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For a Gaussian beam w(x,y,t;k) concentrated on a ray path that is tangent to x=0 at (x,y,t)=(0,0,0) we calculate the \"reflected\" wave z(x,y,t;k) in t > 0 such that w(x,y,t;k)+z(x,y,t;k) satisfies Friedlander's wave equation and vanishes on x=0. These computations are done to leading order in k on the ray path. The interaction of beams with boundaries has been studied for non-tangential beams and for beams gliding along the boundary. We find that the amplitude"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1707.03477","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2017-07-11T22:09:47Z","cross_cats_sorted":[],"title_canon_sha256":"e935b94c9e33e7bf12d0e0b8533e58165628fc33a777faa8d3d0c311bbf409cc","abstract_canon_sha256":"8519078dde415a2fde816afad6c702d021efcfa7ef10a622321697ab92e22420"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:40:25.895676Z","signature_b64":"mPtLTkGpM6GTOHAjh06OcKnCPiB0r6+MzodnIO53BfDSBbUM9f6OyP2SqtD49LS3WWmeu8UKQ0VuwwBQmr2vDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"866ec657ba548985c4b8f7c02155e167b5336c7243577fa6dbebec1bd7e97b43","last_reissued_at":"2026-05-18T00:40:25.895066Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:40:25.895066Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A Grazing Gaussian Beam","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"James Ralston, Neelesh Tiruviluamala","submitted_at":"2017-07-11T22:09:47Z","abstract_excerpt":"We consider Friedlander's wave equation in two space dimensions in the half-space x > 0 with the boundary condition u(x,y,t)=0 when x=0. 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