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  "body": "\n## Summary\n\nLet a [[contact manifold|HBQ8BZ7XB5]] be fibred over a base by a Legendrian fibration.\nThe wave front of a Legendrian map is the image of a [[Legendrian submanifold|Q2DMSTWZYZ]] in this base.\nIn the model case of the space of $1$-jets $J^1(Q) = T^*Q \\times \\R$, it is the projection of the Legendrian submanifold to $Q \\times \\R$.\nThe front is usually a hypersurface with singularities such as cusps.\nAt its smooth points the front determines the Legendrian submanifold through its tangent hyperplanes.\nWave fronts are the contact counterpart of [[caustics|GMRYG4S3G0]], and they describe the moving fronts of waves in geometric optics.\n\n## Explanations\n\n### L1 human\n\nDrop a stone in a pond with a curved bank: the ripple reflected from the bank is a curve that moves and changes shape.\nAt some moments this curve develops sharp corners called cusps.\nThe ripple is a wave front, and it is the shadow of a smooth curve one dimension up.\nThat smooth object records, at each point of the front, also the direction in which the front is facing.\nFor a function $f$ of one variable, the front of its $1$-jet curve $(q, f'(q), f(q))$ is just the graph of $f$.\nThe semicubical cusp $(t^2, \\tfrac23 t^3)$ is a front that is not a graph near $0$.\n",
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