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

7.7 The Rectangular Cavity 139Because acoustic energy cannot escape from a completely closed cavity withrigid walls, standing waves constitute the only appropriate solutions of the waveequation. Insertinginto the wave equationp(x, y, z, t) = X(x)Y (y)Z(z)e iωt (7.30)∇ 2 p = 1 ∂p(7.31)c 2 ∂tand separating variables results in the following set of equations:( d2) ( d2) ( d2)dx + 2 k2 x X = 0,dy + 2 k2 y Y = 0,dz + 2 k2 z Z = 0 (7.32)Here the separation constants are related as follows:k 2 = ω2c 2 = k2 x + k2 y + k2 z (7.33)The boundary conditions of Equation (7.29) stipulate cosine solutions, and Equation(7.30) revises towhere the components of k arep lmn = A lmn cos k xl x cos k ym y cos k zn xe iω lmnt(7.34)k xl = lπ ,L xl = 0, 1, 2,..., k ym = mπ ,L ym = 0, 1, 2,...k zn = nπ ,L zn = 0, 1, 2,..., (7.35)This leads to the quantization of allowable frequencies of vibration√l 2 π 2ω lmnc=L 2 x+ m2 π 2L 2 y+ n2 π 2L 2 z(7.36)The above gives rise to eigenfunctions of Equation (7.34). Each eigenfunctionis characterized by its own eigenfrequency (7.36) specified by the ordered integers(l,m,n). Equation (7.34), which is the solution to the wave equation (7.31), yieldsthree-dimensional standing waves in the cavity with nodal planes parallel to thewalls. The pressure varies sinusoidally between these nodal planes. In the samemanner that a standing wave on a string could be resolved into a pair of waves travelingin opposite directions, we can separate the eigenfunctions in the rectangularcavity into traveling plane waves. This is done by casting the solutions (7.34) intocomplex exponential form and expanding it as a sum of products:p lmn = A ∑lmn eiω lmn t(±k x x±k x y±k z z)(7.37)8where the summation is taken over all permutations of plus and minus signs.There are eight terms in all, each representing a plane wave traveling along the