VRIJE UNIVERSITEIT BRUSSEL Acoustics - the Dept. of ...

12 CHAPTER 1. FUNDAMENTAL CONCEPTS OF ACOUSTICSTransformation into a spherical coordinate system one can obtain (see[19]) the following form of the wave equation :∂ 2 rp∂r 2 + = 1 c 2 ∂ 2 rp∂t 2 (1.20)with r = √ x 2 +y 2 +z 2 .Taking rp as one variable we get a spherical wave equation of the sameform as the equation for plane waves. A general solution of this equation, astraveling waves, is thus :rp(r,t) = p 1 (t−r/c)+p 2 (t+r/c) or p(r,t) = 1 r p 1(t−r/c)+ 1 r p 2(t+r/c)(1.21)withthefirsttermrepresenting adivergent spherical wave andthesecondonea convergent spherical wave. Both waves exhibit the same propagation speedc and their amplitudes decreases with increasing distance r, radially fromthe point source. Convergent waves have so to say no acoustical application,while the divergent waves have . Indeed, this latter is present as soon as thedistance from the sound source becomes larger than the physical dimensionsof that source. If the point source produces an harmonic wave one may writefor the divergent wave :p(r,t) = A rexp(iωt−ikr) (1.22)1.4 Cylindrical sound wavesFor a cylindrical line sound source one can show that the solution can bewritten as (see [19]) :p(r,t) = A √ rexp(iωt−ikr) (1.23)An important conclusion concerning the three simple types of sources is thefollowing :For plane sound waves the sound pressures does not decrease with thedistance.For spherical sound waves the sound pressure decreases linearly withincreasing distance.