXbOcvTTmkmm9rFLgL9ApQVL5D15SfbJW/KKfCY//llLZTXSWXZx92ZaGvfmH11ofv+vKsRdwui36tCZJQzgajYrw9njjElP4c/004eP95vXNpfUZfKMfMH5n5I98gZPwKff/OcbdPMJFPAB7L+v+yDYWq7ZK7XljXpprZ4/xRxchEtQxvtehTW4BevQAt9YNTrGwBiar80P5kfz0yzVNHLNefjDzK8/AaLd2b4=</latexit> ˇ Beven 0 =  jk p 4⇡ , 0, · · · , 0 T <latexit sha1_base64="pxkvuknhk/nTxxjvL3Pbm7o/6zE=">AAADAHichVE7bxNBEJ47XsE8YqBBojnhBCiItRcsgZCQImgo83ISyWusu/XaXnx3e+ytTcJqG0r+AEUqkBAgCkSbNg1/IEV+QkQZJBoK5s6HEETAnG7nm2/nm5ndDdNIZJqQfcc9dvzEyVNTpytnzp47P129cHEtkyPFeJPJSKqNMMh4JBLe1EJHfCNVPIjDiK+Hwwf5/vqYq0zIZFVvpbwdB/1E9AQLNFKd6jZN+FMm4zhIuoaGwrb8tqFxKDcxklE3DvTAm6n5M9ZWKBtwNjR5mnff2o4h9pGhmm9qw8c8wYx7czTiPd2iPRUw83hoDc2eKG0aNBXW3vQM9axHJp6yrtTZT44q0R/oNtbLO6rYrGKDao3USWHeUeCXoAalLcrqW6DQBQkMRhADhwQ04ggCyPBrgQ8EUuTaYJBTiESxz8FCBbUjzOKYESA7xLWPUatkE4zzmlmhZtglwl+h0oNZskfek0PymXwgB+T7X2uZokY+yxb6cKLlaWf6xeWVb/9Vxeg1DH6p/jmzhh7cKWYVOHtaMPkp2EQ/fvbycOXu8qy5Rl6TLzj/K7JPdvEEyfgre7PEl7ehgg/g/3ndR8HafN2/VZ9fatQWGuVTTMEVuAo38L5vwwI8hEVoAnMc57pDHN997r5zP7qfJqmuU2ouwW/m7vwA21LE9A==</latexit> ˚ D0 = D = 1 8⇡2 ⇣ UT 0 U0 ⌘ 1 J0 <latexit sha1_base64="iEDOXFLrWkDjDl1dxo5heWkj2sk=">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</latexit> J0 = [j0(kr1), j0(kr2), · · · , j0(krQ)]T <latexit sha1_base64="49qrYBZ/6YOMPVIDQ//fitxaRiE=">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</latexit> ɿ࣮਺ ɿ௨ৗ͸ෳૉ਺ S1(r, ✓, ) = Q X q=1 Z 2⇡ 0 D(rq, 0) G(r, rq)| ✓0=0 d 0 = 1 X n=0 n X m= n 2⇡jk Q X q=1 ˚ Dm,q P|m| n (0)jn(krq)hn(kr)Y m n (✓, ) <latexit sha1_base64="RiA0IsyMkowjM0T44I31QDvJk/Y=">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</latexit>
  8. ݪཧɿෳ਺ԁܗεϐʔΧΞϨΠΛ༻͍ͨ࣍ݩԻ৔੍ޚ ֎෦Ի৔ͷٿ໘ௐ࿨ల։ ෳ਺ԁܗΞϨΠʹΑΔԻ৔ ෳ਺ԁܗΞϨΠͷۦಈ৴߸ɿٿ໘ௐ࿨εϖΫτϧͷۮ਺੒෼ͷΈ ఏҊ๏ɿԖ௚εϐʔΧରʹΑΔԻ৔Λ૬ࡴ͢Δํࣜ ্࣠ͷʹ഑ஔͨ͠Ԗ௚εϐʔΧରʹΑΔԻ৔ ෳ਺ԁܗΞϨΠͷۦಈ৴߸ 1SPQPTFEGPSNVMBUJPO  S(r,

    ✓, ) = 1 X n=0 n X m= n ˇ Bm n hn(kr)Y m n (✓, ) <latexit sha1_base64="CGN8lKd9T9sqiB0gRW/T9PgOq6Y=">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</latexit> ˇ Bm n + 2⇡jk Q X q=1 ˚ Dm,qjn(krq)P|m| n (0) = 0 <latexit sha1_base64="TiKjBu+6sjWVr0l4xLtD9UbCnQY=">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</latexit> ˚ D even m = 1 2⇡jk ⇣ UT |m| U|m| ⌘ 1 UT |m| ˇ Beven m <latexit sha1_base64="YKPL9ethyxwVPaHP8G5o8gGuD6I=">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</latexit> ɿ७ڏ਺ S1(r, ✓, ) = Q X q=1 Z 2⇡ 0 D(rq, 0) G(r, rq)| ✓0=0 d 0 = 1 X n=0 n X m= n 2⇡jk Q X q=1 ˚ Dm,q P|m| n (0)jn(krq)hn(kr)Y m n (✓, ) <latexit sha1_base64="RiA0IsyMkowjM0T44I31QDvJk/Y=">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</latexit> ɿ௨ৗ͸ෳૉ਺ S(r, ✓, ) = 1 X n=0 jn(kr0)hn(kr)Y 0 n (0, 0)Y 0 n (✓, ) + 1 X n=0 jn(kr0)hn(kr)Y 0 n (⇡, 0)Y 0 n (✓, ) <latexit sha1_base64="m5O+41KEOffqI2choIC45P9Owlg=">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</latexit> = jk 1 X n=0 r (2n + 1) ⇡ jn(kr0)hn(kr)Y 0 n (✓, ) <latexit sha1_base64="fJkVP9NrJMnBW0kkXS0WLTHWjBE=">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</latexit> ˇ Beven 0 = jk p ⇡ h j0(kr0), p 5j2(kr0), · · · , p 2N + 1jN (kr0) iT <latexit sha1_base64="kRUUCMdFEnBvNZokpmYSLDAL3uA=">AAADN3ichVHLahRBFL3dvuL4yKgbwU3jJBJRh+ohYhSEQTeuQl6TBKYmQ3dNzXRl+mV1zZhY1A/4Ay5cKYiIX+DajWvBRVbuRHElEdy48HbPoCTBeJvue+vcc26fqvLTUGSKkB3LPnL02PETEydLp06fOTtZPnd+NUsGkvEGS8JErvtexkMR84YSKuTrqeRe5Id8ze/fz/trQy4zkcQrajvlrcjrxaIrmKcQapc/0Jg/YkkUeXFHU1+YptvSNPKTLVwlYSfyVOBMVdwpY0qUBZz1dU5z7hnT1sRsaKr4ltJ8yGNj7t6gXekxvdk3mmYPpdI0Fbkw5F3V3MwFM32Zp6vXHU0d44xYNw32avt6lHUSle3h1eavuTl1/g+VStELVAtt5EZlpFfQV7lCqqQI52DhjosKjGMhKb8CCh1IgMEAIuAQg8I6BA8yfJrgAoEUsRZoxCRWouhzMFBC7QBZHBkeon389nDVHKMxrvOZWaFm+JcQX4lKB6bJR/Ka7JL35A35Sn79c5YuZuRetjH7Iy1P25NPLi7//K8qwqwg+Ks61LOCLswVXgV6Twsk3wUb6YePn+4u31ma1lfIC/IN/T8nO+Qd7iAe/mAvF/nSMyjhBbj7j/tgsVqrurPV24uzlXp9fBUTcAkuwwye9y2owwNYgAYwa87asHpWYL+1P9mf7S8jqm2NNRdgT9jffwO0p9v5</latexit> ˚ D even 0 = D = 1 2⇡jk ⇣ UT 0 U0 ⌘ 1 UT 0 ˇ Beven 0 <latexit sha1_base64="wfbBqaLZMiZLzg4gqhTfbR6x6aQ=">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</latexit> z <latexit sha1_base64="2nAWqti7tIwnqDQtKHSZtxGcVsA=">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</latexit> ±r0 <latexit sha1_base64="+iofLvTkUcmiNY4B+ZX25jt7MXk=">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</latexit>
  9. ଟ఺੍ޚʹجͮ͘ฏ໘෼ࢄεϐʔΧʹΑΔ࣮ݱ ΞϨΠத৺ͷεϐʔΧ܈ͷ์ࣹԻ৔ΛपΓͷฏ໘෼ࢄεϐʔΧͰ૬ࡴ εϐʔΧ܈ͷԻ৔͸ɹɹฏ໘ ্Լ ͰରশͱͳΔΑ͏ʹ഑ஔ ੍ޚ఺Λ൒ԁ্ʹۉҰʹ഑ஔ ೚ҙͷฏ໘഑ஔ ฏ໘෼ࢄεϐʔΧ Ͱͷ࣮ݱ͕Մೳ "MUFSOBUJWFNFUIPE

     x-y <latexit sha1_base64="RD7RruvvLGLlGceDl+W+qRiQAGM=">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</latexit>
  10. ΞϨΠ഑ஔ ΞϨΠ൒ܘɿɼɼN ैདྷ๏ த৺఺Իݯ ɿ   νϟωϧ ఏҊ๏ Ԗ௚εϐʔΧର

    ɿ   νϟωϧ Ԗ௚εϐʔΧ഑ஔɿœN ଟ఺੍ޚ ฏ໘෼ࢄεϐʔΧ ɿ   νϟωϧ ԁܗΞϨΠʹରͯ͠ฏۉ෼ࢄΛ෇Ճ  ଟ఺੍ޚʹ͓͚Δ੍ޚ఺഑ஔ ੍ޚ఺਺ɿ ൒ܘNͷ൒ٿ্ʹҰ༷഑ஔ  Ի଎ɿNT ࣗ༝Ի৔ ࠶ੜप೾਺ɿ)[ 4JNVMBUJPODPOEJUJPOT 
  11. த৺఺Իݯ഑ஔͷ৔߹ εϐʔΧରΛ༻͍ͨ৔߹ 4JNVMBUJPOSFTVMUT  Իѹ෼෍ܗঢ়ͷ҆ఆԽˠఏҊ๏ͷ༗ޮੑΛ֬ೝ ͲΕ΋ෆ҆ఆͳԻѹ෼෍ܗঢ় ैདྷ๏ɿෳ਺ԁܗΞϨΠ ଟ఺੍ޚɿෳ਺ԁܗΞϨΠ ଟ఺੍ޚɿฏ໘෼ࢄεϐʔΧ ఏҊ๏ɿෳ਺ԁܗΞϨΠ

    ଟ఺੍ޚɿෳ਺ԁܗΞϨΠ ଟ఺੍ޚɿฏ໘෼ࢄεϐʔΧ ଟ఺੍ޚɿฏ໘෼ࢄεϐʔΧ ฏ໘εϐʔΧର
  12. ฏ໘ΞϨΠͱԖ௚εϐʔΧରΛ༻͍ͨ࣍ݩۙ๣ΤϦΞ࠶ੜ Ԗ௚εϐʔΧରͷಋೖʹΑΔ࠶ੜྖҬͷԻѹ෼෍ܗঢ়ͷ҆ఆԽ ଟ఺੍ޚํࣜͰͷ࣮ݱ ܭࢉػγϛϡϨʔγϣϯʹΑΓఏҊ๏ͷ༗ޮੑΛ֬ೝ ँࣙ ຊݚڀͷҰ෦͸+414Պݚඅ+1,ͷॿ੒Λड͚ͨ΋Ͱ͋Δ ͍ΘΏΔج൫$ $PODMVTJPOT  (b)

    Exterior sound field synthesis + (c) Proposed method D D D0 D0 rQ (a) Monopole sound field Residual sound field for r < rQ ˇ B0 n for r > 0 − ˇ B0 n for r > rQ BOE