## One-dimensional Wave Propagation in Rods of Variable Cross Section: A WKBJ Solution |

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a(xi aluminum and brass amplitude ratio approximate solution technique asymptotic power series constant cross sectional area cross sectional radii curves that relate DIMENSIONAL PLANE WAVE dispersion relation electric flux density equation is derived error associated expression of eqn.(28 four piezoelectric materials high the frequency in/sec incident wave IQI Model 501 isotropic elastic rod isotropic nonpiezoelectric Jeffreys Lewis Research Center longitudinal wave speeds lower bound frequency m/sec Moriguchi NASA Nondestructive testing nonpiezoelectric and piezoelectric nonpiezoelectric materials PARTIAL DIFFERENTIAL EQUATION Performing Organization phase velocity piezoelectric and nonpiezoelectric piezoelectric rods piezoelement PLANE WAVE PROPAGATION propagation in isotropic Propagation in Rods rearranging give relate the amplitude relates the ratio respective locations rod of variable scale length second term shown in Figs solution is valid stratified media Substituting transducer IQI Model transition point ultrasonic and acoustic values variable cross section wave is high wave number Wentzel WKBJ approximate solution WKBJ solution x-direction