waves, which have been the subject of many studies. In particular, radial standing solutions of the equation were investigated in [@Kevrekidis2011]. The existence of standing waves in deep waters were proved in [@Iooss2005]. In the shallow-water regime standing wave patterns have been found in the context of the Boussinesq equations by M. <span style="font-variant:small-caps;">Chen</span> & G. <span style="font-variant:small-caps;">Iooss</span> [@Chen2005; @Chen2006; @Chen2008]. 2-D bi-periodic travelling wave solutions to the Euler equations were studied by W. <span style="font-variant:small-caps;">Craig</span> & D. <span style="font-variant:small-caps;">Nicholls</span> (2002) [@Craig2002]. Recently, 2-D wave patterns of the free surface were investigated experimentally by D. <span style="font-variant:small-caps;">Henderson</span> *et al.* (2010) [@Henderson2010] along with a theoretical stability analysis.

In this work, symbolic dynamics and associated techniques of substitutions are exploited to investigate the structure of standing wave patterns (see [@Morse1940; @Lind1995; @Berthe2000]). In physics, their application in studies of dynamical systems led to unveiling the structure of quasi-periodic tilings (see [@Senechal1995]). Symbolic dynamics allows coding the nonlinear behavior of a complex system and pattern formation by means of 1-D or 2-D words of a finite alphabet. Such approach was applied to code the non-periodic trajectories on a unit circle with a particular partition on two intervals (see [@Morse1940]). Coding of finite, 