1. Mathematical modeling of wave phenomena#
Waves are everywhere – from the ripples in your coffee cup to the way your favorite song travels through the air. But what exactly is a wave? It turns out, putting a finger on it is trickier than one might think.
Wikipedia defines a wave as
…a propagating dynamic disturbance (change from equilibrium) of one or more quantities.
As general as this sounds this definition already dictates the following necessary specifications for any physical wave model [1]:
Geometry: Where is the dynamic disturbance in question modeled?
Constituents: What are the physical quantities? What are the primal physical quantities to be modeled?
Balance relations: How are the physical quantities related by basic energetic or kinematic principles?
Material laws: How relations between the physical quantities are determined by the specific materials?
Parameters: Which material data are required for the model?
External forces, boundary and initial data: How is the initial configuration and which external forces and conditions on the domain boundaries drive the model?
Simplifying assumptions: Assumptions on the balance relations and material laws to derive meaningful, “simple” equations
In the following we will specify everything stated above to derive the governing partial differential equations (PDEs) of some wave models.