A note on the Pohhmme feqeny eqtion SCIENCE AND TECHNOLOGY - Reseh Jonl - Volme 6 - Univesity of Mitis Rédit Mitis. A NOTE ON THE POCHHAMMER FREQUENCY EQUATION by F.R. GOLAM HOSSEN Deptment of Mthemtis Flty of Siene Univesity of Mitis Rédit Mitis. (Reeived Apil 999 - Aepted Jne 999 ABSTRACT The Pohhmme feqeny eqtion fo both onvex nd onve ylinde is nlysed in the limiting se hen the popgtion onstnt tends to eo. It is fond tht belo etin vle of the dis the displements involve ombintion of odiny nd modified Bessel fntions insted of jst one type of Bessel fntions. This vle of the dis is llted nd displyed fo vios mteils. Keyods : Pohhmme popgtion onstnt. 9
F. R. Golm Hossen INTRODUCTION The popgtion of time-hmoni ves in n elsti il ylinde of infinite extent is govened by the Pohhmme eqtion. The tditionl deivtion of this eqtion is thogh the Helmholt deomposition of the Nvie eqtion of lssil elstiity (Kolsky 963; Ahenbh 984. The displements e fist oked ot nd imposing the ondition of no stess t the sfe then yields the dispesion eltion o feqeny eqtion knon s the Pohhmme feqeny eqtion. Oing to its omplexity only limiting ses e slly onsideed. Ths fo the se hen the velength beomes smll Bnoft (94 hs shon tht the phse veloity ppohes tht of Ryleigh sfe ves. In this note e obtin the Pohhmme eqtion ithot sing the Helmholt deomposition fo both the inteio nd exteio poblems (i.e. fo both onvex ylinde nd onve ylinde o ylindil vity nd onside the limiting se hen the dis nd ve nmbe e both smll. It is fond tht thee exists vle of the dis belo hih the displements involve ombintion of the odiny nd modified Bessel fntions insted of jst the ltte. The itil dis t hih this flip-ove os is llted fo diffeent mteils nd displyed gphilly. THE GOVERNING EQUATIONS The stting point fo the tetment of elsti modes of popgtion in il ylinde in the bsene of body foes is the integtion of the Nvie eqtion ( λ µ.u µ U = ρ U& ( hee U is the displement veto λ nd µ e the Lmé onstnts nd ρ is the mss density. We ok in ylindil pol oodintes ( R θ Z nd denote the omponents of U by U V W nd the dis of the ylinde by A. We shll onside only longitdinl ves hteied by the pesene of only the displement omponents U nd W both ith symmety bot the -xis so tht thee is no θ dependene. It is lso onvenient to non-dimensionlie the vibles in ode tht the poblem hs the smllest nmbe of pmetes ovetly involved. The dimensionless vibles e intoded s follos :
A note on the Pohhmme feqeny eqtion ( ( ; / Z W U R A t T ρω µ = ω = hee ω is the ngl feqeny of the ve hih is onsideed to be imposed in this poblem. We lso intode mteil pmete α defined by /( µ λ = µ α hih is elted to Poisson s tio ν by n - n - =. We note tht 5. α. In tems of these dimensionless vibles the govening eqtions e ( t = α α α ( ( t = α α α. (b The onditions fo ttion-fee sfe beome ( = α (3 =. (3b THE FREQUENCY EQUATION FOR THE INTERIOR PROBLEM We e inteested in the popgtion of n infinite tin of sinsoidl ves long the ylinde sh tht the displement t eh point is simple hmoni fntion of s ell s of t. Aodingly e seek soltions of the fom
F. R. Golm Hossen = U exp[ i( k t] (4 (4b hee the mplitdes U nd W e fntions of only. Sbstittion of eqns. (4 into eqns. ( eslts in the folloing opled diffeentil eqtions : (5 d W dw du α α p W = ik( α[ U ] (5b d d d hee = p k α nd q = k. Eqns. (5 e of the modified Bessel type nd sine fo the inteio poblem the field vibles shold be finite t the ente of the ylinde thei soltions involve only I nd I the modified Bessel fntions of the fist kind. It is esily shon tht the displements e given by [ C I ( p C I ( ] = exp[ i( k t] q (6 k q = i exp[ i( k t] CI ( p CI ( q (6b p k hee C C e bity onstnts. The bondy onditions (3 then yield system of homogeneos eqtions hih fo onsisteny eslts in the Pohhmme feqeny eqtion : p I( p I ( p I( q (k I ( q I( q 4k I ( q I( p pq = I ( p. (7 We no nlyse eqn. (7 in the limiting se hen the popgtion onstnt k is smll. To this end e pose petbtion expnsion fo k of the fom k = k k O( 4. Using expnsions of Bessel fntions fo smll gments nd the bove k in eqn. (7
A note on the Pohhmme feqeny eqtion e obtin k ~ α 3 4α ( α 6( α(3 4α... k. (8 Eqn. (8 shos tht k nd k e lys less thn one so tht thee is vle of belo hih q beomes imginy; in othe ods thee is itil dis belo hih the modified Bessel fntions get hnged into the stndd ones. To detemine e exmine the feqeny eqtion s k edes tods nity; odingly e set the folloing expnsion fo k : k = ε Κ ε. Upon sbstittion into Eqn. (7 e obtin I ( X α = 6 X (9 I( X hee X = α. By ssigning sitble vles to X α nd hene n be obtined. The vition of ith the mteil pmete α is displyed in the gph belo (Fig.. Hene fo vles of less thn nd of k gete thn α the displements s given in eqns. (6 then involve ombintion of modified nd odiny Bessel fntions of the fist kind. Inteio Fig.. Vition of the itil dis ith the mteil pmete α. 3
F. R. Golm Hossen THE FREQUENCY EQUATION FOR THE EXTERIOR PROBLEM Fo the exteio poblem the soltions mst be bonded t infinity so tht the displements no involve K nd K the modified Bessel fntions of the seond kind. The oesponding feqeny eqtion fo longitdinl ves beomes p K( p K ( p K( q (k K ( q K( q 4k K ( q By sing simil nlysis s bove the itil dis the folloing eqtion K ( X α = X. K ( X K( p pq = K ( p. ( n be detemined fom The vition of ith α is displyed in Fig.. We note tht fo ptil mteil the itil dis is mh smlle fo the onve ylinde thn fo the onvex one. REFERENCES ACHENBACH J. D. (984. Wve Popgtion in Elsti Solids. Elsevie The Nethelnds. BANCROFT D. (94. The veloity of longitdinl ves in ylindil bs. Physil Revie 59 588-593. KOLSKY H. (963. Stess Wves in Solids. Dove Pblitions In. Ne Yok. 4