An analysis of the effect of the discontinuity in a bifurcated circular guide upon plane longitudinal waves

The problem of t heoreticall y acco un ting for t he scatterin g of sound b y a se mi-infi nite circul a r t ube wit h a s mall di a meter inser ted axiaUy, as a measurin g probe, in to a larger t ube of i nfini te le ngth , is sol ved ill a r igo rous a nd explicit m a nner when t he incid ent plane waves are r estri cted to t he lowest propagatin g mode.The investigat ion follows the met hods outlined b y J .Schwinger and consists in sol ving H elmholtz's eq uation su bject to boundary condi tions on t he aco ustic ve locity poten ti al at r)g id wall s. " T i th t he aid of Green 's theore m a nd the Green's fun ction for a poin t source in t he large guid e, an in tegra l rep resentation of t he des ired velocity potent ial is obtain ed.T he imposit ion of t he bo undary co ndi t ion leads d irectly to a homogeneous in tegral eq uation of t he Wie ner-H o pf type, whi ch is solved by tra nsform tech niq ues.T he re ults of t his solu t ion are shown to be r elated to a dista nce d, whi ch is t he length the inner gui de mus t be extended to acco unt for t he di st urba nce occ urrin g in t h e immediate vicini ty of the d iscont inuity caused by t he excitation of hi gher mode fi elds.Another satisfactor y method of repre en t ing th e e ffect of the di scon tinui ty is b y t he usc of eq ui val ent circui ts, by means of whi ch a discon t inui ty in a wave guid e is replaced by a l umped parameter network in a set of t ransmission lines.The equi valent circui t for t his networ k is deriv ed, and t he one esse ntia l circui t p a ra meter is related to d a nd to t he expli cit solu t io n of t he integral eq uation .N umeri cal res ul ts for d are g ive n a s a fun ction of se veral differen t guide r atios and drivin g frequencies of t he inciden t wa ves.The res ul ts are in co nsiderable disagree men t wi th approx imation m ethods pre viously repor ted a nd in d icate t h at t hese met hods a re unsatisfacto ry in t he present probl em.'Present Address, H ughes Aircraft Co., Culver City, Calif.I T he preparatio n of this paper was sponsored (in part) by thc Office of Naval Research.T he conte nt of t his paper was submitted as partial fulfillment of the requ irements for tbe degree or doctor o[ philosophy at the U niversity o[ CaliCornia at Los Angeles.' F igures in brackets indicate the literature references at the end of tbis papcr.