continuity e.q.,equilibrium e.q. ,N-S , constitutive e.q. for elastic solid and compressible fluid in the space-frequency domain

continuity e.q.,equilibrium e.q. ,N-S , constitutive e.q. for elastic solid and compressible fluid in the space-frequency domain, as shown in the following Fig.

i don't understand how can this author deduce? why v=iwu? is this assumption?  why is this necessary?    thanks for your help and guidance!