  Regular nested word languages (a.k.a. visibly pushdown languages) strictly
extend regular word languages, while preserving their main closure and
decidability properties. Previous works have shown that considering languages
of 2-nested words, i.e. words enriched with two matchings (a.k.a. 2-visibly
pushdown languages), is not as successful: the corresponding model of automata
is not closed under determinization. In this work, inspired by homomorphic
representations of indexed languages, we identify a subclass of 2-nested words,
which we call 2-wave words. This class strictly extends the class of nested
words, while preserving its main properties. More precisely, we prove closure
under determinization of the corresponding automaton model, we provide a
logical characterization of the recognized languages, and show that the
corresponding graphs have bounded treewidth. As a consequence, we derive
important closure and decidability properties. Last, we show that the word
projections of the languages we define belong to the class of linear indexed
languages.
