Euler equations

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What's the relationship between the incompressible, free surface euler equations and the euler equations?

Are the latter just the former when the free surface is identically zero?

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Euler equations is a term for a class of PDEs from fluid dynamics. Their prominent common feature is absence of viscosity.

The PDEs within this class can be classified further as:

  1. Incompressible fluid (constant density, zero divergence of velocity) or compressible fluid (all others)
  2. Fluid with free surface (think of water waves splashing) or without free surface (think of water flowing in a pipe).