Picking up from Volume 10

Chapter 1 of Volume 10 set out the myoelastic–aerodynamic theory: muscles set the folds’ position, length and tension, but each opening and closing is a self-sustained oscillation of airflow and elastic tissue (Berg 1958); and “Bernoulli suction” alone cannot sustain it. This chapter asks the next question: what exactly makes the air do net positive work on the tissue in each cycle? Titze’s 1988 small-amplitude theory answers it and yields a formula for the phonation threshold pressure (Titze 1988).

Why a symmetric shape gains no energy

The result is easy to compute. Let the entry (lower) and exit (upper) half-areas of the glottis move sinusoidally with the same amplitude, the upper one lagging by a phase $\varphi$. In Titze’s formula the mean intraglottal pressure is $P_g = P_s(1 – a_2/a_1)$ (with the pressure above the glottis taken as zero), acting on the medial surface of the folds; integrating it over the mean displacement of that surface for one cycle gives the net work done by the airflow on the tissue.

# Net aerodynamic work per cycle on the folds in a surface-wave glottis (after Titze
# 1988). Entry (lower) and exit (upper) half-areas move sinusoidally; the exit lags by
# phi:
#   a1 = a0 + A sin(wt),  a2 = a0 + A sin(wt - phi)
# Mean intraglottal pressure (supraglottal pressure taken as zero): Pg = Ps (1 - a2 / a1).
# Work per cycle W = integral of Pg dx, x = (a1 + a2)/2 the mean medial-surface
# displacement. All values illustrative.
import math

PS = 800.0                     # Pa, subglottal pressure (about 8 cm H2O)
A0, AMP = 1.0, 0.4             # mean and amplitude of the half-areas (arbitrary units)

def work_per_cycle(phi, n=4000, tilt=0.0):
    # tilt > 0 gives a fixed convergent prephonatory shape (entry wider than exit)
    w, dt = 0.0, 2 * math.pi / n
    for i in range(n):
        t = i * dt
        a1 = A0 + tilt + AMP * math.sin(t)
        a2 = A0 - tilt + AMP * math.sin(t - phi)
        pg = PS * (1 - a2 / a1)
        dx = AMP * (math.cos(t) + math.cos(t - phi)) / 2 * dt     # dx/dt * dt
        w += pg * dx
    return w

print("phase lag of upper edge | net work per cycle (Pa x area units)")
for deg in (0, 15, 30, 60, 90, 120):
    print(f"   {deg:4d} deg             | {work_per_cycle(math.radians(deg)):+8.1f}")
print(f"   -30 deg (upper leads) | {work_per_cycle(math.radians(-30)):+8.1f}"
      "   energy flows from tissue to air")
print(f"fixed convergent shape (entry 1.2, exit 0.8), no lag: "
      f"{work_per_cycle(0.0, tilt=0.2):+.1f}  (no net work)")
best = max(range(0, 181, 5), key=lambda d: work_per_cycle(math.radians(d), 1000))
print(f"largest energy transfer in this sketch at a lag of about {best} deg")
# phase lag of upper edge | net work per cycle (Pa x area units)
#       0 deg             |     +0.0
#      15 deg             |   +108.8
#      30 deg             |   +211.1
#      60 deg             |   +371.7
#      90 deg             |   +438.8
#     120 deg             |   +388.2
#    -30 deg (upper leads) |   -211.1   energy flows from tissue to air
# fixed convergent shape (entry 1.2, exit 0.8), no lag: +0.0  (no net work)
# largest energy transfer in this sketch at a lag of about 90 deg

Three points stand out. First, with no phase lag the net work is exactly zero; even when the glottis has a fixed convergent shape (entry 1.2, exit 0.8), as long as the two margins move together the net work is still zero. This is what “a symmetric or unchanging shape gains no energy” means: the pressure is then a function of displacement alone, and its integral around a closed path must vanish. Second, when the upper margin lags, the net work is positive, largest in this sketch at a lag of about 90°; when the upper margin leads, the work is negative and energy flows from the tissue back into the air, acting as extra damping. Third, the Bernoulli effect has not disappeared, since the intraglottal pressure does change with shape, but what sets the direction of energy flow is the timing between changes of shape and motion.

Threshold pressure: checking the formula against the original

The lowest subglottal pressure at which vibration can just be sustained is the phonation threshold pressure. In Titze’s theory it is the pressure at which the “negative damping” supplied by the airflow exactly cancels the damping of the tissue.

The framework’s $P_{\text{th}} = 2k_tBc\xi_0/T$ keeps the factor 2 of the general expression but replaces $\xi_{01}^2/(\xi_{01}+\xi_{02})$ by $\xi_0$ outright; for a rectangular glottis this doubles the threshold. The program below compares the two versions with the same illustrative parameters and looks at the effect of glottal shape and tissue properties.

# Phonation threshold pressure in Titze's (1988) small-amplitude surface-wave theory.
# General form (as reproduced by Zhang 2010; Fulcher & Scherer 2011):
#     Pth = 2 kt B c xi01^2 / [T (xi01 + xi02)]
# kt transglottal pressure coefficient (~1.1); B damping per unit medial area (N s/m^3); c
# mucosal-wave speed (m/s); T medial-surface thickness (m); xi01, xi02 prephonatory
# half-widths at entry (lower) and exit (upper). Rectangular glottis: Pth = kt B c xi0 /
# T. Parameter values illustrative, chosen to give about 3 cm H2O at baseline.
KT, B, C, T = 1.1, 800.0, 1.0, 0.003
PA_PER_CMH2O = 98.07

def pth(xi01, xi02, b=B, c=C, t=T, kt=KT):
    return 2 * kt * b * c * xi01 ** 2 / (t * (xi01 + xi02))

xi0 = 0.001                                   # 1 mm
p_rect = pth(xi0, xi0)
p_short = KT * B * C * xi0 / T                 # rectangular special case
p_author = 2 * KT * B * C * xi0 / T            # the outline's "2 kt B c xi0 / T"
print(f"rectangular glottis, general formula : {p_rect:6.1f} Pa"
      f" = {p_rect / PA_PER_CMH2O:4.2f} cm H2O")
print(f"rectangular glottis, kt B c xi0 / T  : {p_short:6.1f} Pa (identical)")
print(f"'2 kt B c xi0 / T' would give        : {p_author:6.1f} Pa (twice too high)")
print("shape at the same mean half-width of 1 mm (entry / exit, mm):")
for e, x in ((1.0, 1.0), (1.2, 0.8), (1.4, 0.6), (0.9, 1.1)):
    p = pth(e / 1000, x / 1000)
    print(f"  {e:.1f} / {x:.1f}  Pth = {p / PA_PER_CMH2O:4.2f} cm H2O"
          f"  ({p / p_rect:4.2f} x rectangular)")
print("one factor at a time from the rectangular baseline (illustrative multipliers):")
cases = (("tissue damping B x 0.85 (better hydrated?)", {"b": 0.85 * B}),
         ("tissue damping B x 1.25 (drier?)", {"b": 1.25 * B}),
         ("mucosal wave 1.3 x faster (stiffer cover)", {"c": 1.3 * C}),
         ("thinner medial surface, T x 0.7", {"t": 0.7 * T}))
for name, kw in cases:
    p = pth(xi0, xi0, **kw)
    print(f"  {name:43s} {p / PA_PER_CMH2O:4.2f} cm H2O ({p / p_rect - 1:+.0%})")
# rectangular glottis, general formula :  293.3 Pa = 2.99 cm H2O
# rectangular glottis, kt B c xi0 / T  :  293.3 Pa (identical)
# '2 kt B c xi0 / T' would give        :  586.7 Pa (twice too high)
# shape at the same mean half-width of 1 mm (entry / exit, mm):
#   1.0 / 1.0  Pth = 2.99 cm H2O  (1.00 x rectangular)
#   1.2 / 0.8  Pth = 4.31 cm H2O  (1.44 x rectangular)
#   1.4 / 0.6  Pth = 5.86 cm H2O  (1.96 x rectangular)
#   0.9 / 1.1  Pth = 2.42 cm H2O  (0.81 x rectangular)
# one factor at a time from the rectangular baseline (illustrative multipliers):
#   tissue damping B x 0.85 (better hydrated?)  2.54 cm H2O (-15%)
#   tissue damping B x 1.25 (drier?)            3.74 cm H2O (+25%)
#   mucosal wave 1.3 x faster (stiffer cover)   3.89 cm H2O (+30%)
#   thinner medial surface, T x 0.7             4.27 cm H2O (+43%)

The baseline parameters ($k_t = 1.1$, $B = 800$ N·s/m³, $c = 1$ m/s, $\xi_0 = 1$ mm, $T = 3$ mm) give about 293 Pa, or about 3.0 cm H₂O, the same baseline as Volume 10. For a rectangular glottis the general form and the simplified form agree exactly; “$2k_tBc\xi_0/T$” gives 587 Pa.

The effect of shape matches Titze’s prediction. Keeping the mean half-width at 1 mm, a convergent glottis with entry 1.2 mm and exit 0.8 mm raises the threshold to 1.44 times the rectangular value, and 1.4/0.6 mm raises it to 1.96 times; a slightly divergent glottis (0.9/1.1 mm) lowers it to 0.81 times. Note, though, that this is a consequence of a small-amplitude theory applied to the prephonatory shape: in a clearly divergent glottis the point where the flow separates moves in real airflow, the small-amplitude formula may no longer hold, and one cannot conclude that “the more divergent, the easier the onset”.

As for tissue properties, 25 % more damping raises the threshold by 25 % and 15 % less lowers it by 15 % (the threshold is proportional to $B$); a 30 % faster mucosal wave (a stiffer cover) raises it by 30 %; and thinning the medial surface to 0.7 times raises it by 43 %, which is in the same direction as the experience that thresholds are higher at high pitch (elongated folds have thinner medial surfaces).

“Hydration lowers viscosity”: stating the evidence accurately

Singing teachers often say, “drink plenty of water; moist folds start more easily.” Real research lies behind this, but the form of the evidence needs to be stated clearly.

So this book puts it this way: hydration affects how easily phonation starts, most clearly at high pitch; researchers explain this through tissue viscosity, but changes in viscosity have not been measured directly in people. “Damping × 0.85” or “× 1.25” in the program are only illustrations; there are no reliable figures for how much a real change in hydration changes $B$. The book gives no advice on how much to drink, nor on nebulisers or medicines; consult a doctor about voice problems.

Chapter summary

The energy of self-sustained vocal fold oscillation comes from the phase difference between the lower and upper margins: it makes the glottis more convergent while opening and more divergent while closing, so that the pressure “pushes at the right moment”; no unchanging or synchronously moving shape gains net energy. Checked against Titze’s 1988 general form, the threshold pressure is $2k_tBc\xi_{01}^2/[T(\xi_{01}+\xi_{02})]$, reducing to $k_tBc\xi_0/T$ for a rectangular glottis as in Volume 10; the framework’s $2k_tBc\xi_0/T$ is twice too high. The threshold rises with damping, mucosal-wave speed and convergence, and falls as the medial surface thickens. The evidence that hydration affects ease of onset comes from excised canine larynges and small human experiments, appears mainly at high pitch, and treats tissue viscosity as an explanation rather than a measurement. The next chapter turns to the last term of the theory: how the vocal tract changes the threshold in return.