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Acoustic and elastic wave equations are routinely used in geophysical andengineering studies to simulate the propagation of waves, with a broad range ofapplications, including seismology, near surface characterization,non-destructive structural evaluation, etc. Finite difference methods remainpopular choices for these simulations due to their simplicity and efficiency.In particular, the family of finite difference methods based on thesummation-by-parts operators and the simultaneous-approximation-terms techniquehave been proposed for these simulations, which offers great flexibility inaddressing boundary and interface conditions. For the applications mentionedabove, surface of the earth is usually associated with the free surfaceboundary condition. In this study, we demonstrate that the weakly imposed freesurface boundary condition through the simultaneous-approximation-termstechnique can have issue when the source terms, which introduces abruptdisturbances to the wave field, are placed too close to the surface. Inresponse, we propose to build the free surface boundary condition into thesummation-by-parts finite difference operators and hence strongly andautomatically impose the free surface boundary condition to address this issue.The procedure is very simple for acoustic wave equation, requiring resetting afew rows and columns in the existing difference operators only. For the elasticwave equation, the procedure is more involved and requires special design ofthe grid layout and summation-by-parts operators that satisfy additionalrequirements, as revealed by the discrete energy analysis. In both cases, theenergy conserving property is preserved. Numerical examples are presented todemonstrate the effectiveness of the proposed approach.

Strongly imposing the free surface boundary condition for wave equations with finite difference operators