We start with the solution for the rigid-tube model of Lambossy. A rigorous derivation of the solution for blood volume flow rate, flow velocity and shear force has been published by Painter et al. 13. The velocity of flow at distance r from the central axis of the artery is

u ⌣ = [ A ⌣ / ( ρ i ω ) ] e i ω t [ 1 − J 0 ( i 3 / 2 { ω ρ / μ } 1 / 2 r ) / J 0 ( i 3 / 2 α ) ] . MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=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@65FD@

The volume flow rate is

Q ⌣ = [ π A ⌣ R 2 e i ω t / ( i ω ρ ) ] [ − J 2 ( i 3 / 2 α ) / J 0 ( i 3 / 2 α ) ] . MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafmyuaeLbaqbacqGH9aqpcqGGBbWwcqaHapaCcuWGbbqqgaafaiabdkfasnaaCaaaleqabaGaeGOmaidaaOGaemyzau2aaWbaaSqabeaacqWGPbqAcqaHjpWDcqWG0baDaaGccqGGVaWlcqGGOaakcqWGPbqAcqaHjpWDcqaHbpGCcqGGPaqkcqGGDbqxcqGGBbWwcqGHsislcqWGkbGsdaWgaaWcbaGaeGOmaidabeaakiabcIcaOiabdMgaPnaaCaaaleqabaGaeG4mamJaei4la8IaeGOmaidaaOGaeqySdeMaeiykaKIaei4la8IaemOsaO0aaSbaaSqaaiabicdaWaqabaGccqGGOaakcqWGPbqAdaahaaWcbeqaaiabiodaZiabc+caViabikdaYaaakiabeg7aHjabcMcaPiabc2faDjabc6caUaaa@5CE6@

We define I(i^3/2α) = [-J₂(i^3/2α)/J₀(i^3/2α)]/(iωρ) and write

Q ⌣ = π A ⌣ R 2 e i ω t I ( i 3 / 2 α ) . MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafmyuaeLbaqbacqGH9aqpcqaHapaCcuWGbbqqgaafaiabdkfasnaaCaaaleqabaGaeGOmaidaaOGaemyzau2aaWbaaSqabeaacqWGPbqAcqaHjpWDcqWG0baDaaGccqWGjbqscqGGOaakcqWGPbqAdaahaaWcbeqaaiabiodaZiabc+caViabikdaYaaakiabeg7aHjabcMcaPiabc6caUaaa@4379@

We define P_Q(i^3/2α) as J₂(i^3/2α)/[(i^3/2α)²/8] and note that as R goes to zero the imaginary part of P_Q(i^3/2α) vanishes and |P_Q(i^3/2α)| goes to 1. Substitution now gives

Q ⌣ = [ π A ⌣ R 4 / ( 8 μ ) ] e i ω t P Q ( i 3 / 2 α ) / J 0 ( i 3 / 2 α ) . MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafmyuaeLbaqbacqGH9aqpcqGGBbWwcqaHapaCcuWGbbqqgaafaiabdkfasnaaCaaaleqabaGaeGinaqdaaOGaei4la8IaeiikaGIaeGioaGJaeqiVd0MaeiykaKIaeiyxa0Laemyzau2aaWbaaSqabeaacqWGPbqAcqaHjpWDcqWG0baDaaGccqWGqbaudaWgaaWcbaGaemyuaefabeaakiabcIcaOiabdMgaPnaaCaaaleqabaGaeG4mamJaei4la8IaeGOmaidaaOGaeqySdeMaeiykaKIaei4la8IaemOsaO0aaSbaaSqaaiabicdaWaqabaGccqGGOaakcqWGPbqAdaahaaWcbeqaaiabiodaZiabc+caViabikdaYaaakiabeg7aHjabcMcaPiabc6caUaaa@578E@

Therefore, I(i^3/2α) is approximately R²/(8μ) when α² ≪ 8 and is approximately 1/(iωρ) when α² ≫ 8.

The shear force (per unit area) at the inner wall of the tube is

-μ [∂u/∂r|_{r = R} = -[A⌣ MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafmyqaeKbaqbaaaa@2CFB@μ/(ρiω)]e^iωti^3/2(α/R)J_-1(i^3/2α)/J₀(i^3/2α), where i^3/2(α/R)J_-1(i^3/2α) = dJ₀(i^3/2[ωρ/μ]^1/2r)/dr|_{r = R}. We note that -J_-1(i^3/2α) = J₁(i^3/2α), which is written as (i^3/2α/2)P_S(i^3/2α). Substitution now gives the expression for the shear force (per unit area), [A⌣ MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafmyqaeKbaqbaaaa@2CFB@e^iωtR/2]P_S(i^3/2α)/J₀(i^3/2α). This expression is further simplified to

− μ [ ∂ u / ∂ r | r = R = [ A ⌣ R / 2 ] e i ω t + i θ P S − i θ J 0 | P S ( i 3 / 2 α ) | / | J 0 ( i 3 / 2 α ) | . MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=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@72CA@

The ratio of shear force per unit length, -2πRμ[∂u/∂r|_{r = R}, to volume flow rate is denoted K.

We now replace the harmonic pressure gradient by the exponential gradient Ae^at. A solution for this exponential pressure-gradient model is generated by substituting A for A⌣ MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafmyqaeKbaqbaaaa@2CFB@ and a for iω in Equations (4), (5) and (6). Note that, with these substitutions, the Bessel functions become real-valued series in which each term is a positive real number. The solutions for flow rate and shear force per unit area in the exponential pressure wave model are, respectively,

Q = [πAR⁴/(8μ)]e^atP_Q(iβR)/J₀(iβR)

and

-μ[∂ u/∂r]|_{r = R} = [AR/2]e^atP_S(iβR)/J₀(iβR),

where β =(aρ/μ)^1/2. The function I(iβR) is equal to Q divided by the force gradient πR²Ae^at. When

aρR²/μ << 8,

P_Q(iβR), P_S(iβR) and J₀(iβR) are approximately equal to 1, I(iβR) is approximately R²/(8μ), and the function K is approximately equal to 8μ/(ρR²). When aρR²/μ ≫ 8, I(iβR) is approximately equal to 1/(aρ), and K is approximately equal to 0.

Now consider the fluid motion when the force function, F = πR²P, is

F = fe^ate^-bz,

where a, b and f are positive real-valued constants. At time t, this expression defines the force function of distance z along the tube. A point defined by a particular value of z can be traced backward in time to a point at z = 0 on the force function at time t-z/c. In the absence of damping as a result of shear forces, the value of the force function would be identical for these two points. Therefore, fe^ate^-bz = fe^a(t-z/c), so that b = z/c in the absence of damping.

It is assumed that the ratio of flow rate to force gradient is constant in our model of a segment of an artery that does not contain a branch. An equivalent assumption is that the value of R² does not vary significantly with decreasing pressure along the segment. As a consequence, it follows that flow rat, Q, is proportional to e^a(t-z/c) in the absence of damping, i.e. the flow wave is described by Q₀e^a(t-z/c), where Q₀ is the flow rate at t = 0 and z = 0.

In the absence of damping, there is no loss of momentum per unit length in the velocity wave. When there is damping, Equations (7) and (8) imply that the point on the velocity wave at z = 0 at t = 0 loses momentum at specific rate -K as it travels distance z = t/c in time t. Therefore, momentum, velocity and flow rate are reduced by the factor e^-Kz/c as the flow wave moves a distance equal to z, and volume flow rate is proportional to Q₀e^a(t-z/c) e^-Kz/c. By analogy to the rigid-tube model where flow rate is proportional to the force gradient -∂F/∂z for small R, it is assumed that, in the model where small changes in R are allowed, force gradient is also proportional to volume flow rate (for small R). Therefore force is likewise proportional to e^a(t-z/c) e^-Kz/c, and we write

F = fe^ate^{-z(a + K)/c}.

The force gradient is

-∂F/∂z = [(a + K)/c]fe^ate^{-z(a + K)/c}.

Substitution into Equation (7) gives

Q = [(a + K)/c]fe^ate^-z(az+K)/cI(iβR).

For an incompressible fluid in a cylindrical elastic tube, conservation of mass requires that

-∂Q/∂z = 2πR(∂R/∂t).

Substitution into Equation (12) gives

[(a + K)/c]²fe^ate^{-z(a + K)/c} I(iβR)- [(a + K)/c]fe^ate^-z(az+K)/c(∂I(iβR)/∂R)(∂R/∂z) = 2πR(∂R/∂t).

For an elastic tube with wall thickness equal to h and elastic modulus equal to E,

P = Eh(R-R₀)/R².

Consequently, F = πEh(R - R₀), and ∂F/∂z = πEh(∂R/∂z), which is rewritten as

-[(a + K)/c]fe^ate^{-z(a + K)/c} = πEh(∂R/∂z).

Similarly,

afe^at e^-z(az+K)/c = πEh(∂R/∂t).

Combining Equations (16) and (17) with Equation (14) gives a description of the flow-pressure coupling (FPC) associated with the pulse wave. The FPC equation will be solved for the PWV. This will be simplified by considering two cases. The first is when aρR²/μ << 8. I(iβR) is approximated as R²(8μ), and the function K is approximated as 8μ/(ρR²). Combining Equation (16) and (17) with Equation (14) leads to

Eh/(2ρR) + 2fe^ate^-z(az+K)/c/(πρR²) = c²aK/(a + K)².

Equation (18) is rewritten as

Eh/(2ρR)[1 + 4fe^ate^{-z(a + K)/c}/(πREh)] = c²aK/(a + K)²,

and combining this equation with Equations (10) and (15) leads to

Eh/(2ρR)[1 + 4(R-R₀)/R] = c²aK/(a + K)².

When

4(R - R₀)/R ≪ 1,

the above equation is approximated as

c² ≈ [Eh/(2ρR)](a + K)²/(aK).

which is rewritten as

c 2 / c 0 2 ≈ a / K + 2 + K / a . MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaem4yam2aaWbaaSqabeaacqaIYaGmaaGccqGGVaWlcqWGJbWydaqhaaWcbaGaeGimaadabaGaeGOmaidaaOGaeyisISRaemyyaeMaei4la8Iaem4saSKaey4kaSIaeGOmaiJaey4kaSIaem4saSKaei4la8IaemyyaeMaeiOla4caaa@3ED2@

Because a/K = aρR²/(8μ) is assumed to be much smaller than 1,

c ≈ c₀ [2+8μ/(aρR²)]^1/2.

This result is not in agreement with Womersley's prediction that the PWV is approximately proportional to c₀ multiplied by R when aρR²/μ << 8.

For the case where aρR²/μ ≫ 8, I(iβR) is approximately equal to 1/(aρ), and K is approximately equal to 0. Substitution into Equation (14) leads to the Korteweg-Moens expression,

c² ≈ Eh/(2ρR).

The mass of the arterial wall and surrounding tissue was not included in the above analysis. The effect of this mass on the PWV can be assessed by writing a differential equation for its acceleration caused by the difference in fluid pressure in the tube and the force per unit area of the inner wall on the fluid. The solution leads to the approximation

P ≈ (a²[R-R₀]h_sρ_s + Eh)(R-R₀)/R²,

where h_s and ρ_s are the thickness and density, respectively, of the wall and surrounding tissue accelerated by the increasing arterial pressure.

There are many published estimates of the modulus E for mammalian elastic arteries. Estimates from studies of the change in arterial radius with changes in pressure are usually between 10⁶ dynes/cm² and 10⁷ dynes/cm² 1415. Therefore, the term a²R²h_sρ_s may be small compared with Eh unless the rate of rise in the arterial pulse is very steep.

Womersley 9 replaced the rigid tube of Lambossy with an elastic tube that expands or contracts in response to increasing or decreasing pressure of blood. If the tube is not tethered to surrounding tissue, it also moves axially in response to frictional force of blood on the inner wall. Womersley denoted the radial displacement of the inner wall of the tube by ξ and the longitudinal displacement by ς. It is assumed that ξ and ς are described by harmonic functions:

ξ = D₁ exp[iω(t - z/c)],

ς = E₁ exp [iω(t-z/c)].

The pressure of the fluid p is also assumed to be a harmonic function

p = p₁ exp [iω(t-z/c)].

If c is a positive real number, this equation imposes a direction of flow for the pulse wave.

In Equations (35) and (36) of the article by Womersley 9, the boundary conditions for the fluid in contact with the tube's inner wall are described by:

i ω E 1 = C 1 + A 1 ρ 0 c , MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaemyAaKMaeqyYdCNaemyrau0aaSbaaSqaaiabigdaXaqabaGccqGH9aqpcqWGdbWqdaWgaaWcbaGaeGymaedabeaakiabgUcaRKqbaoaalaaabaGaemyqae0aaSbaaeaacqaIXaqmaeqaaaqaaiabeg8aYnaaBaaabaGaeGimaadabeaacqWGJbWyaaGccqGGSaalaaa@3D63@

i ω D 1 = 1 2 i ω R c { C 1 F 10 ( α ) + A 1 ρ 0 c } , MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaemyAaKMaeqyYdCNaemiraq0aaSbaaSqaaiabigdaXaqabaGccqGH9aqpjuaGdaWcaaqaaiabigdaXaqaaiabikdaYaaadaWcaaqaaiabdMgaPjabeM8a3jabdkfasbqaaiabdogaJbaakiabcUha7jabdoeadnaaBaaaleaacqaIXaqmaeqaaOGaemOray0aaSbaaSqaaiabigdaXiabicdaWaqabaGccqGGOaakcqaHXoqycqGGPaqkcqGHRaWkjuaGdaWcaaqaaiabdgeabnaaBaaabaGaeGymaedabeaaaeaacqaHbpGCdaWgaaqaaiabicdaWaqabaGaem4yamgaaOGaeiyFa0NaeiilaWcaaa@4F19@

where ρ₀ is the density of the fluid, C₁ is a constant and F₁₀(α) is 2J₁(i^3/2α)/[i^3/2αJ₀(i^3/2α)]. In Equations (37) and (38) of the article, the boundary conditions for the wall of the tube in contact with the fluid are described by:

− ω 2 D 1 = A 1 h ρ − B ρ { σ R ( − i ω E 1 c ) + D 1 R 2 } , MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaeyOeI0IaeqyYdC3aaWbaaSqabeaacqaIYaGmaaGccqWGebardaWgaaWcbaGaeGymaedabeaakiabg2da9KqbaoaalaaabaGaemyqae0aaSbaaeaacqaIXaqmaeqaaaqaaiabdIgaOjabeg8aYbaakiabgkHiTKqbaoaalaaabaGaemOqaieabaGaeqyWdihaaOGaei4EaSxcfa4aaSaaaeaacqaHdpWCaeaacqWGsbGuaaGccqGGOaakjuaGdaWcaaqaaiabgkHiTiabdMgaPjabeM8a3jabdweafnaaBaaabaGaeGymaedabeaaaeaacqWGJbWyaaGccqGGPaqkcqGHRaWkjuaGdaWcaaqaaiabdseaenaaBaaabaGaeGymaedabeaaaeaacqWGsbGudaahaaqabeaacqaIYaGmaaaaaOGaeiyFa0NaeiilaWcaaa@54C9@

− ω 2 E 1 = − μ h ρ R { − C 1 2 i 3 α 2 R 2 F 10 ( α ) + 1 2 A 1 ρ 0 c ω 2 R 2 c 2 } − B ρ { ω 2 E 1 c 2 + σ R i ω D 1 c } , MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaeyOeI0IaeqyYdC3aaWbaaSqabeaacqaIYaGmaaGccqWGfbqrdaWgaaWcbaGaeGymaedabeaakiabg2da9KqbaoaalaaabaGaeyOeI0IaeqiVd0gabaGaemiAaGMaeqyWdiNaemOuaifaaOGaei4EaSxcfa4aaSaaaeaacqGHsislcqWGdbWqdaWgaaqaaiabigdaXaqabaaabaGaeGOmaidaaOGaemyAaK2aaWbaaSqabeaacqaIZaWmaaGccqaHXoqydaahaaWcbeqaaiabikdaYaaakiabdkfasnaaCaaaleqabaGaeGOmaidaaOGaemOray0aaSbaaSqaaiabigdaXiabicdaWaqabaGccqGGOaakcqaHXoqycqGGPaqkcqGHRaWkjuaGdaWcaaqaaiabigdaXaqaaiabikdaYaaadaWcaaqaaiabdgeabnaaBaaabaGaeGymaedabeaaaeaacqaHbpGCdaWgaaqaaiabicdaWaqabaGaem4yamgaaOGaeqyYdC3aaWbaaSqabeaacqaIYaGmaaqcfa4aaSaaaeaacqWGsbGudaahaaqabeaacqaIYaGmaaaabaGaem4yam2aaWbaaeqabaGaeGOmaidaaaaakiabc2ha9jabgkHiTKqbaoaalaaabaGaemOqaieabaGaeqyWdihaaOGaei4EaSxcfa4aaSaaaeaacqaHjpWDdaahaaqabeaacqaIYaGmaaGaemyrau0aaSbaaeaacqaIXaqmaeqaaaqaaiabdogaJnaaCaaabeqaaiabikdaYaaaaaGccqGHRaWkjuaGdaWcaaqaaiabeo8aZbqaaiabdkfasbaadaWcaaqaaiabdMgaPjabeM8a3jabdseaenaaBaaabaGaeGymaedabeaaaeaacqWGJbWyaaGccqGG9bqFcqGGSaalaaa@7DFE@

where ρ is the density of the tube, σ is Poisson's ratio and B is E/(1-σ²).

Womersley interprets Equations (27)–(30) as a system of homogenous linear equations with variables A₁, C₁, D₁ and E₁. Setting the determinant of coefficients equal to 0 gives the equation for c. A solution for c is easily found when σ = 0 and when the tube is tethered to surrounding structures so that E₁ = 0. Combining Equations (27) and (28) gives

D 1 = − 1 2 R C 1 c { 1 − F 10 ( α ) } . MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaemiraq0aaSbaaSqaaiabigdaXaqabaGccqGH9aqpcqGHsisljuaGdaWcaaqaaiabigdaXaqaaiabikdaYaaadaWcaaqaaiabdkfasjabdoeadnaaBaaabaGaeGymaedabeaaaeaacqWGJbWyaaGccqGG7bWEcqaIXaqmcqGHsislcqWGgbGrdaWgaaWcbaGaeGymaeJaeGimaadabeaakiabcIcaOiabeg7aHjabcMcaPiabc2ha9jabc6caUaaa@43BE@

Combining Equations (27) and (29) gives

(-ω²hρ + Eh/R²)D₁ = -ρ₀cC₁,

and dividing by Equation (31) gives

c² = {-ω²Rhρ/(2ρ₀) + Eh/(2ρ₀R)}{1-F₁₀(α)}.

Substituting -J₂(i^3/2α)/J₀(i^3/2α) for 1-F₁₀(α) gives

c² = {-ω²Rhρ/(2ρ₀) + Eh/(2ρ₀R)}{-J₂(i^3/2α)/J₀(i^3/2α)}.

Because ω²Rhρ/(2ρ₀) is small compared to Eh/(2ρ₀R), we have

c 2 / c 0 2 ≈ − J 2 ( i 3 / 2 α ) / J 0 ( i 3 / 2 α ) . MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaem4yam2aaWbaaSqabeaacqaIYaGmaaGccqGGVaWlcqWGJbWydaqhaaWcbaGaeGimaadabaGaeGOmaidaaOGaeyisISRaeyOeI0IaemOsaO0aaSbaaSqaaiabikdaYaqabaGccqGGOaakcqWGPbqAdaahaaWcbeqaaiabiodaZiabc+caViabikdaYaaakiabeg7aHjabcMcaPiabc+caViabdQeaknaaBaaaleaacqaIWaamaeqaaOGaeiikaGIaemyAaK2aaWbaaSqabeaacqaIZaWmcqGGVaWlcqaIYaGmaaGccqaHXoqycqGGPaqkcqGGUaGlaaa@4B33@

Womersley interprets the imaginary part of 1/c multiplied by ω as the coefficient in the exponential damping function of the wave as a function of distance. The exponential damping coefficient of time is therefore equal to the real part of c times the imaginary part of 1/c multiplied by ω. When α² << 8, the real part of c is approximately c₀α/4, and the imaginary part of 1/c is approximately 4/(c₀α). Therefore, the technique of Womersley leads to the prediction that the coefficient for the damping with time is approximately equal to ω and that this coefficient is approximately independent of the radius in small arteries. This does not make sense because Equations (5) and (6) show that for α² << 8 the damping coefficient is approximately 8μ/(ρR²), the expression used in the exponential pressure gradient model to describe damping of pulse waves in small arteries.

There is another solution for the PWV in the above equations from the paper of Womersley. Combining equations (27) and (30) for the case where σ = 0 and E₁ = 0 leads to

c² = ω²/[iα²F₁₀(α)],

which can not be correct because it predicts that c does not approach the Korteweg-Moens velocity for large values of R.

The equations of Womersley do not contain an expression for the damping caused by shear force between the inner wall of the tube and the moving liquid. If the expression for shear force is added to the Womersley model, Equation (31) becomes

i ω D 1 = R 2 ( i ω + K ) 2 A 1 i ω ρ 0 c 2 { 1 − F 10 ( α ) } , MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaemyAaKMaeqyYdCNaemiraq0aaSbaaSqaaiabigdaXaqabaGccqGH9aqpjuaGdaWcaaqaaiabdkfasbqaaiabikdaYaaadaWcaaqaaiabcIcaOiabdMgaPjabeM8a3jabgUcaRiabdUealjabcMcaPmaaCaaabeqaaiabikdaYaaacqWGbbqqdaWgaaqaaiabigdaXaqabaaabaGaemyAaKMaeqyYdCNaeqyWdi3aaSbaaeaacqaIWaamaeqaaiabdogaJnaaCaaabeqaaiabikdaYaaaaaGccqGG7bWEcqaIXaqmcqGHsislcqWGgbGrdaWgaaWcbaGaeGymaeJaeGimaadabeaakiabcIcaOiabeg7aHjabcMcaPiabc2ha9jabcYcaSaaa@53FB@

and Equation (32) becomes

(-ω²hρ + Eh/R²)D₁ = iωA₁/(iω).

When ω²ρ ≪ E/R², the above equations are combined and approximated by

c 2 = E h 2 ρ 0 R ( i ω + K ) 2 i ω { 1 − F 10 ( α ) } / ( i ω ) MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaem4yam2aaWbaaSqabeaacqaIYaGmaaGccqGH9aqpjuaGdaWcaaqaaiabdweafjabdIgaObqaaiabikdaYiabeg8aYnaaBaaabaGaeGimaadabeaacqWGsbGuaaWaaSaaaeaacqGGOaakcqWGPbqAcqaHjpWDcqGHRaWkcqWGlbWscqGGPaqkdaahaaqabeaacqaIYaGmaaaabaGaemyAaKMaeqyYdChaaOGaei4EaSNaeGymaeJaeyOeI0IaemOray0aaSbaaSqaaiabigdaXiabicdaWaqabaGccqGGOaakcqaHXoqycqGGPaqkcqGG9bqFcqGGVaWlcqGGOaakcqWGPbqAcqaHjpWDcqGGPaqkaaa@53E1@

When α² << 8, {1-F₁₀(α)}/(iω) is closely approximated by 1/K. Furthermore, replacing {1-F₁₀(α)}/(iω) by 1/K and replacing iω by the constant a gives Equation (21). Therefore, it appears that the difference between Equation (21) and the corresponding expression derived by Womersely for c² in the tethered, thin-walled tube model is due largely to his omission of the expression for damping due to friction between the wall of the tube and blood moving downstream in the pressure wave.