Cargese Cours1 pdf - ESPCI Paris · 2019. 8. 28. · AAACxHicjVHLSsNAFD2Nr1pfVZdugkVwVZIq6M6CIC5bsA+oRZLptIZOHkwmQin6A27128Q/0L

?<latexit sha1_base64="pgRW1/f8nr/FpnZWoo62PsBNreU=">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</latexit>


Asymptotic homogenization<latexit sha1_base64="u4boJcmpqAeatoGvcpmWkm487bk=">AAAC33icjVHBTttAEH2YtlBKS9re2suqERIna20Smkg9pOqlxyA1JBIgZG+WsIrttex1pRQhceOGuPYHei1/g/oH8BfMbh0JDlG7lu3ZN++93ZmJ80SVhvM/S97yk6fPVlafr71Yf/lqo/H6zV6pq0LIgdCJLkZxVMpEZXJglEnkKC9klMaJHMbTLzY//C6LUunsm5nl8jCNJpk6ViIyBB013n0uZ2lutFGCnehUT2SmfrgcO2o0uc/DTrsVMu6Hbd4NuhS0edDdabHA5241Ua++btzgAGNoCFRIIZHBUJwgQknPPgJw5IQd4pSwgiLl8hJnWCNtRSxJjIjQKX0ntNuv0Yz21rN0akGnJPQWpGTYJI0mXkGxPY25fOWcLbrI+9R52rvN6B/XXimhBieE/ks3Z/6vztZicIyOq0FRTblDbHWidqlcV+zN2YOqDDnkhNl4TPmCYuGU8z4zpyld7ba3kcvfOqZF7V7U3Ap39pY04PkU2eJgL/SDbT/cDZu9T/WoV/EeH7BF8/yIHr6ijwF5n+MXfuPai7wL79K7+kv1lmrNWzxa3s97Tn6azA==</latexit>

Homogeneous medium<latexit sha1_base64="l9FDRxWxiFhEPhn1ixiE4GJcDU0=">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</latexit>











interface<latexit sha1_base64="4qN+FRES7D04kaXgJhHkrTvMAGI=">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</latexit>




1) For which kind of structures ?<latexit sha1_base64="k/v7kFklU/h++aaSVZnfCxz2AAc=">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</latexit>

3) Examples (local resonance or not)<latexit sha1_base64="k8k4v/Ke3nDFy3ks4FEdRdHJe14=">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</latexit>

2) Propagation and boundary layer e↵ects<latexit sha1_base64="eNczI/IOVCpQx4blUb3sMGK+Pa8=">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</latexit>


interface<latexit sha1_base64="4qN+FRES7D04kaXgJhHkrTvMAGI=">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</latexit>




periodic structures<latexit sha1_base64="gGvESjMrSQzIDUw+EJ3BD0Qz7sI=">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</latexit>





k = 2⇡/�<latexit sha1_base64="ZdPMRa6XHzh9a6RWRuJm+WJllAA=">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</latexit>

h<latexit sha1_base64="Kf6jojaTLedPp6aYDrV3fc6o2xI=">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</latexit>

e<latexit sha1_base64="fKgjUWlhndpgSHPja9pQkMwxKcM=">AAACxHicjVHLSsNAFD2Nr1pfVZdugkVwFdJasN0VBHHZgn1ALZJMp3XoNAnJRChFf8Ctfpv4B/oX3hlTUKTohCR3zj3nzNx7/UiKRLnuW85aWV1b38hvFra2d3b3ivsHnSRMY8bbLJRh3PO9hEsR8LYSSvJeFHNv6kve9ScXOt+953EiwuBazSI+mHrjQIwE8xRBLX5bLLlOpV5zz+r276DsuGaVkK1mWHzFDYYIwZBiCo4AimIJDwk9fZThIiJsgDlhMUXC5DkeUCBtSixODI/QCX3HtOtnaEB77ZkYNaNTJL0xKW2ckCYkXkyxPs02+dQ4a3SZ99x46rvN6O9nXlNCFe4I/Uu3YP5Xp2tRGKFmahBUU2QQXR3LXFLTFX1z+1tVihwiwnQ8pHxMMTPKRZ9to0lM7bq3nsm/G6ZG9Z5l3BQf+pY04MUU7eVBp+KUz5xKq1pqVLNR53GEY5zSPM/RwBWaaBvvJzzjxbq0pJVY6RfVymWaQ/xY1uMnEZqPvQ==</latexit>

kh =<latexit sha1_base64="6oR5MM0u9/Wq6DVJHQlQXE4WVkY=">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</latexit>





k = 2⇡/�<latexit sha1_base64="ZdPMRa6XHzh9a6RWRuJm+WJllAA=">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</latexit>

e<latexit sha1_base64="fKgjUWlhndpgSHPja9pQkMwxKcM=">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</latexit>









typical wavelength � spacing of the structure<latexit sha1_base64="/DILowx0rwFXBVymb0W5uURZXUM=">AAAC93icjVHLThsxFD1MaQn0QYAlG4ukUlfRJK0ESyQ2XaYSSZAgQh7jTCw8D9keUIT4j+66q7rtD7Bt/wDxB/AXXJtB4iHUejQz5557z7Gvb1JqZV0cX81Fr+Zfv1loLC69fff+w3JzZXVoi8oIORCFLsxewq3UKpcDp5yWe6WRPEu0HCXHOz4/OpHGqiLfdbNSjjOe5mqiBHdEHTZ7RFKg2Sk/kVrmqZuy9kGatpktuVB5yooJc1PJrDOVcJWR7LDZijtxWOw56NaghXr1i+YlDnCEAgIVMkjkcIQ1OCw9++giRkncGGfEGUIq5CXOsUTaiqokVXBij+mbUrRfsznF3tMGtaBdNL2GlAwfSVNQnSHsd2MhXwVnz77kfRY8/dlm9E9qr4xYhymx/9LdV/6vzvfiMMFW6EFRT2VgfHeidqnCrfiTswddOXIoifP4iPKGsAjK+3tmQWND7/5uechfh0rP+ljUtRVu/ClpwN2n43wOhr1O93On963X2v5Sj7qBdWzgE81zE9v4ij4G5P0dF/iDv9Es+hH9jH7dlUZztWYNj1b0+xapY6N9</latexit>

h<latexit sha1_base64="1s9K20MusN/DXg5qgRUhUjA0NQ0=">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</latexit>

� = 2⇡/k<latexit sha1_base64="e1buJI4CwhTbAm/uZrHvWp2TLGY=">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</latexit>





for kh ⌧ 1<latexit sha1_base64="mND8mBWy+0Y31YsHO6/CAd6ow/E=">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</latexit>













Other strategies for finite wavelengths ! Course of Bojan Guzina<latexit sha1_base64="3yuRKiD79njvs62rE5Gl1WOa3QE=">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</latexit>









Other strategies for finite wavelengths ! Course of Bojan Guzina<latexit sha1_base64="3yuRKiD79njvs62rE5Gl1WOa3QE=">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</latexit>


Other strategies for quasiperiodic structures<latexit sha1_base64="g95Gm+0ZPwpdcxe3sIzDEXtiHsM=">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</latexit>

! Course of Sebastien Guenneau<latexit sha1_base64="3DXXVyFtV2Rk7CHswAzzOYlPSAY=">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</latexit>












In its most classical form, the homogenization interrogates<latexit sha1_base64="HG6ZqYs+dmU1az4xEDX0q9HSVS4=">AAADF3icjVFNaxUxFD0dv2qt+tSlm+Cj4KI85r0WdFlwU3dVfG2hFsmkd+aFZiZDkinU4g/xn7jrTty6Kt24tf4Lb9J5UJGiGSa5Ofeck9zcojXahzw/X8hu3Lx1+87i3aV7y/cfPBw8erztbecUTZU11u0W0pPRDU2DDoZ2W0eyLgztFIevYn7niJzXtnkXjlvar2XV6FIrGRj6MHj7uhE6eFFbH4Qy0ntOGVFaV6+KMCMxs7WtqNEfk0DoJpBztpKBvFihsiQV9BGJ1tlWVr3pMB+t53GIv4PxKK35EP3YsoMzvMcBLBQ61CA0CBwbSHj+9jBGjpaxfZww5jjSKU/4hCXWdswiZkhGD3mueLfXow3vo6dPasWnGP4dKwVWWGOZ5ziOp4mU75JzRK/zPkme8W7HvBa9V81owIzRf+nmzP/VxVoCSrxMNWiuqU1IrE71Ll16lXhzcaWqwA4tYzE+4LzjWCXl/J1F0vhUe3xbmfIXiRnRuFc9t8OveEtu8LyL4vpgezIar40mbybDjfW+1Yt4imd4zv18gQ1sYgtT9v6CH/iJi+xzdpp9zb5dUrOFXvMEf4zs+28GRrGW</latexit>

e↵ective propagation<latexit sha1_base64="AnUUw7rBoIsMrzeNbwSgCbKOecg=">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</latexit>



e↵ective parameters aeff (tensor) and beff (scalar)<latexit sha1_base64="H0Ct7zR3h1BZB5RVTd40zZ53kks=">AAADI3icjVFNT1QxFD08BREBR126aZwxgc3kzUCiSxI2LjFxgITBSV/nPmh4X2n7iJPJ/Bz/iTt2xsS4cA9u9Qd4W98kGiDYl9fennvOaW9vUmXaujj+thDdu7+49GD54cqj1bX1x60nT/dtWRtFA1VmpTlMpKVMFzRw2mV0WBmSeZLRQXK26/MH52SsLot3blLRcS5PCp1qJR1Do9Z7SlNSTp+TqKSROTkmi84wl+7UplM5Gw0dfXDTodPFRDB51hEbjgpbmk0hi7EQneQmilUyk2Zz1GrH3e3YD3E96HXDGrfRjL2y9RVDjFFCoUYOQgHHcQYJy98ReohRMXaMKWOGIx3yhBlWWFszi5ghGT3j+YR3Rw1a8N572qBWfErGv2GlwEvWlMwzHPvTRMjXwdmjt3lPg6e/24TXpPHKGXU4ZfQu3Zz5vzpfi0OK16EGzTVVAfHVqcalDq/iby7+qsqxQ8WYj8ecNxyroJy/swgaG2r3bytD/jIwPer3quHWuPK35AbPuyhuD/b73d5Wt/+2397Zblq9jOd4gQ3u5yvs4A32MGDvT/iBn/gVfYwuos/Rlz/UaKHRPMM/I/r+G6iGtX8=</latexit>

a and b vary in space<latexit sha1_base64="/pszchpj+TpB4rdIPbqRoQseAus=">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</latexit>



constant in space<latexit sha1_base64="lArihoJvv/6KucoZVMnywz2wdMk=">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</latexit>


div(arp)� b@2p

@t2= 0

<latexit sha1_base64="6VlcEZt1KVFGnjEhTLhRZfG7UtU=">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</latexit>

div(aeffrp)� beff@2p

@t2= 0

<latexit sha1_base64="1yLsWth50cz9JfvNjKg1pTRlOus=">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</latexit>













The (classical) homogenization in the bulk assumes an infinite medium





. . .<latexit sha1_base64="J6DPfzsKIzov9cixPbkCulk6PY0=">AAACynicjVHLSsNAFD2Nr1pfVZdugq3gKqStYLsruHHhooJ9QFskSac1NC9mJkIp7vwBt/ph4h/oX3hnTEGRohOS3Dn3nDNz73WTwBfStt9yxsrq2vpGfrOwtb2zu1fcP+iIOOUea3txEPOe6wgW+BFrS18GrJdw5oRuwLru9ELlu/eMCz+ObuQsYcPQmUT+2PccSVC3PBjFUpRviyXbqjbqdq1h/g4qlq1XCdlqxcVXDDBCDA8pQjBEkBQHcCDo6aMCGwlhQ8wJ4xT5Os/wgAJpU2IxYjiETuk7oV0/QyPaK0+h1R6dEtDLSWnihDQx8TjF6jRT51PtrNBl3nPtqe42o7+beYWEStwR+pduwfyvTtUiMUZd1+BTTYlGVHVe5pLqrqibm9+qkuSQEKbiEeU5xZ5WLvpsao3QtaveOjr/rpkKVXsv46b4ULekAS+maC4POlWrUrOq19VS8ywbdR5HOMYpzfMcTVyihbau8gnPeDGuDG7MjPkX1chlmkP8WMbjJ6ofkfA=</latexit>. . .

<latexit sha1_base64="J6DPfzsKIzov9cixPbkCulk6PY0=">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</latexit>

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and the e↵ective parameters are deduced from so-called ”cell problems”








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cell problem (bulk)<latexit sha1_base64="KeDifXK9/mckSkmlOJUyqIRlyAg=">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</latexit>










. . .<latexit sha1_base64="fCNaoJA1KgXnBw+UgdDIArWclQU=">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</latexit>


elementary problem (boundary)<latexit sha1_base64="Y2uY0WCkdwYjXmHUZsdkOe0HuHQ=">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</latexit>






A particular case is that of small thickness of the structure<latexit sha1_base64="f762cJPVK1yzmCW2VI6NFUTMOiE=">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</latexit>



















Other strategies of homogenization ! Course of Berangere Delourme<latexit sha1_base64="+pQVOPgGD8Q6zM6m44EAe2Zopoc=">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</latexit>






2 di↵erent questions that have to be addressed<latexit sha1_base64="08LhRnhZ+0BniRhRkavG7V2ZB/c=">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</latexit>




Asymptotic analysis : ⌘ = kh ! 0<latexit sha1_base64="MJuQTBoTjuAfs4DvgYruD+TVw40=">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</latexit>



Asymptotic analysis : ⌘ = kh ! 0<latexit sha1_base64="MJuQTBoTjuAfs4DvgYruD+TVw40=">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</latexit>

We can expect resonances of the resulting medium:<latexit sha1_base64="6vFvQIIXBcziYztE5Nhx66m5EvA=">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</latexit>

• if resonances take place in the resulting structure<latexit sha1_base64="GCEwbdAxggy05z8bmmrUCw0PTpk=">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</latexit>

Faraday cage, FP interferometer · · ·<latexit sha1_base64="vFzppt8PVzeXoxE4MnFFwifB3eE=">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</latexit>

• if a single inclusion supports resonances<latexit sha1_base64="iuqmumrsmxHaU2TZHG9oj0arfnk=">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</latexit>

subwavelength resonance: Mie, Helmholtz, Minnaert · · ·<latexit sha1_base64="h89cW0N3TaBceFxBEnKxDnRrqtg=">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</latexit>

Appropriate scalings are needed to encapsulate the resonance<latexit sha1_base64="xboCyfii72/IcWUp0l3lk8KOURM=">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</latexit>

for instance Mie<latexit sha1_base64="Tbhtzd8VsZmO7es0FnYY5zqBmqY=">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</latexit> kh = ⌘ ⌧ 1 but k0h = O(1) ! c0/c = O(⌘)

<latexit sha1_base64="a1JsEln/ebmVFxh9coAGdmIWXE0=">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</latexit>

! Course of Kim Pham<latexit sha1_base64="cB+8QGNqQcn9o6zT7SijcL/fuE0=">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</latexit>

! Course of Claude Boutin<latexit sha1_base64="t59avDzE7SExdgoV2+i8FsHhpLA=">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</latexit>


classical homogenization: e↵ective wave equation<latexit sha1_base64="09qEzGcLy0qJGPfPx0UGouMHv5E=">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</latexit>

properties of the e↵ective parameters<latexit sha1_base64="jYOuhOD/ZqlcxSPXKkOU4tod/DY=">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</latexit>

Higher order homogenization, including transmission conditions<latexit sha1_base64="cplRRS3UZbYvbaNYk9xGUNrMdB8=">AAADA3icjVFNTxUxFD2MX4gfPHXppvHFxIWZ9A0Q39sQEjYsMfEBCRIy0ynzGmbaSdshAeKSf+LOnXHLH3CrifEf6L/wts5LYEG0k2lvzz3ntLe3aGvlPOc/F5Jbt+/cvbd4f+nBw0ePlwdPnu4401khp8LUxu4VuZO10nLqla/lXmtl3hS13C2ON0N+90Rap4x+509bedDklVZHSuSeoMPB+paqZtIyY0uaZ6YxldTqLGZfM6VF3ZVKV8zbXLtGuWDEhNGlCgx3OBjydDLhq6trjKdrPMuyMQV8JRtPRmyU8jiG6Me2GfzAe5QwEOjQQELDU1wjh6NvHyNwtIQd4JwwS5GKeYkPWCJtRyxJjJzQY5or2u33qKZ98HRRLeiUmn5LSoaXpDHEsxSH01jMd9E5oDd5n0fPcLdTWoveqyHUY0bov3Rz5v/qQi0eRxjHGhTV1EYkVCd6ly6+Srg5u1KVJ4eWsBCXlLcUi6icvzOLGhdrD2+bx/yvyAxo2Iue2+F3uCU1eN5FdnOwk6WjlTR7mw03eN/qRTzHC7yifr7BBrawjSl5f8RXfMP35CL5lHxOvvylJgu95hmujeTyDySgqhw=</latexit>

Acoustic case<latexit sha1_base64="enN8mnkcWb9c3njmTW9aP4g+klU=">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</latexit>

Acoustic case<latexit sha1_base64="enN8mnkcWb9c3njmTW9aP4g+klU=">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</latexit>

2D elastic case<latexit sha1_base64="SFO+i45KqHdX+skCBSioEHWQwg0=">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</latexit>

Homogenization of an array of beams on the top of an elastic half-space<latexit sha1_base64="39a3vV/pye0nYGBbUSw4d9HquHc=">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</latexit>


div(aeffrp)� beff@2p

@t2= 0

<latexit sha1_base64="1yLsWth50cz9JfvNjKg1pTRlOus=">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</latexit>

div(arp)� b@2p

@t2= 0

<latexit sha1_base64="6VlcEZt1KVFGnjEhTLhRZfG7UtU=">AAADAXicjVFNSxxBEH2OMfEz2ZijlyaLoAeX2c1CAokg5OLRQFYFR6Wnt1cbe2eanh6JLHvKP/HmTXLNH/CqF/EfJP8i1Z1ZUUS0h5l+9arem66u1GhVuDi+GYvGX0y8fDU5NT0zO/f6Te3t/GaRl1bIjsh1brdTXkitMtlxymm5bazk/VTLrfToq89vHUtbqDz77k6M3O3zg0z1lOCOqP3al8TJH27QVcfDJZ5kPNWcmeWVNPmc9CwXg8Rw6xTXey1mhrcRc3ut4Wq8X6vHjXbsF3sImo2wx3VUayOvXSNBFzkESvQhkcER1uAo6NlBEzEMcbsYEGcJqZCXGGKatCVVSargxB7R94CinYrNKPaeRVAL+oum15KSYZE0OdVZwv5vLOTL4OzZx7wHwdOf7YT2tPLqE+twSOxTulHlc3W+F4cePoUeFPVkAuO7E5VLGW7Fn5zd6cqRgyHO4y7lLWERlKN7ZkFThN793fKQ/xMqPetjUdWW+OtPSQMeTZE9DjZbjeaHRutbu77WrkY9iQW8xxLN8yPWsI4NdMj7FBe4xFX0MzqLzqNf/0ujsUrzDvdW9PsfUG6nzw==</latexit>

Maurel & Pham<latexit sha1_base64="N/9O85EVldYU4fD3W99F3aF35uE=">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</latexit>

Conversion of surface waves in a forest of trees,a homogenization approach

Kim Pham,

Jean-Jacques Marigo,

Agnes Maurel,

Sebastien Guenneau,

Institut des Sciences de la Mecanique et Applications Industrielles,ENSTA, Palaiseau - France

Laboratoire de Mecanique des Solides,Polytechnique, Palaiseau - France

Institut Langevin, ESPCI, Paris - France

Institut Fresnel, Marseille - France



Problem setting: Anti-plane elasticity



• Guiding layer over a substrate


Problem setting:

• Forest of trees

Anti-plane elasticity




How do Love waves propagate and interact with trees ?


• Forest of trees


How do Love waves propagate and interact with trees ?




x

z

0

Ht

�H`


(⇢t, µt,'t)

(⇢`, µ`)

(⇢s, µs)

x

z

0

Ht

�H`


(⇢t, µt,'t)

(⇢`, µ`)

(⇢s, µs)

x

z

0

Ht

�H`

in the trees, layer, substrate

� = µru, ⇢@2u

@t2= div�,


(⇢t, µt,'t)

(⇢`, µ`)

(⇢s, µs)

x

z

0

Ht

�H`


� = µru, ⇢@2u

@t2= div�,

continuity of u and � · n between two elastic media

Neumann b.c. � · n = 0 elastic medium/air


(⇢t, µt,'t)

(⇢`, µ`)

(⇢s, µs)

x

z

0

Ht

�H`


� = µru, ⇢@2u

@t2= div�,




(⇢t, µt,'t)

(⇢`, µ`)

(⇢s, µs)

x

z

0

Ht

�H`

(⇢`, µ`)

(⇢s, µs)

x

z

0

Ht

�H`


in the layer, substrate

� = µru, ⇢@2u

@t2= div�,



� = µru, ⇢@2u

@t2= div�,


(⇢t, µt,'t)

(⇢`, µ`)

(⇢s, µs)

x

z

0

Ht

�H`

(⇢`, µ`)

(⇢s, µs)

x

z

0

Ht

�H`



in the e↵ective medium (trees)

� = µru, ⇢@2u

@t2= div�,



� = µru, ⇢@2u

@t2= div�,

� = µt

✓0 00 't

◆ru, ⇢t

@2u

@t2= div�,


(⇢t, µt,'t)

(⇢`, µ`)

(⇢s, µs)

x

z

0

Ht

�H`

(⇢`, µ`)

(⇢s, µs)

x

z

0

Ht

�H`

e↵ective b.c.




� = µru, ⇢@2u

@t2= div�,



� = µru, ⇢@2u

@t2= div�,

� = µt

✓0 00 't

◆ru, ⇢t

@2u

@t2= div�,


(⇢t, µt,'t)

(⇢`, µ`)

(⇢s, µs)

x

z

0

Ht

�H`

(⇢`, µ`)

(⇢s, µs)

x

z

0

Ht

�H`

e↵ective b.c.




• z = Ht, �z = �Le

@�z

@z

� = µru, ⇢@2u

@t2= div�,



� = µru, ⇢@2u

@t2= div�,

� = µt

✓0 00 't

◆ru, ⇢t

@2u

@t2= div�,


(⇢t, µt,'t)

(⇢`, µ`)

(⇢s, µs)

x

z

0

Ht

�H`

(⇢`, µ`)

(⇢s, µs)

x

z

0

Ht

�H`

e↵ective j.c.

e↵ective b.c.




• z = Ht, �z = �Le

@�z

@z

� = µru, ⇢@2u

@t2= div�,



� = µru, ⇢@2u

@t2= div�,

� = µt

✓0 00 't

◆ru, ⇢t

@2u

@t2= div�,


(⇢t, µt,'t)

(⇢`, µ`)

(⇢s, µs)

x

z

0

Ht

�H`

(⇢`, µ`)

(⇢s, µs)

x

z

0

Ht

�H`

e↵ective j.c.

e↵ective b.c.

JuK = B1�z + B2@u

@x

J�zK = B2@�z

@x� B3

@2u

@x2+ ⇢e

@2u

@t2




• z = Ht,

• z = 0,

�z = �Le

@�z

@z

� = µru, ⇢@2u

@t2= div�,



� = µru, ⇢@2u

@t2= div�,

� = µt

✓0 00 't

◆ru, ⇢t

@2u

@t2= div�,


(⇢t, µt,'t)

(⇢`, µ`)

(⇢s, µs)

x

z

0

Ht

�H`

(⇢`, µ`)

(⇢s, µs)

x

z

0

Ht

�H`

e↵ective j.c.

e↵ective b.c.

JuK = B1�z + B2@u

@x

J�zK = B2@�z

@x� B3

@2u

@x2+ ⇢e

@2u

@t2



• z = Ht,

• z = 0,

�z = �Le

@�z

@z

� = µru, ⇢@2u

@t2= div�,

� = µt

✓0 00 't

◆ru, ⇢t

@2u

@t2= div�,


(⇢t, µt,'t)

(⇢`, µ`)

(⇢s, µs)

x

z

0

Ht

�H`

(⇢`, µ`)

(⇢s, µs)

x

z

0

Ht

�H`

e↵ective j.c.

e↵ective b.c.


(⇢t, µt,'t)

(⇢`, µ`)

(⇢s, µs)

x

z

0

Ht

�H`

x

z

0

Ht

�H`



x

z

0

Ht

�H`

x

z

0

Ht

�H`


Results in the harmonic regime


Results in the harmonic regime x

z

0

Ht

�H`



z

0

Ht

�H`

Guided waves for the homogenized problem



z

0

Ht

�H`


u(x, z) =

8><

>:

e↵szei�x, z < �H`,

(A cos k`z +B sin k`z)ei�x, �H` < z < 0

(C cos ktz +D sin ktz)ei�x, 0 < z < Ht



↵s =

s

�2 � !2

c2s

k` =

s!2

c2`� �2

kt =!

ct

x

z

0

Ht

�H`


u(x, z) =

8><

>:






1� µ`k`

µs↵s

tan k`H` �µtkt

µ`k`

't tan ktHt

✓tan k`H` +

µ`k`

µs↵s

◆= 0.

Dispersion relation:

↵s =

s

�2 � !2

c2s

k` =

s!2

c2`� �2

kt =!

ct

x

z

0

Ht

�H`

(without foliage)


u(x, z) =

8><

>:






1� µ`k`

µs↵s

tan k`H` �µtkt

µ`k`

't tan ktHt

✓tan k`H` +

µ`k`

µs↵s

◆= 0.


↵s =

s

�2 � !2

c2s

k` =

s!2

c2`� �2

kt =!

ct

x

z

0

Ht

�H`

(without foliage)



1� µ`k`

µs↵s

tan k`H` �µtkt

µ`k`

't tan ktHt

✓tan k`H` +

µ`k`

µs↵s

◆= 0.


x

z

0

Ht

�H`

(without foliage)

↵s =

s

�2 � !2

c2s

k` =

s!2

c2`� �2

kt =!

ct



z

0

Ht

�H`

↵s =

s

�2 � !2

c2s

k` =

s!2

c2`� �2

kt =!

ct1� µ`k`

µs↵s

tan k`H` �µtkt

µ`k`

't tan ktHt

✓tan k`H` +

µ`k`

µs↵s

◆= 0.Dispersion relation:



z

0

Ht

�H`

↵s =

s

�2 � !2

c2s

k` =

s!2

c2`� �2

kt =!

ct1� µ`k`

µs↵s

tan k`H` �µtkt

µ`k`

't tan ktHt

✓tan k`H` +

µ`k`

µs↵s




1� µ`k`

µs↵s

tan k`H` �µtkt

µ`k`

't tan ktHt

✓tan k`H` +

µ`k`

µs↵s


Love waves

↵s =µ`k`

µs

tan k`H`.

↵s =

s

�2 � !2

c2s

k` =

s!2

c2`� �2

kt =!

ct

x

z

0

Ht

�H`

�=!/c

�=!/c

s

!

�0

⇡�c

e

2⇡�c

e

0

Ht=

0



1� µ`k`

µs↵s

tan k`H` �µtkt

µ`k`

't tan ktHt

✓tan k`H` +

µ`k`

µs↵s


Spoof plasmons

↵s =µtkt

µs

't tan ktHt.

Love waves

↵s =µ`k`

µs

tan k`H`.

↵s =

s

�2 � !2

c2s

k` =

s!2

c2`� �2

kt =!

ct

x

z

0

Ht

�H`

�=!/c

�=!/c

s

!

�0

⇡�c

e

2⇡�c

e

0 0

�=!/c

s

BG

�

⇡ct

2H

⇡ct

H

3⇡ct

2H

Ht=

0

H`=

0



1� µ`k`

µs↵s

tan k`H` �µtkt

µ`k`

't tan ktHt

✓tan k`H` +

µ`k`

µs↵s


Spoof plasmons

↵s =µtkt

µs

't tan ktHt.

↵s =

s

�2 � !2

c2s

k` =

s!2

c2`� �2

kt =!

ct

x

z

0

Ht

�H`

0

�=!/c

s

BG

�

⇡ct

2H

⇡ct

H

3⇡ct

2HH

`=

0



1� µ`k`

µs↵s

tan k`H` �µtkt

µ`k`

't tan ktHt

✓tan k`H` +

µ`k`

µs↵s


↵s =

s

�2 � !2

c2s

k` =

s!2

c2`� �2

kt =!

ct

x

z

0

Ht

�H`

f(Hz)

SPPs

Hybrid wave

Love wave

� (m�1) ⇡/`00

40

80

without foliage



1� µ`k`

µs↵s

tan k`H` �µtkt

µ`k`

't tan ktHt

✓tan k`H` +

µ`k`

µs↵s


↵s =

s

�2 � !2

c2s

k` =

s!2

c2`� �2

kt =!

ct

x

z

0

Ht

�H`

2

18

10

� (m�1)

�Love ⇡/`!/cs !/c`

Ht(m

)

without foliage

2

18

� (m�1)

⇡/`!/cs !/c`

Ht(m

)



Forest of trees with a gradient of heights

2

18

� (m�1)

⇡/`!/cs !/c`

Ht(m

)



without foliage

2

18

� (m�1)

⇡/`!/cs !/c`

Ht(m

)

Comsol multiphysics (S. Guenneau)



without foliage

2

18

� (m�1)

⇡/`!/cs !/c`

Ht(m

)


2

18

� (m�1)

⇡/`!/cs !/c`

Ht(m

)



without foliage


2

18

� (m�1)

⇡/`!/cs !/c`

Ht(m

)



without foliage


2

18

� (m�1)

⇡/`!/cs !/c`

Ht(m

)



without foliage


2

18

� (m�1)

⇡/`!/cs !/c`

Ht(m

)



without foliage





(a)


(b) homogenized solution


kH = 100, kh = 1, ke = 0.05<latexit sha1_base64="rd8buajARV8jtffU/DSjwK30T8g=">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</latexit>

H/e = 2000<latexit sha1_base64="W8ywGn9CbAloqaW+iN8GuwEGudo=">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</latexit>



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Cargese Cours1 pdf - ESPCI Paris · 2019. 8. 28. · AAACxHicjVHLSsNAFD2Nr1pfVZdugkVwVZIq6M6CIC5bsA+oRZLptIZOHkwmQin6A27128Q/0L