Design and measurements of variably

nonuniform acoustic resonators

Bruce Denardo and Miguel Bernard

Department of Physics and Astronomy, and National Center for Physical Acoustics

University of Mississippi, University, MS 38677

We describe the design, construction, and acoustical measurements of resonators with nonuniform cross-sectional areas that are easily altered to yield different resonance frequencies. These resonators are useful as educational demonstrations of symmetry breaking and of an effect of nonuniformity upon standing waves. Resonators that yield two, three, and four pitches are considered, where the relative frequencies are designed to correspond to musical intervals. Agreement is within 2% in all cases and 1% for most. The data reveal a breakdown of the theory, which is shown to be a result of additional kinetic energy, and thus effective inertia, near a discontinuity in cross-sectional area. The data also reveal that an appropriate end correction for our case is not that for a thin-walled tube but, rather, an infinite flange. © 1996 American Association of Physics Teachers.

I. INTRODUCTION

In a previous article¹, we established the motivation, demonstrations, and basic theory regarding acoustic resonators whose resonance frequencies are changed when the nonuniformity in cross-sectional area is altered. The resonators are of two types: single-piece "slappers" whose ends are slapped to generate sound (Fig. 1a), and double-piece "clappers" whose halves are clapped together while one set of ends rests on a flat surface (Fig. 1b). By "slapping" we mean that the demonstrator uses a hand or large rubber stopper to abruptly close one end and to leave it closed, thus forming a resonator with one end open and one closed. The fundamental resonance frequency is changed if the other end of a single-piece resonator is slapped, or one or both halves of a double-piece resonator are inverted. The effect can be readily demonstrated,^1,2 and can be physically understood by a variety of ways:¹ continuous deformation to the extreme cases of a Helmholtz resonator and weakly coupled resonators, Rayleigh’s energy method, or adiabatic invariance. On an elementary level, the resonators are useful as demonstrations of symmetry breaking.

In the present article, we establish sets of dimensions that yield frequencies whose ratios correspond to musical intervals. The production of musical intervals aids in the listener’s recognition of the change in frequency, makes the effect more dramatic, and demonstrates the utility of theory while offering a test of it. We will comment on the construction of the resonators and compare the measured frequency intervals to the musical values. Despite the simple appearance of the resonators, accurate design and construction must be done carefully.

Several references have come to our attention since our first article on this subject appeared. Our resonators are similar to R. M. Sutton’s "trick" whistle,³ which yields different frequencies depending upon which end is blown. Piecewise-uniform resonators have been considered in connection with the observation that, as a solid rod is inserted into an organ pipe, the resonance frequency is first lower and then higher than the original frequency.⁴ A piecewise-uniform geometry has also been considered as a source of frequency shifts of woodwind instruments.⁵ Monitoring the change in acoustic resonance frequency in ducts offers a potential method for blockage detection.^6,7 Certain piecewise-cylindrical and piecewise-conical resonators have the musically desirable property that the eigenfrequencies are harmonically related.⁸ In higher dimensions, shifts in resonance frequencies due to nonuniformities caused by levitated objects⁹ and other obstructions¹⁰ have been investigated.

The fundamental frequency of either piecewise-uniform resonator in Fig. 2 is¹

where c is the speed of sound, L is the length of the resonator, and A₁ and A₂ are the cross-sectional areas of the sides with the closed and open ends, respectively. Switching the boundary conditions inverts the ratio of the areas in Eq. (1), and thus leads to a different frequency. Eq. (1) ignores the open-end correction,¹¹ which lowers the frequency because the pressure-release point for the open end of a resonator effectively occurs some distance outside the end. We neglected this effect in our original design and construction, and found that the resonators were noticeably out-of-tune by as much as a musical quartertone or 3%. The current design includes an end correction (although an incorrect one, as will be shown), which reduces most deviations to less than 1%. The data clearly reveal a breakdown of Eq. (1), however, which we will show is a result of the spatial extent of the motion near a discontinuity in cross-sectional area.

In the equal-tempered chromatic musical scale, an octave (an interval from one frequency to twice that frequency) is split into twelve intervals that are equally-spaced with respect to the logarithm of the frequency.¹² The frequencies of the musical notes in an octave are then f_n = 2^n/12 f_o, where n = 0,1,2,...,12. The relative frequencies f_n/f_o are listed in Table I. It is convenient not to label these by the index n, but by the number of the note of the major scale, which is the sequence {1,2,3,4,5,6,7,8} in Table I. A note between two consecutive major scale notes is labeled by a flattening (^b) of the higher note. The single-piece resonators (Fig. 1a) are designed such that the higher frequency is either note 3 (major third ), 5 (perfect fifth), or 8 (octave) compared to each lower frequency. We thus label these resonators as 1-3, 1-5, and 1-8, respectively. We will show that the double-piece resonators (Fig. 1b), which have an intermediate frequency corresponding to the inversion of either half, are then 1-2-3, 1-3-5, and 1-5-8, respectively. We will also investigate four-pitch resonators, which have dissimilar halves so that the degeneracy of the intermediate frequency is lifted. We consider two such resonators that together yield a harmonic minor scale: 1-2-3^b-4 and 5-6^b-7-8.

II. APPARATUS

In the design of the resonators, we employ the simplest nonuniformity: a step in the cross-sectional area (Fig. 2). No apparent advantage is gained with more complicated nonuniformities. We utilize the simple case of circular cross sections (or bi-circular for intermediate orientations of the three-pitch resonators and all orientations of the four-pitch resonators). This geometry has the advantages of being calculable to higher accuracy than a geometry that is not piecewise-uniform,¹ and of being the easiest to machine. Furthermore, we select the step to lie halfway between the ends of the resonator. This has the advantages both of yielding the greatest frequency shift for a fixed discontinuity in area,¹ and of corresponding to an invertible expression (1) for the frequency when the end correction is neglected.

A. Two-pitch resonators

The two-pitch resonators (Fig. 2a) are chosen to have length 6 in., outer diameter 3 in., and characteristic inner diameter 1 in. Because the wavelength is much greater than the wall thickness, we assume that the open end is approximately unflanged (thin-walled). The end correction¹¹ then amounts to the addition of the length L = d/2 to the geometrical length L, where d is the open-end diameter and = 0.61. This modifies the frequency f in Eq. (1) in two ways. First, because it is inversely proportional to the length, f will be reduced as a result of the end correction. Second, because the end correction causes the step to effectively no longer lie at the middle, f will be altered. However, f is at an extremum when the step is at the middle, so its value changes negligibly if the step’s location is perturbed. Ignoring this effect and incorporating the first effect into Eq. (1), we find that the frequency of the resonator represented by Fig. 2a for the case of circular cross sections is

where d₁ and d₂ are the diameters of the closed-end and open-end sides of the resonator, respectively.

Switching the boundary conditions inverts the ratio of the diameters and thus leads to two possible frequencies f+ and f-, where f+ > f-. We will conveniently denote the larger value of d₁ and d₂ as d₊, and the smaller as d-. From Eq. (2), the ratio of the two frequencies is

The resonator design requires values of the diameters such that the ratio f+/f- is a desired musical value in Table I. Eq. (3) cannot be inverted unless the end correction is neglected ( = 0). Because this effect is small (less than 3% for all the resonators in this study), we first solve for d± in the case = 0, and then use these as starting values in a numerical search with Eq. (3). We uniquely determine d± by arbitrarily choosing the average value to be 1 in., and determine the diameters to the nearest 0.001 in. The search can be quickly accomplished with only a hand calculator. The resultant values for three single-piece resonators are listed in Table II. A tolerance of ± 0.001 in. in the diameters implies a deviation of f+/f- from a musical value by approximately ± 0.25%. We consider this to be acceptable because successive frequencies (musical semitones) in Table I differ by approximately 6%.

B. Three-pitch resonators

If we imagine cutting a single-piece resonator in half (Fig. 1b), a new configuration which mixes the diameters is now possible. The cross-sectional area in this case is the same on both sides of the step, and is reduced at the location of the step. Because this variation occurs at only one point, however, the resonator is theoretically uniform.¹ The frequency (2) of the fundamental of the new configuration is thus f_o = c/[4(L+dL)]. To determine the end correction for this noncircular case, note that the diameter of the circle whose area equals that of the end is the root-mean-square (rms) value of the individual diameters:

so the end correction is L = d_rms/2.¹¹

The intermediate frequency f_o lies approximately halfway between f+ and f-,¹ and the musical notes 2, 3, and 5 also lie approximately halfway between 1 and 3, 1 and 5, and 1 and 8, respectively. Hence, a possible design of 1-2-3, 1-3-5, and 1-5-8 resonators is to cut in half the 1-3, 1-5, and 1-8 resonators. To inquire into the accuracy of this design, we use the diameters of the two-pitch resonators (Table II) to calculate the ratio of f_o to f-, and compare this value to the desired musical value. From Eq. (2), the ratio is

.

The deviations of these values from the musical values are all found to be less than 1%, which we consider to be acceptable for demonstrations. We thus choose the design of the 1-2-3, 1-3-5, and 1-5-8 resonators to be specified by the two-pitch diameters in Table II.

C. Four-pitch resonators

By constructing dissimilar halves of a resonator (Fig. 2b), we can produce four different frequencies. It is then possible to design two resonators of different length such that the longer yields the first four notes of a musical scale, and the shorter yields the latter four notes. Using the symmetry of the frequency splittings¹ as a guide, however, we find that we do not have the freedom to approximate a major scale or natural minor scale {1,2,3b,4,5,6b,7b,8}. In fact, the symmetry restricts the possibilities among standard scales to the harmonic minor scale (Fig. 3.) It should be noted that the symmetry exists in the absence of end corrections. Furthermore, the symmetry is with respect to the frequency, whereas the desired musical symmetry is with respect to the logarithm of the frequency. It is thus not immediately clear whether a sufficiently accurate design of four-pitch resonators is possible.

The simplest design is to continue with the step of each half at the middle. Eq. (3) thus still applies, where the effective diameters are the four possible rms combinations (4) of the semicircular diameters a_1,2 and b_1,2. As before, we denote the larger and smaller values of a_1,2 as a±, and similarly for b_1,2. The square of the effective diameters of the configuration in Fig. 2b, and when one half is inverted, are then

where we assume without loss of generality that u₊ > v₊ > v- > u-. We design each resonator such that the ratio of the first and fourth frequencies, and the second and third, are desired musical values. The intervals between the first and second, as well as the third and fourth, will then only approximate the musical intervals. The values of u± are thus chosen such that the two frequencies correspond to the relative notes 1-4 for the longer resonator and 5-8 for the shorter resonator. Both of these intervals are a perfect fourth (P4). The values of v± are chosen such that a minor second (m2) occurs for the longer resonator, and a minor third (m3) for the shorter resonator. Hence, Eq. (5) represents four equations in the four unknowns a± and b±. However, the equations are not consistent unless a constraint is satisfied: the total area of both ends must be independent of the configuration, or . This is not satisfied if the mean values of u₊ and u-, and v₊ and v-, are chosen to be 1 in., as in the two-pitch and three-pitch cases. We can satisfy the constraint by choosing u_rms = v_rms = 1 in., and are also free to conveniently choose a_rms = b_rms = 1 in. The solution is then

,

where all quantities are all in inches. The resultant values for the shorter resonator, which is arbitrarily chosen to have length L_II = 6 in., are given in Table III.

The length L_I of the longer resonator (Fig. 3) must be chosen such that f1 = f5/, where f1 and f5 are the lowest frequencies of the long and short resonators, respectively, and where = 1.4983 from Table I. This cannot be determined analytically but can be successively approximated, as we now show. If the end correction is ignored, the length is L_I = L_II = 8.990 in. Substituting this value into Eq. (3), we determine the u± diameters such that a P4 interval occurs. We then use these values with the shorter resonator u± diameters, which we determined earlier, in the expression for the ratio of the lowest frequencies of the two resonators, which follows from Eq. (2). Equating this ratio to and solving for L_I yields the improved value 9.143 in., which is 2% greater than the original value. To continue our iterative approach, we substitute this value of L_I into Eq. (3) and determine new values of u±. However, these values are found to differ by less than 0.0001 in. from the previous values, so there is no need to continue the iteration. The values of v± are then computed with the improved value of L_I. The resultant dimensions of the four-pitch resonators are listed in Table III. The frequencies corresponding to the intervals 1-2, 3^b-4, 5-6^b, and 7-8, which only approximate the musical intervals, do not deviate more than 1% from the musical values.

A nonrigorous but plausible approach leads to a formula for L_I. Let f_I and f_II be the frequencies of uniform resonators of length L_I and L_II with diameter d = 1 in. The frequencies are expected to lie very near the middle of each P4 interval (Fig. 3). The relationship f₁/f_I = f₅/f_II is thus expected to hold to a good approximation. Combining this with the desired ratio f5/f1 = , and the fact that f_I is inversely proportional to L_I + d/2 and similarly for f_II, gives

which yields a value within 0.001 in. of the above value, L_I = 9.143 in.

D. Construction of the resonators

All of the resonators were made from clear acrylic, and were precision-machined to the dimensions in Tables II and III for the unflanged case. The tolerance was ± 0.001 in. Clear acrylic is advantageous because it is easily machined, and because it allows audiences to see the inner geometry either directly or on an overhead projector. Furthermore, we experimented with both aluminum and maple double-piece resonators, and found that acrylic produces the best sound with regard to increased quality factor and the reduction of unwanted frequencies. We did not discern any acoustical difference between the single-piece resonators, as expected.

The holes in each single-piece resonator were drilled and then bored. The steps were carefully machined to be perpendicular to the axis, with as little rounding of the corners as possible because the resonant frequencies are sensitive to the geometry of the step.⁴

The three-pitch resonators were not constructed by cutting a single-piece resonator in half, because the resultant loss of material would alter the cross sectional areas. Instead, we constructed all of the double-piece resonators by machining two identical single-piece resonators, and cutting these off-center to eventually produce one double-piece resonator with concentric halves. The contacting surfaces must be very flat for satisfactory resonances to be obtained. This flatness was obtained by milling the surfaces, and then fine-sanding them on a precision-flat granite block. The ends of the clappers must also be flat, and must be perpendicular to the contacting surfaces. To ensure the flatness of the surface required for clappers, a thick acrylic base was milled and then sanded as above.

III. ACOUSTICAL MEASUREMENTS

There are several approximations in the theory:¹ the dissipation does not alter the frequency, the end correction does not effectively alter the location of the step if it is at the middle, the radiation geometry is unflanged, and the motion near the step is effectively one-dimensional. Whereas the errors due to the first two can be shown to be negligible, the validity of the latter two are difficult to examine analytically. Our care in the design and construction of the resonators was partly due to a desire to experimentally probe these assumptions.

The experiments were conducted in an anechoic room, which substantially improved the signal. The sound was detected by a microphone along the resonator axis and roughly 12 in. from the open end, and was captured by a digital oscilloscope. Fig. 4 shows a typical time record of a clapped double-piece resonator resting on our flat base. Similar time records occur for single-piece resonators. By plotting the logarithm of the absolute value of time record data, we find that the decay is linear and that the typical quality factor is roughly 50. For accuracy, the single-piece resonators were slapped with a large hard rubber stopper rather than with a hand. The clapping was accurately accomplished by arranging one edge of each half to be in contact, and using the thumbs to act as hinges. These slapping and clapping methods were found to yield highly reproducible results.

We accurately measured the frequency by determining the time for a large number (typically 30) of zero crossings in a time record. The precision of these frequency values is dictated not by the precision of the zero crossings but, rather, by the presence of higher modes in the beginning of the record and by the decline of the signal-to-noise ratio toward the end. By measuring the frequency over various intervals and different trials, we empirically deduced that the uncertainty of a measurement is roughly ± 0.2 Hz. (Throughout this article, uncertainties represent approximate worst-case values.) At room temperature and for our typical frequency of 500 Hz, temperature variations of only ± 0.2^°C and relative humidity variations of roughly ± 10% are not negligible at this level of precision,¹³ so we ensured that the temperature and humidity remained sufficiently constant while we gathered data.

Table IV shows frequency data for two-pitch resonators whose dimensions are given by the unflanged values in Table II. For each resonator, the relative frequency (ratio of the upper frequency to the lower frequency) is compared to the desired musical value in Table I. The deviations from the musical values are all less than 0.5%, which is satisfactory because a musical semitone (the increment in Table I) is a 6% difference in frequency. However, because the machining tolerance of ± 0.001 in. and the frequency uncertainty of ± 0.2 Hz cause an overall uncertainty in the relative frequency of only ± 0.3%, the measured deviations of the relative frequencies from the predicted values are not conclusively within the predicted uncertainty. The deviations indicate a negative trend whose magnitude increases for greater step sizes.

Table V shows frequency data for three-pitch resonators whose dimensions are given by the unflanged values in Table II. For each resonator, the relative value of the higher of the two upper frequencies is predicted to coincide with a musical value, identical to the two-pitch case. These observed values indicate the same negative decreasing trend as in the two-pitch case.

The intermediate relative frequency of each three-pitch resonator is not predicted to exactly coincide with a musical value as a result of a lack of freedom in the design parameters (Sec. II.B). Both the measured and predicted deviations of these values from the musical values are shown in Table V. (The deviation of a measured value from the predicted value is the difference of these two deviations.) The intermediate frequencies show a significant negative decreasing trend beyond the uncertainty of ± 0.3%. The greatest deviation occurs for the 1-5-8 resonator, whose intermediate relative frequency is 2.1% below the predicted value.

Table VI shows frequency data for a set of four-pitch resonators whose dimensions are given by the unflanged values in Table III. The relative frequencies of the 1-2-3^b-4 resonator are all within 1.0% of the musical values, and are all within the expected tolerance of ± 0.3% from the predicted values. The relative frequencies of the 5-6^b-7-8 resonator are also within 1.0% of the musical values, but are all approximately 0.5% below the predicted values. This error arises because the two resonators do not match with regard to the musical scale; the length according to the design of the 1-2-3^b-4 resonator (Sec. IIC) is evidently less than the correct value.

IV. SYSTEMATIC ERRORS

The greatest error in Sec. III is the negative deviation of the relative intermediate frequencies of the three-pitch resonators. Because the cross-sectional area in this case has a constant value except at the step, the theory predicts no frequency shift compared to the uniform resonator of the same length and cross-sectional area. A lowering of the frequency is expected, however, because the alteration of the motion increases the kinetic energy and thus the effective inertia, while not affecting the potential energy or stiffness.^1,14 To confirm this, we placed cellophane tape over successively greater amounts of area at the step of the 1-5-8 resonator in its intermediate-frequency configuration, until the original cross-sectional area at the step was halved. The measured frequencies indeed continually decreased, with the final value being 5% lower than the original value.

The lowering of the frequency as a result of additional kinetic energy caused by a discontinuity should also occur for the lowest and highest frequencies, but is not evident in the analysis in Table V because this deals with frequencies normalized to the lowest frequency. What is required is a comparison to the absolute frequency (2). To do this, we accurately calculated the speed of sound by including the effects of temperature, humidity, and frequency.¹³ As a precaution against the occurrence of unknown errors, we first compared the theoretical and experimental frequencies for a single-piece uniform resonator (designated as 1-1), which had been constructed as part of the demonstrations.¹ To our surprise, the experimental frequency was 1.8% below the predicted value, with an overall uncertainty of only ± 0.1%. We confirmed the experimental value by using a loudspeaker to excite the resonator. The deviation led us to suspect that the thickness of the resonators (typically 1 in.) was not negligible, even though it is much smaller than the wavelength (typically 30 in.). This is confirmed by several articles^15,16 which reveal that the end correction increases rapidly with wall thickness, from the infinitesimal (unflanged) value of = 0.61 to the infinitely-flanged value of = 0.82. These results, our own calculations,¹⁷ and the fact that the 1.8% deviation in the frequency of the uniform resonator is nearly eliminated when an infinitely-flanged end is assumed, indicate that the value of e for our geometry is near the latter. This slightly disagrees with an empirical investigation¹⁸ that predicts = 0.74 for our case of an outer-to-inner-radius of roughly 2. Carrying out the design assuming an infinitely-flanged end leads to the dimensions listed second in Tables II and III.

The physical reason for the flanged end in our case is that the motion near the opening is sensitive to the nature of the boundary there, depending on the thickness compared to the radius rather than the wavelength.¹⁶ Because the motion near the opening dictates the acoustic reactance and thus the end correction, it is reasonable that our case can be approximated by an infinite flange.

Our mistake with the end correction also resolves the mismatch error of the two four-pitch resonators (Sec. III). An infinitely-flanged end yields a value of L_I in Eq. (6) that is 0.6% greater than the value for an unflanged end. This is very near the observed shift of relative frequencies of the 5-6^b-7-8 resonator from the desired values.

Fig. 5 compares the theoretical and experimental frequencies of the uniform and three-pitch resonators, where an infinitely flanged end is assumed in the theory. The remaining -0.2% systematic deviation of the uniform resonator may be due to the compliance of the rubber stopper. The other data show a systematic negative deviation that increases for resonators with greater step size, and is approximately symmetrical (the lowest and highest frequencies of a resonator have approximately the same fractional deviation). This trend is additional confirmation of the presence of increased kinetic energy near the step. The symmetry accounts for the smallness of the errors of the frequency ratios in Table IV and the uppermost frequency ratios in Table V.

The additional inertia due to a step discontinuity in cross-sectional area has been analytically considered in the approximation that the wavefront in the section of smaller diameter is planar (fictitious piston).¹⁹ The result is represented in terms of an end correction; the additional inertia m is the density multiplied by the volume associated with the end correction. Application of Rayleigh’s energy method¹ for this perturbation yields the relationship 0 = E = 2(f/f)KE + m[v²]/2, where E and KE are the total and kinetic energies of the mode, f is the frequency, and [v²] is the change in the square of the velocity at the discontinuity. This predicts identical shifts for the lowest and highest frequencies, and yields f/f = -0.14%, -0.43%, and 1.07% for the 1-2-3, 1-3-5, and 1-5-8 resonators, respectively, compared to the observed shifts of -0.37%, -0.78%, and -1.35%, which are averages over the lowest and highest frequencies of each three-pitch resonator in Fig. 5. Due to the piston approximation, the calculation is expected to underestimate the additional inertia. The experimental results are in accord with this, and show that the approximation is better for greater discontinuities.

The intermediate-frequency configurations of the three-pitch resonators have a point (rather than step) discontinuity in cross-sectional area. It is plausible that these configurations are acoustically similar to a uniform resonator with a thin aperture located at the middle, if the diameter of the aperture equals the diameter at the discontinuity. However, twice the frequency shift is expected in the aperture case because the extent of the disturbance is doubled. A theory exists for the kinetic energy due to the flow through an aperture, in the approximation that this flow is unaffected by the tube walls.¹⁴ Using Rayleigh’s energy method (see above) to estimate the frequency shift caused by this effective inertia, and halving the result, yields values of -1.2%, -2.2%, and -3.7% for the 1-2-3, 1-3-5, and 1-5-8 resonators, respectively, compared to the observed values of -0.52%, -1.2%, and -3.1% in Fig. 5. There is reasonable agreement for the 1-5-8 resonator, which has the greatest discontinuity. Because the aperture area is not small compared to the tube cross-sectional area for the other resonators, the theory is expected to overestimate the additional inertia, which is indeed the case.

V. CONCLUSIONS

We have investigated closed-open piecewise-uniform resonators that are slapped or clapped to produce sound. The configurations can be easily altered, which changes the resonance frequency because the nonuniformity changes. A theoretical design has led to sets of dimensions that yield relative frequencies corresponding to the following notes of a musical major scale: 1-3, 1-5, 1-8, 1-2-3, 1-3-5, and 1-5-8. In addition, two resonators yield a complete harmonic minor scale 1-2-3^b-4 and 5-6^b-7-8, which has the required symmetry. The resonators are readily machined, but this must be done carefully to ensure musical accuracy and a reasonably high quality factor.

We have measured the frequencies, and have found that all ratios are within 2% of the musical values, and most are within 1%. According to our design, some ratios will deviate by as much as 1% due to the absence of freedom in choosing some of the frequencies. A search involving perturbations of the piecewise-uniform geometry may yield resonators that more closely yield musical intervals.

The acoustical measurements reveal two systematic deviations from the theory. First, the design used in the construction of the resonators assumed the acoustical end correction corresponding to a thin-walled (unflanged) end, because the wavelength is much greater than the thickness. The data, however, indicate that the end is effectively infinitely flanged. To our knowledge the open-end correction has not yet been thoroughly tabulated due to the difficulty of the mathematics. The dimensions of the resonators based on a flanged end correction are listed in Tables II and III. These rather than the unflanged values should be used in the construction of the resonators.

The second systematic deviation is a result of a breakdown of the assumption that the motion is one-dimensional except in a negligibly small region about a discontinuity in cross-sectional area. That the disturbance is in fact spatially extended causes a lowering of the frequency because the increase in kinetic energy effectively increases the inertia. This is most dramatic in the case of a configuration that is uniform except at a single location where one part of the resonator is laterally displaced and rotated 180^o relative to the other, for which the theory predicts zero shift in frequency. The approximate inertia caused by an aperture¹⁴ accounts for our observation of the lowering of the frequency in the case of the greatest discontinuity, where this correction applies. For the cases of a step discontinuity in cross-sectional area, the lowering of the frequency is roughly consistent with an approximate theory¹⁹ for the inertia correction at the step.

ACKNOWLEDGMENTS

We are especially grateful for the assistance of model-maker G. Harrell and student S. Alkov (Naval Postgraduate School, Monterey, California). The assistance of student D. Woolworth and comments of Dr. R. Raspet (University of Mississippi) are also acknowledged. We are grateful for the many comments of Dr. C. J. Nederveen (Netherlands), particularly in regard to the end correction at an open end of a thick tube and the inertia correction at a step discontinuity in cross-sectional area inside a tube. This work was supported in part by a joint fellowship from the American Society for Engineering Education and the Office of Naval Technology.

¹Bruce Denardo and Steven Alkov, "Acoustic resonators with variable nonuniformity," Am. J. Phys. 62, 315-321 (1994).

²Bruce Denardo, "Acoustic resonators with variable nonuniformity," J. Acoust. Soc. Am. 95, Pt. 2, 2935 (1994).

³Arthur H. Benade, Horns, Strings, and Harmony (Doubleday, Garden City, NY, 1960), pp. 149-151.

⁴Arthur Tabor Jones, "Resonances in certain nonuniform tubes," J. Acoust. Soc. Am. 10, 167-172 (1939), and references therein.

⁵Cornelis J. Nederveen, Acoustical Aspects of Woodwind Instruments (Frits Knuf, Amsterdam, 1969), pp. 59-60.

⁶M. Antonopoulos-Domis, "Frequency dependence of acoustic resonances on blockage position in a fast reactor subassembly wrapper," J. Sound. Vib. 72, 443-450 (1980).

⁷Qunli Wu and Fergus Fricke, "Determination of blocking locations and cross-sectional area in a duct by eigenfrequency shifts," J. Acoust. Soc. Am. 87, 67-75 (1990).

⁸J.-P. Dalmont and J. Kergomard, "Lattices of sound tubes with harmonically related eigenfrequencies," Acta Acustica 2, 421-430 (1994).

⁹E. Leung, C. P. Lee, N. Jacobi, and T. G. Wang, "Resonance frequency shift of an acoustic chamber containing a rigid sphere," J. Acoust. Soc. Am. 72, 615-620 (1982).

¹⁰Mary E. Smith, Thomas W. Moore, and Howard W. Nicholson, Jr., "Wave phenomena in an acoustic resonant chamber," Am J. Phys. 42, 131-136 (1974).

¹¹Allan D. Pierce, Acoustics (Acoustical Society of America, Woodbury, NY, 1989), p. 348-349.

¹²Ref. 11, pp. 59-60.

¹³CRC Handbook of Chemistry and Physics, 75th edition, edited by David R. Lide, (CRC Press, Boca Raton, Florida, 1994), pp. 14-35 to 14-38. The deviation of temperature from 20oC was included by multiplying by the square root of the ratio of the absolute temperature to the absolute temperature corresponding to 20oC.

¹⁴Philip M. Morse and K. Uno Ingard, Theoretical Acoustics (McGraw-Hill, New York, 1968), pp. 480-482.

¹⁵Y. Ando, "On the sound radiation from semi-infinite circular pipe of certain wall thickness," Acustica 22, 219-225 (1969).

¹⁶M. C. A. M. Peters, A. Hirschberg, A. J. Reijnen, and A. P. J. Wijnands, "Damping and reflection coefficient measurements for an open pipe at low Mach and low Helmholtz numbers," J. Fluid Mech. 256, 499-534 (1993).

¹⁷M. Bernard and B. Denardo, "Re-computation of Ando’s approximation of the end correction for a semi-infinite circular pipe," to be published in Acustica.

¹⁸A. H. Benade and J. S. Murray, "Measured end corrections for woodwind toneholes," J. Acoust. Soc. Am. 41, 1609 (abstract) (1967). See also Ref. 5, p. 63.

¹⁹Uno Ingard, "On the theory and design of acoustic resonators," J. Acoust Soc. Am. 25, 1037-1061 (1953). The values of the end corrections were obtained from Fig. 3 in this article.