The wave equation in an acoustic medium

Let x_i, where i=1,2,3 be three orthogonal directions and 73#73the position vector in a coordinate system associated with the three directions. Let us define 452#452 as the mass per volume unit in the acoustic medium, 453#453 as the velocity of the acoustic medium and K as the bulk modulus of the acoustic medium. The second law of dynamics states that mass 454#454 acceleration = force = - pressure gradient:  

 455#455 (184)

Energy can be stored by compression and volume variation. If  

 456#456 (185)

we say that the flow diverges (the volume changes). This leads to a pressure variation, proportional to the divergence of the velocity:  

 457#457 (186)

The wave equation in an acoustic medium can be deduced from () and () as follows. Derivate () with respect to time:  

 458#458 (187)

Divide () by 459#459 and derivate it with respect to the axis x_i:  

 460#460 (188)

Plug () in ():  

 461#461 (189)

Approximation: 459#459 is a constant that does not depend on the position vector. By denoting the acoustic waves propagation velocity through the medium by v, where  

 462#462 (190)

we obtain the acoustic wave equation:  

 463#463 (191)