THE SCIENCE AND APPLICATIONS OF ACOUSTICS - H. H. Arnold ...

144 7. Pipes, Waveguides, and Resonators7.9 Boundary Condition at the Driving Endof the WaveguideAn impedance match must be made at the waveguide entry z = 0 so that thepermissible mode solutions comply with the acoustic behavior of the active surface.If we know the pressure or velocity distribution of the driving source, these can becorrelated with the behavior of p(x, y, 0,t) for the pressure or z ·⃗u(x, y, 0, t) forthe velocity. If the pressure distribution of the source is given, then the boundarycondition becomesp(x, y, z, t) = P(x, y)e iωt (7.53)However, the left-hand side of Equation (7.53) can also be expressed as a superpositionof the normal modes of the waveguide, i.e.,p(x, y, z, t) = ∑ A lm cos k xl x cos k ym ye i(ωt−k zt)(7.54)l,mSetting z = 0, Equation (7.53) becomesP(x, y) = e iωt ∑ A lm cos k xl x cos k ym y (7.55)With Equation (7.55) we can establish the values of A lm by applying Fourier’stheorem, first for the x direction and then for the y direction.˘7.10 Rigid-Walled Circular WaveguideThe treatment of the rigid-walled circular waveguide of radius r = a is fairlystraightforward, beginning withP ml = A ml J m (k ml r) cos(mθ) e (iωt−k zt)where (r, θ, z) are the customary cylindrical coordinates, J m is the mth-orderBessel function, and√ω2k z =c − 2 k2 lmwhere k ml is found from the boundary conditions. Because r ·∇p = 0atthewallr = a,k ml = j ml′awhere jlm ′ are the zeros of dJ m(z). When the values of kdzml are determined, theapplicable results developed for the rectangular waveguide may be applied bysubstituting the values of k ml for the circular waveguide. As with the rectangularunit, the (0,0) mode of the circular waveguide is a plane wave that propagates withphase velocity c p = c for all ω>0. For frequencies below f 1,1 , only plane wavescan propagate in a rigid-walled circular waveguide.