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

162 8. Acoustic Analogs, Ducts, and FiltersThe presence of one orifice turns a pipe into a high-pass filter. If the radiusof the orifice is increased, the attenuation of the low-frequency components alsoincreases. If a pipe contains several orifices separated by only a fraction of a wavelength,these orifices can be treated as a group acting with their equivalent parallelimpedance. But if the distances between the orifices constitute an appreciable portionof the wavelength, the system becomes analogous to an electrical network offilters or to a transmission line which has a number of impedances shunted acrossit, spaced apart at wide intervals. Waves reflected from these different orificesare then out of phase with respect to each other, and Equation (8.30) no longerremains valid. Electrical filter theory must then be utilized to compute the transmissioncoefficient. As a rule, a number of orifices strategically placed apart canattenuate at low frequencies more effectively than a single orifice of equal totalarea.The sound power transmission coefficient T pg into a single orifice is approximatedbyT pg=2k 2 A[ (2AkL′) ] 2(8.31)π+ 1πa 2The filtering action of an orifice is principally that of the reflection of energy backtoward the source, not so much the loss of acoustic energy out of the pipe throughthe orifice into the ambient medium.A common example of the application of orifices is the control of the behaviorof a wind instrument such as a flute or a saxophone. When such an instrument isplayed in its fundamental register, all or nearly all of the orifices some distancebeyond the mouthpiece are kept open by the player. The diameters of these orificesnearly equal the bore of the tube, essentially shortening the effective length of theinstrument. The acoustic energy reflected from the first open orifice generates apattern of standing waves between the first open orifice and the mouthpiece. Theflute behaves like an open pipe, with the wavelength approximately equal to twicethe distance between the first orifice and the opening of the mouthpiece. A clarinetor a saxophone contains a vibrating reed at the mouthpiece, which approximatesthe conditions of the closed end of a tube. In this case the wavelength will equalnearly four times the distance from the reed to the first open orifice.Both the reed-type (clarinet, saxophone, coronet, etc.) and tubular instruments(flute, recorder, piccolo, etc.) contain a number of harmonics, those of the reedinstruments being primarily odd harmonics (characteristic of closed pipes). Whenhigher notes are played on either type of instruments, the fingering of these holesbecome more complicated, with some orifices beyond the first orifice closed andsome others opened. The fingering of these orifices control the standing wavespatterns which correspond to specific notes.3. Band-Pass Filters. A side branch in the form of a long pipe rigidly capped atits far end or a fully enclosed Helmholtz resonator (shown in Figure 8.8) contains