OPTIMIZING DIAMETER, LENGTH, AND WATER IMMERSION IN FLOW RESISTANT TUBE VOCALIZATION

Abstract

Objective:

The objective was to quantify the range of airflow resistance and oral pressure attainable with variation of length, diameter, and water immersion depth of tubes and straws.

Study Design:

Pressure flow equations for tubes, determined previously for variable tube geometries, were used to calculate oral pressure ranges. Human subjects were then recruited to use the variable tube geometries to produce oral pressures, which were quantified with commercial manometers.

Results:

Nomograms for airflow resistances and oral pressures are plotted as a function of tube length, tube diameter, and water insertion depth.

Conclusions:

It is shown that tube diameters in the range of 2.5-3.0 mm with tube lengths of 10 – 40 cm produce oral pressures in the range of 10 – 40 cm H₂O. Insertion of the distal end into water adds a pressure in the amount of the depth of insertion. Maximum power transfer with different tube geometries is discussed.

INTRODUCTION

Flow resistant tube (FRT) and water-resistant tube (WRT) vocalization are two of the many methods used to improve the efficiency of sound production in speaking and singing. It belongs to a larger group of interventions known as semi-occluded vocal tract (SOVT) exercises. The exercises have a long history, and there are many variations. When a flow-resistant tube (or straw) is used, there is always the question about the length and the diameter of the tube. Furthermore, if the distal end of the tube is inserted into water, there is an additional question about the depth of insertion. The answers have not been rapidly forthcoming because there are multiple objectives in voice training and therapy. One objective is to optimize the glottal configuration with a steady supraglottal pressure. A lower phonation threshold pressure is achieved with parallel vocal fold surfaces, which a steady supraglottal pressure can facilitate (Titze, 1988; Chan et al., 1997). Another objective is to lower the glottal airflow resistance for maximum power transfer from the source to the vocal tract. A lower glottal resistance allows for better impedance matching between the glottis to the vocal tract (Titze, 2021). Yet another objective is to increase the inertive reactance of the vocal tract, which enhances source-filter interaction by lowering formant frequencies (Story et al., 2000). Many of the benefits are obtained simultaneously by creating a positive intraoral pressure with an oral semi-occlusion. Here we focus on geometric details of flow resistant tubes for control of intraoral pressure, the key variable underlying all the benefits. Flow-resistant tubes were included in the Maxfield et al. (2014) study of oral pressures obtainable with various semi-occlusions. Da Silva et al. (2019) quantified the viscous and kinetic resistances of flow-resistances with analytical equations. Kapsner-Smith et al. (2015), Guzman et al. (2016), Guzman et al. (2017a), Meerschman et al. (2019), Kang et. al. (2020), and Ford (2021) reported on the efficacy of FRT and WRT exercises in voice therapy. Apfelbach and Guzman (2021) have provided an integrative review of the effects of SOVT exercises.

The results of these studies provide support that FRT and WRT exercises facilitate the clinical objectives of improved voice quality by reducing roughness, effort, and improving vocal comfort. These effects are accomplished by optimizing vocal tract source-filter interaction and lowering phonation threshold pressure. Kang (2020) showed that mean flow increased with a decreased closed quotient after FRT exercises, showing promise of clinical relevance for high demand voice users who fatigue. Ford (2021) recently commented on improved shimmer measures, which might relate to improving vibratory characteristics and reduced vocal strain, along with an improved self-perception of vocal loudness. Finally, Guzman (2016) showed increases in glottal area and larger amplitudes of vibration with increased subglottal pressure. This supports the objective that Verdolini et al. (1998) reported for resonant voice therapy, namely barely adducted vocal folds. It clearly applies to patients with pressed voice who are at risk for vocal trauma. Guzman et al (2017a) also showed that FRT resulted in a wider pharynx and lower larynx The lower larynx increases inertive reactance in the vocal tract. This adjustment likely improved the efficiency and the power transfer to the vocal tract by matching impedances between the source and the filter.

The questions to be answered in this investigation are: (1) what is the trade-off between tube diameter and tube length in producing an oral pressure range, (2) what tube diameter and tube length combinations produce a lip resistance that approximates the glottal resistance, (3) what tube diameter and tube length combinations produce oral pressures above 20 cm H₂O for high resistance training, and (4) how does immersion of the distal end in water modify the oral pressure? This paper builds on the work of Da Silva et al. (2019) in that some human subject measurements are included to quantify the range of oral pressures achievable with flow-resistant tubes.

METHODS

Computations of flow resistance and oral pressure were measured for a variety of tube diameters D, length L, and volume airflow U, based on an empirical equation published by Smith and Titze (2016).

$$\begin{gathered} R=\left(3.7631\times {10}^{-7}\frac{L}{D^{4.4997}}+1.0268\times {10}^{-6}\frac{1}{D^{4.0416}}\right)U \\ +\left(3.9913\times {10}^{-9}\frac{L}{D^{5.0089}}+8.0169\times {10}^{-7}\frac{1}{D^{3.7696}}\right) \end{gathered}$$ (1)

In this equation, developed from bench-top measurements, the length is in m, the diameter is in m, the flow is in L/s, and the resulting resistance is in Pa per L/s. The equation is reported to have an accuracy of 2% over a range of diameters 1.8 to 9.7 mm, a range of lengths from 3.0 to 24 cm, and a range of flows from 0.01 to 1.5 L/s. In the bench-top measurements, each tube was connected upstream to a wider tube of 1.9 cm diameter, which approximated an oral diameter. Calibrated pressure and flow meters were installed upstream, while the distal end of each tube remained open to atmospheric pressure. Input pressures to the straw were calibrated up to 6.89 kPa.

The first term in the equation is known as a kinetic resistance, which occurs primarily at tube entry, where a sudden area contraction impedes the airflow. This term varies linearly with airflow U. The second term is independent of U. It is known as a viscous resistance. Air viscosity and air density are included in the numerical constants in the equation. Da Silva et al. (2019) developed analytical equations for these terms that explicitly show the dependency on viscosity and density of air, as well as airflow and geometry of the tubes.

A MATLAB script was written to vary D, L, and U over ranges that correspond to those used in voice therapy and training. Resistances and oral pressures were computed for typical tube sizes used clinically and commercially available. For tubes that were to be immersed in water in practice, a hydrostatic pressure was added to represent the boundary condition at the distal end of the tube. The ranges of calculated pressures (resistances from Eq. 1 multiplied by the flow U) were compared to those obtained from human subjects with two hand-held commercial manometers, one for low pressures (PrimAtü 10 Low-Cost Low Range Differential Air Pressure Transducer, range 0 - 20 cm H₂O) and one for higher pressures (MA-Line, model MA 12835, range 0 – 100 cm H₂O).

A plastic tube (3 mm inside diameter) was inserted between the lips on one end and to the manometer input at the other end. This tube, which drew no airflow, was placed between the lips in addition to the experimental tube that released the flow. There was no attempt to obtain an exact point-by-point pressure comparison between different tube dimensions because the airflows were not measured. The objective was to obtain a range of pressures over a small range of airflows (0 – 0.4 L/s) that are typical for mean flows in semi-occluded vocal tract phonation.

Eight human subjects were recruited (four male, four female). Subjects were roughly age matched across a total age range of 35 to 81 years. All subjects reported normal vocal function on the day of collection, although three (one female, two males) reported a history of voice disorder for which treatment had been sought. Recruitment and data collection protocols were reviewed and approved by the University of Utah Institutional Review Board (IRB_00056031). With the tube between their lips and the distal end submerged in water at a depth specified by the experimental design, subjects were asked to sustain phonation at a comfortable pitch first as softly as possible while maintaining voicing, and again as loudly as possible without discomfort. Subjects wore nose-clips to prevent air leakage through the nose and no attempt was made to control pitch.

RESULTS

The first result was to compare the relative contributions of the kinetic resistance to the viscous resistance, the first term versus the second term in Eq. (1). Fig. 1 shows the results for 4 different diameters in the separate subplots. Solid lines are for the kinetic term and dashed lines for the viscous term. Each family of three curves is for tube lengths of 4 cm, 12 cm, and 20 cm (bottom to top). Airflow is plotted along the horizontal axis. Note that the kinetic term varies linearly with airflow, while the viscous term is constant with airflow. For all diameters, there is a cross-over between the kinetic resistance and the viscous resistance. This occurs at an airflow of 0.2 L/s, a typical mean airflow in human phonation. For airflows below 0.2 L/s, the viscous resistance is dominant, whereas for airflows greater the 0.2 L/s the kinetic resistance is dominant. The two resistances are additive to get the total tube resistance. As a comparison to human airway resistances, typical glottal resistances are in the range of 2.0 - 8.0 kPa per L/s (Konnai et al., 2017). This range is equivalent to the range achieved with a 3.5-4.0 mm diameter tube of any length, as long as the airflow is on the order of 0.1-0.2 L/s, as shown in parts (c) and (d). To the contrary, for the smallest 2.5 mm diameter tube in part (a), the viscous component is so large that the tube resistance exceeds a typical glottal resistance for any length and any airflow.

Fig. 1.

Kinetic (solid lines) and viscous (dashed lines) resistance components versus airflow for four different tube diameters. The families of three curves are for 4, 12, and 20 cm tube lengths (bottom to top).

Fig. 2 shows the total resistance (kinetic + viscous) as a function of tube length. Separate subplots are for different airflows, and the family of three curves is for different tube diameters. Consider part (a), the case of 0.1 L/s airflow. For a 2.5 mm diameter tube (top line), a length increase from 2 cm to 40 cm more than triples the resistance, from 10 to 35 kPa per L/s. A similar tripling is also seen for the 3 mm diameter, from 8 to 24 kPa per L/s. For the 6 mm diameter, the resistance becomes very low for all tube lengths. These are the diameters for so-called “drinking straws”. They are useful to match a low glottal resistance in breathy voice, which is on the order of 1-2 kPa per L/s (Konnai et al., 2017). Low resistance tubes are often useful as an introduction to SOVT exercises for those who feel discomfort with high oral resistance and low airflows. Ultimately, however, the benefit of SOVT exercises is greatly increased with high resistance tubes or straws because they can alter the airway geometry and the posture of the vocal folds for maximum power transfer (Titze, 2021; Titze et al., 2021).

Fig. 2.

Airflow resistance as a function of tube length, with tube diameter as a parameter on the family of curves. The four panels are for 0.1, 0.2, 0.3, and 0.4 L/s airflows.

Note the insensitivity of resistance to airflow rate in comparison to diameter and length. Comparing parts (a) to (d), quadrupling the airflow rate from 0.1 to 0.4 L/s only doubles the resistance. This insensitivity was the motivation for not measuring airflow along with pressure for the human subjects, a considerably more complicated procedure.

Fig. 3 shows the introduction of yet another parameter, namely depth of immersion of the distal end of the tube into water. Oral pressure is now the dependent variable instead of resistance because hydrostatic pressure adds directly to trans-tube pressure. Trans-tube pressure was obtained by multiplying the resistance in Eq. (1) by airflow U. All pressures are reported in cm H₂O (10 cm H₂O = 0.98 kPa) because the hydrostatic pressure is directly visible in a water glass or bottle. Airflow was adjusted to match a range of pressures produced by the human subjects for soft and loud phonation over 3 diameters D,

$$\text{For}D=0.0025\text{mm},U=0.040\mathrm{L}∕\mathrm{s}\text{soft},U=0.10\mathrm{L}∕\mathrm{s}\text{loud}$$ (2)

$$\text{For}D=0.0030\text{mm},U=0.045\mathrm{L}∕\mathrm{s}\text{soft},U=0.15\mathrm{L}∕\mathrm{s}\text{loud}$$ (3)

$$\text{For}D=0.0040\text{mm},U=0.050\mathrm{L}∕\mathrm{s}\text{soft};U=0.20\mathrm{L}∕\mathrm{s}\text{loud}$$ (4)

$$\text{For}D=0.0060\text{mm},U=0.070\mathrm{L}∕\mathrm{s}\text{soft};U=0.50\mathrm{L}∕\mathrm{s}\text{loud}$$ (5)

Fig. 3.

Oral pressure versus tube length for soft phonation (dashed lines) and loud phonation (solid lines) for four depths of water insertion (0, 2, 5, 10 cm). (a) 2.5 mm diameter, (b) 3.0 mm diameter, (c) 4.0 mm diameter, (d) 6.0 mm diameter. Airflows were adjusted as shown in Eqs. (2)-(5).

The hydrostatic pressures were 2, 5, and 10 cm H₂0, the depth of immersion into water. In Fig. 3, dashed lines are for soft phonation and solid lines for loud phonation into the tube. For the 2.5 diameter tube in part (a), the range of pressures achievable is 2 – 40 cm H₂O, depending on length, diameter, and depth of immersion of the tube. For the 3.0 mm diameter tube in part (b), the range of achievable pressures is 1 – 35 cm H₂O.

For the 4-6 diameter tubes in parts (c) and (d), the pressure scale in the graphs has been changed. There are no pressures above 20 cm H₂O for any condition. For soft phonation (dashed lines), the hydrostatic pressure is the only relevant pressure. The oral pressure is simply the hydrostatic pressure in cm H₂O. This is evident from the vertical axis intercepts of the dashed lines. For loud phonation (solid lines), the trans-tube pressure can match or exceed the hydrostatic pressure for long tube lengths.

Fig. 4 shows the ranges of oral pressures obtained by four females and four males as measured with the commercial manometers described in the Methods. Part (a) shows the ranges for males phonating through a 2.5 mm tube. Here the amount of water insertion is shown with data points (“o” for soft and * for loud). There were not enough data points to draw curves across length. Without water insertion (no open circles or asterisks), the mean oral pressure ranged from 2 – 5 cm H₂O for soft voice (dashed line) and from 24 – 31 cm H₂O for loud voice (solid line). The addition of water immersion (2 cm, 5 cm, and 10 cm, bottom to top data points) increased the pressure range. As shown, a 35.5 cm straw length produced an oral pressure range between 4 – 45 cm H2O when soft and loud conditions are combined. Part (b) shows the same results for females. There is little difference for the open-ended tubes, but females did not gain any pressure from water immersion for loud phonation (asterisks).

Fig. 4.

Oral pressure versus tube length for soft and loud phonations averaged over sex (a) Four males, 2.5 mm diameter tube, (b) Four females, 2.5 mm diameter tub. Data points are for 10 cm immersion of a 37.5 cm length tube. (c) Four males, 6 mm diameter tube, (d) Four females, 6 mm diameter tube. Data points are for 2 cm and 5 cm immersion into water of a 17.75 cm length tube and an additional 10 cm immersion for a 37.5 cm length tube.

Parts (c) and (d) show results for the 6.0 mm diameter tube. Note first that the pressure scale on the vertical axis is reduced from 50 cm H₂O to 25 cm H₂O, as in Fig. 3. Soft phonation with an open distal end (dashed lines, no water immersion) offered oral pressures less than 2.0 cm H₂O for all lengths, across both genders. Adding water immersion (open circles for soft and asterisks for loud) raised the pressures in proportion to the immersion depth. For loud phonation, the open-ended tube produced a 6 – 10 cm H₂O pressure range for males and a 7 – 12 cm H₂O range for females. With water immersion, both males and females were able to increase the pressure roughly in proportion to the immersion depth (open circles for soft and asterisks for loud).

DISCUSSION

This article is a follow-up to the experiments conducted by Maxfield et al. (2014), Smith and Titze (2016), and Da Silva et al. (2019) on the resistance of tubes and straws for semi-occluded vocal tract exercises. The data are intended for direct application in clinics and voice studios where questions arise regularly about the best geometry of tubes or straws. The first question, what is the trade-off between tube diameter and tube length for a flow-resistant tube (FRT), has been answered quantitatively. For small diameter tubes (less than 3 mm, commonly known as stirring straws) and correspondingly low airflows (on the order of 0.1 L/s or less), increasing the length from 4 cm to 20 cm doubles the oral pressure. More generally, the rate of increase of oral pressure is about 0.5 cm H₂O per cm increase in length according to Fig. 3. In comparison, a small reduction of diameter, from 3.0 mm to 2.5 mm, increases the pressure at a rate of about 4 cm H2O per 0.5 mm diameter for a 10 cm tube length. Thus, reduction in diameter is much more effective than increasing the length for raising the resistance and oral pressure. Narrow tubes should be the choice for widening the airway and spreading the top of the vocal folds for better laryngeal posturing (Titze et al 2021).

The second question, what tube diameter and tube length combinations produces lip resistances that approximate the glottal resistance, has also been answered. For large diameter tubes (more than 5 mm), tube length adds little resistance at flows that are typical for vocalization. Long tubes with large diameters (e.g., drinking straws or glass tubes) are useful for lowering formant frequencies and raising supraglottal inertance (Story et al., 2000), but they do not build up much intraoral pressure without water insertion. They can, however, match the glottal resistance in soft or breathy voice when the vocal tract offers no other semi-occlusion. Exercises may begin with these large diameter tubes to get the initial feel of maximum power transfer from the source to the vocal tract. For small diameters, a narrow 2.5 mm diameter tube with a short length (less than 4 cm) and a low airflow rate (0.1 L/s) can balance the glottal resistance. Likewise, a 3 mm diameter tube of 10 cm length and slightly greater airflow can also balance the glottal resistance. It is hypothesized, but not fully investigated, that the higher resistance tubes invite alternative semi-occlusions in the lower vocal tract (e.g., the larynx canal or the lower pharynx) when the mouth is opened (Titze et al., 2021). These semi-occlusions may then be able to maintain maximum power generation and transmission for a full inventory of vocal tract shapes (Titze 2021).

The third question, what tube diameter and tube length combinations produce oral pressures above 20 cm H₂O for high resistance training, has also been addressed. Here there are multiple trade-offs between length, diameter, and airflow. A 2.5 mm diameter tube of 20 cm length produce 20 cm H₂O with a low airflow (0.1 L/s). Blowing harder (more airflow) will increase the pressure to about 40 cm H₂O, but mean airflows much greater than about 0.2 L/s are difficult to sustain for more than a few seconds (e.g., 1.0 L of lung volume expelled in 4 s yields a flow of 0.25 L/s).

The final question, how does immersion of the distal end in water modify the oral pressure, is answered very simply. The water depth in cm adds directly to the aerodynamic trans-tube pressure, a result previously reported by Da Silva et al. (2019). A 10 cm immersion combined with a 2.5 mm diameter and a 20 cm tube length can produce an oral pressure of 40 cm H₂O or more with airflow rates of 0.1-0.2 L/s.

The clinical relevance of high resistance WRT exercises investigated by Guzman et al (2017b) was that better vocal fold contact quotients resulted from deeper submersion of tubes in water. Water resistance may also increase respiratory and glottal effort to compensate for non-ideal supraglottal resistance. Supraglottal resistance may accomplish the clinical objective of increasing respiratory conditioning while maintaining efficiency in vocal fold and vocal tract posturing. This conditioning may apply to both hyperfunctional and hypofunctional dysphonias. Relevance is noted for patients that show difficulty with the pressures needed to increase pharyngeal areas, a lowered larynx, or maintaining an ideal glottal configuration. Guzman et al (2018) studied the geriatric population and noted an improvement in adduction and subglottal pressure after high resistance WRT training (8 cm H₂O immersion). The training also improved radiated sound pressure level. Calvache et al (2020) showed that SOVT exercises with a tube submerged in 10 cm of water showed significant differences between pre and post values relating to increased vocal economy for normal and dysphonic conditions of hyperfunction. Phonation into tubes submerged deep into water had the highest increases in vocal economy, acoustic output, vocal fold adduction, and effortless voice production.

CONCLUSION

Selection of tubes and straws for semi-occluded vocal tract exercises is no longer a trial- and-error process. Flow resistances and oral pressures can be obtained from nomograms provided in this paper. The selection process may differ from client to client, and from beginning to end of therapy or training sessions. For high-resistance training, a long narrow tube (2.5 – 3.0 mm diameter and 20 - 40 cm length) can be combined with 10 cm of water immersion to raise the oral pressure to 40 – 50 cm H₂O. The lung pressure has to rise above that pressure if phonation is to be sustained.

There were limitations in this study. The airflows were not measured when the subjects produced their phonations. They were estimated to be in the 0.04 - 0.2 L/s range, except for one condition for which the airflow was estimated to be 0.5 L/s to match the pressures produced. It was shown, however, that oral pressure was least sensitive to airflow in comparison to tube length, diameter, and water immersion. Another limitation was the subject sample size. There were only eight subjects in the study. Finally, the bubbles produced with water immersion made depth slightly uncertain. These less-than-ideal conditions are faced daily in clinical applications. All conditions tested with the subjects required less than 20 minutes per subject.