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Transcript of modeling-speed-sound (1)
IntroductionDerivation of 1D Wave Equation for Variable Cross-Sectional Area
Analytical Approach in Cylindrical Coordinates2D Numerical Approach in Cartesian Coordinates
Future Work
Modeling the Speed of Sound Through aCorrugated Tube
Tyler Markham
University of North Carolina at Asheville
December 1, 2014
Tyler Markham Modeling the Speed of Sound Through a Corrugated Tube
IntroductionDerivation of 1D Wave Equation for Variable Cross-Sectional Area
Analytical Approach in Cylindrical Coordinates2D Numerical Approach in Cartesian Coordinates
Future Work
Introduction
Goals:
1 Lower frequencies than expected2 Applications3 We want do do this simply
Tyler Markham Modeling the Speed of Sound Through a Corrugated Tube
IntroductionDerivation of 1D Wave Equation for Variable Cross-Sectional Area
Analytical Approach in Cylindrical Coordinates2D Numerical Approach in Cartesian Coordinates
Future Work
Introduction
Goals:1 Lower frequencies than expected
2 Applications3 We want do do this simply
Tyler Markham Modeling the Speed of Sound Through a Corrugated Tube
IntroductionDerivation of 1D Wave Equation for Variable Cross-Sectional Area
Analytical Approach in Cylindrical Coordinates2D Numerical Approach in Cartesian Coordinates
Future Work
Introduction
Goals:1 Lower frequencies than expected2 Applications
3 We want do do this simply
Tyler Markham Modeling the Speed of Sound Through a Corrugated Tube
IntroductionDerivation of 1D Wave Equation for Variable Cross-Sectional Area
Analytical Approach in Cylindrical Coordinates2D Numerical Approach in Cartesian Coordinates
Future Work
Introduction
Goals:1 Lower frequencies than expected2 Applications3 We want do do this simplyTyler Markham Modeling the Speed of Sound Through a Corrugated Tube
IntroductionDerivation of 1D Wave Equation for Variable Cross-Sectional Area
Analytical Approach in Cylindrical Coordinates2D Numerical Approach in Cartesian Coordinates
Future Work
Volume of a Conic SectionConceptual FrameworkConservation of MassApplying the Definition of the DerivativeWrapping it Up
Volume of a Conic Section
For our derivation, let A = πr2 and A0 = πR2. Call this volumeV0.
Tyler Markham Modeling the Speed of Sound Through a Corrugated Tube
IntroductionDerivation of 1D Wave Equation for Variable Cross-Sectional Area
Analytical Approach in Cylindrical Coordinates2D Numerical Approach in Cartesian Coordinates
Future Work
Volume of a Conic SectionConceptual FrameworkConservation of MassApplying the Definition of the DerivativeWrapping it Up
Conceptual Framework
Relationships we need to keep in mind:
1 The gas density changes with displacement
2 Changes in density ⇔ changes in pressure
3 Changes in cross-sectional area ⇔ changes in density
4 Changes in cross-sectional area ⇔ changes in displacement
5 Changes in pressure instigate the motion of the gas
Tyler Markham Modeling the Speed of Sound Through a Corrugated Tube
IntroductionDerivation of 1D Wave Equation for Variable Cross-Sectional Area
Analytical Approach in Cylindrical Coordinates2D Numerical Approach in Cartesian Coordinates
Future Work
Volume of a Conic SectionConceptual FrameworkConservation of MassApplying the Definition of the DerivativeWrapping it Up
Conceptual Framework
Relationships we need to keep in mind:
1 The gas density changes with displacement
2 Changes in density ⇔ changes in pressure
3 Changes in cross-sectional area ⇔ changes in density
4 Changes in cross-sectional area ⇔ changes in displacement
5 Changes in pressure instigate the motion of the gas
Tyler Markham Modeling the Speed of Sound Through a Corrugated Tube
IntroductionDerivation of 1D Wave Equation for Variable Cross-Sectional Area
Analytical Approach in Cylindrical Coordinates2D Numerical Approach in Cartesian Coordinates
Future Work
Volume of a Conic SectionConceptual FrameworkConservation of MassApplying the Definition of the DerivativeWrapping it Up
Conceptual Framework
Relationships we need to keep in mind:
1 The gas density changes with displacement
2 Changes in density ⇔ changes in pressure
3 Changes in cross-sectional area ⇔ changes in density
4 Changes in cross-sectional area ⇔ changes in displacement
5 Changes in pressure instigate the motion of the gas
Tyler Markham Modeling the Speed of Sound Through a Corrugated Tube
IntroductionDerivation of 1D Wave Equation for Variable Cross-Sectional Area
Analytical Approach in Cylindrical Coordinates2D Numerical Approach in Cartesian Coordinates
Future Work
Volume of a Conic SectionConceptual FrameworkConservation of MassApplying the Definition of the DerivativeWrapping it Up
Conceptual Framework
Relationships we need to keep in mind:
1 The gas density changes with displacement
2 Changes in density ⇔ changes in pressure
3 Changes in cross-sectional area ⇔ changes in density
4 Changes in cross-sectional area ⇔ changes in displacement
5 Changes in pressure instigate the motion of the gas
Tyler Markham Modeling the Speed of Sound Through a Corrugated Tube
IntroductionDerivation of 1D Wave Equation for Variable Cross-Sectional Area
Analytical Approach in Cylindrical Coordinates2D Numerical Approach in Cartesian Coordinates
Future Work
Volume of a Conic SectionConceptual FrameworkConservation of MassApplying the Definition of the DerivativeWrapping it Up
Conceptual Framework
Relationships we need to keep in mind:
1 The gas density changes with displacement
2 Changes in density ⇔ changes in pressure
3 Changes in cross-sectional area ⇔ changes in density
4 Changes in cross-sectional area ⇔ changes in displacement
5 Changes in pressure instigate the motion of the gas
Tyler Markham Modeling the Speed of Sound Through a Corrugated Tube
IntroductionDerivation of 1D Wave Equation for Variable Cross-Sectional Area
Analytical Approach in Cylindrical Coordinates2D Numerical Approach in Cartesian Coordinates
Future Work
Volume of a Conic SectionConceptual FrameworkConservation of MassApplying the Definition of the DerivativeWrapping it Up
Conceptual Framework
Relationships we need to keep in mind:
1 The gas density changes with displacement
2 Changes in density ⇔ changes in pressure
3 Changes in cross-sectional area ⇔ changes in density
4 Changes in cross-sectional area ⇔ changes in displacement
5 Changes in pressure instigate the motion of the gas
Tyler Markham Modeling the Speed of Sound Through a Corrugated Tube
IntroductionDerivation of 1D Wave Equation for Variable Cross-Sectional Area
Analytical Approach in Cylindrical Coordinates2D Numerical Approach in Cartesian Coordinates
Future Work
Volume of a Conic SectionConceptual FrameworkConservation of MassApplying the Definition of the DerivativeWrapping it Up
Volume of a Conic Section
The volume of this conic section is given by the following equation:
V0 =π
3
(r2 + rR+R2
)∆x
=1
3
(A+
√A0A+A0
)∆x
Let A = A0 +Ae, where Ae represents a small change incross-sectional area.
Tyler Markham Modeling the Speed of Sound Through a Corrugated Tube
IntroductionDerivation of 1D Wave Equation for Variable Cross-Sectional Area
Analytical Approach in Cylindrical Coordinates2D Numerical Approach in Cartesian Coordinates
Future Work
Volume of a Conic SectionConceptual FrameworkConservation of MassApplying the Definition of the DerivativeWrapping it Up
Volume of a Conic Section
Since(AeA0
)2� 1, utilizing the relationship A = A0 +Ae and
approximating the Taylor series for√
1 + AeA0
gives us an initial
volume of
V0 =1
3
(2A0 +Ae +A0
√1 +
AeA0
)∆x
=1
3
(3A0 +
3
2Ae
)∆x
=
(A0 +
1
2Ae
)∆x.
Tyler Markham Modeling the Speed of Sound Through a Corrugated Tube
IntroductionDerivation of 1D Wave Equation for Variable Cross-Sectional Area
Analytical Approach in Cylindrical Coordinates2D Numerical Approach in Cartesian Coordinates
Future Work
Volume of a Conic SectionConceptual FrameworkConservation of MassApplying the Definition of the DerivativeWrapping it Up
Volume of a Conic Section
Let A = A(x+ ∆x) and A0 = A(x).
Plugging this into our expression for V0 yields
V0 =1
2(A(x+ ∆x) +A(x)) ∆x. (1)
Tyler Markham Modeling the Speed of Sound Through a Corrugated Tube
IntroductionDerivation of 1D Wave Equation for Variable Cross-Sectional Area
Analytical Approach in Cylindrical Coordinates2D Numerical Approach in Cartesian Coordinates
Future Work
Volume of a Conic SectionConceptual FrameworkConservation of MassApplying the Definition of the DerivativeWrapping it Up
Conservation of Mass
Let ρ = ρ0 + ρe. This gives us
ρ0V0 = ρV
ρ0 (A(ξ) +A(x)) = (ρ0 + ρe) (A(ξ + χs) +A(x+ χ))]Deltax
(∂χ
∂x+ 1
),
where ξ = x+ ∆x, χs = χ(x+ ∆x, t) is the shifted displacement,and χ = χ(x, t) is the displacement.
Tyler Markham Modeling the Speed of Sound Through a Corrugated Tube
IntroductionDerivation of 1D Wave Equation for Variable Cross-Sectional Area
Analytical Approach in Cylindrical Coordinates2D Numerical Approach in Cartesian Coordinates
Future Work
Volume of a Conic SectionConceptual FrameworkConservation of MassApplying the Definition of the DerivativeWrapping it Up
Conservation of Mass
With the approximation that ρe∂χ∂x � 1, we find that
ρe = −ρ0[∂χ
∂x+∂A
∂x
(χs + χ
A(ξ + χs) +A(x+ χ)
)]. (2)
When there is no change in cross-sectional area, we getFeynman’s expression back.
Changes in density ⇔ changes in displacement ⇔ changes inarea.
Tyler Markham Modeling the Speed of Sound Through a Corrugated Tube
IntroductionDerivation of 1D Wave Equation for Variable Cross-Sectional Area
Analytical Approach in Cylindrical Coordinates2D Numerical Approach in Cartesian Coordinates
Future Work
Volume of a Conic SectionConceptual FrameworkConservation of MassApplying the Definition of the DerivativeWrapping it Up
Conservation of Mass
With the approximation that ρe∂χ∂x � 1, we find that
ρe = −ρ0[∂χ
∂x+∂A
∂x
(χs + χ
A(ξ + χs) +A(x+ χ)
)]. (2)
When there is no change in cross-sectional area, we getFeynman’s expression back.
Changes in density ⇔ changes in displacement ⇔ changes inarea.
Tyler Markham Modeling the Speed of Sound Through a Corrugated Tube
IntroductionDerivation of 1D Wave Equation for Variable Cross-Sectional Area
Analytical Approach in Cylindrical Coordinates2D Numerical Approach in Cartesian Coordinates
Future Work
Volume of a Conic SectionConceptual FrameworkConservation of MassApplying the Definition of the DerivativeWrapping it Up
Conservation of Mass
With the approximation that ρe∂χ∂x � 1, we find that
ρe = −ρ0[∂χ
∂x+∂A
∂x
(χs + χ
A(ξ + χs) +A(x+ χ)
)]. (2)
When there is no change in cross-sectional area, we getFeynman’s expression back.
Changes in density ⇔ changes in displacement ⇔ changes inarea.
Tyler Markham Modeling the Speed of Sound Through a Corrugated Tube
IntroductionDerivation of 1D Wave Equation for Variable Cross-Sectional Area
Analytical Approach in Cylindrical Coordinates2D Numerical Approach in Cartesian Coordinates
Future Work
Volume of a Conic SectionConceptual FrameworkConservation of MassApplying the Definition of the DerivativeWrapping it Up
Applying the Definition of the Derivative
Since ∆x is small, we can say
χ(x+ ∆x, t)− χ(x, t) =∂χ
∂x∆x
χ(x+ ∆x, t) + χ(x, t) = 2χ(x, t) +∂χ
∂x∆x
χs + χ = 2χ+∂χ
∂x∆x.
A very simlar method can be use to also show
A(ξ + χs) +A(x+ χ) = 2A+ (∆x+ χs + χ)∂A
∂x
= 2A+
(∆x+ 2χ+
∂χ
∂x∆x
)∂A
∂x.
Tyler Markham Modeling the Speed of Sound Through a Corrugated Tube
IntroductionDerivation of 1D Wave Equation for Variable Cross-Sectional Area
Analytical Approach in Cylindrical Coordinates2D Numerical Approach in Cartesian Coordinates
Future Work
Volume of a Conic SectionConceptual FrameworkConservation of MassApplying the Definition of the DerivativeWrapping it Up
Preliminary Result
Plugging this back into (2), neglecting combinations of ∆x, ρe,∂A∂x , and ∂χ
∂x , we find (2) approximates to
ρe = −ρ0A
∂
∂x(χA). (3)
Tyler Markham Modeling the Speed of Sound Through a Corrugated Tube
IntroductionDerivation of 1D Wave Equation for Variable Cross-Sectional Area
Analytical Approach in Cylindrical Coordinates2D Numerical Approach in Cartesian Coordinates
Future Work
Volume of a Conic SectionConceptual FrameworkConservation of MassApplying the Definition of the DerivativeWrapping it Up
Wrapping it Up
Applying Newton’s Second Law to our diagram:
Fnet = P (x, t)A(x)− P (x+ ∆x, t)A(x+ ∆x)
Fnet = ma =1
2ρ0(A(x+ ∆x) +A(x))
∂2χ
∂t2
Tyler Markham Modeling the Speed of Sound Through a Corrugated Tube
IntroductionDerivation of 1D Wave Equation for Variable Cross-Sectional Area
Analytical Approach in Cylindrical Coordinates2D Numerical Approach in Cartesian Coordinates
Future Work
Volume of a Conic SectionConceptual FrameworkConservation of MassApplying the Definition of the DerivativeWrapping it Up
Wrapping it Up
Again, for small ∆x and ∂A∂x , we can say
∂
∂x(PA) = −1
2ρ0
(2A+ ∆x
∂A
∂x
)∂2χ
∂t2
= −ρ0A∂2χ
∂t2.
Tyler Markham Modeling the Speed of Sound Through a Corrugated Tube
IntroductionDerivation of 1D Wave Equation for Variable Cross-Sectional Area
Analytical Approach in Cylindrical Coordinates2D Numerical Approach in Cartesian Coordinates
Future Work
Volume of a Conic SectionConceptual FrameworkConservation of MassApplying the Definition of the DerivativeWrapping it Up
Wrapping it Up
Let P = P0 + Pe. Using Feynman’s relation Pe = Λρe, where
Λ =(∂P∂ρ
)0, and plugging it into our last equation:
∂
∂t2(χA) = v2s
∂
∂x2(χA), (4)
where Λ = v2s and vs is the phase speed of sound in air.
Tyler Markham Modeling the Speed of Sound Through a Corrugated Tube
IntroductionDerivation of 1D Wave Equation for Variable Cross-Sectional Area
Analytical Approach in Cylindrical Coordinates2D Numerical Approach in Cartesian Coordinates
Future Work
The Cylindrical Wave EquationSeparation of VariablesProblems With This Method
The Cylindrical Wave Equation
Tyler Markham Modeling the Speed of Sound Through a Corrugated Tube
IntroductionDerivation of 1D Wave Equation for Variable Cross-Sectional Area
Analytical Approach in Cylindrical Coordinates2D Numerical Approach in Cartesian Coordinates
Future Work
The Cylindrical Wave EquationSeparation of VariablesProblems With This Method
The Cylindrical Wave Equation
3D Wave Equation:
∇2P =1
v2s
∂2P
∂t2
Ignoring the φ term for ∇2 in cylindrical coordinates we have:
∂2P
∂r2+
1
r
∂P
∂r+∂2P
∂z2=
1
v2s
∂2P
∂t2.
Tyler Markham Modeling the Speed of Sound Through a Corrugated Tube
IntroductionDerivation of 1D Wave Equation for Variable Cross-Sectional Area
Analytical Approach in Cylindrical Coordinates2D Numerical Approach in Cartesian Coordinates
Future Work
The Cylindrical Wave EquationSeparation of VariablesProblems With This Method
Separation of Variables
Let P (r, z, t) = R(r)Z(z)T (t). It follows that
1
R
∂2R
∂r2+
1
rR
∂R
∂r+
1
Z
∂2Z
∂z2=
1
v2sT
∂2T
∂t2(5)
Let both sides be equal to −k2. Note the terms with r dependencyand z dependency must both be constant as well. Call theseconstants kr and kz respectively, where k =
√k2r + k2z .
Tyler Markham Modeling the Speed of Sound Through a Corrugated Tube
IntroductionDerivation of 1D Wave Equation for Variable Cross-Sectional Area
Analytical Approach in Cylindrical Coordinates2D Numerical Approach in Cartesian Coordinates
Future Work
The Cylindrical Wave EquationSeparation of VariablesProblems With This Method
Three Ordinary Differential Equations
The partial differential equation splits into the following simplerODEs:
r2∂2R
∂r2+ r
∂R
∂r+ k2rr
2R = 0 (6)
∂2Z
∂z2= −k2zZ (7)
∂2T
∂t2= −ω2T. (8)
We define ω = vsk.
Tyler Markham Modeling the Speed of Sound Through a Corrugated Tube
IntroductionDerivation of 1D Wave Equation for Variable Cross-Sectional Area
Analytical Approach in Cylindrical Coordinates2D Numerical Approach in Cartesian Coordinates
Future Work
The Cylindrical Wave EquationSeparation of VariablesProblems With This Method
Boundary Conditions and Pressure Over Time
Applying the conditions of
n̂z · ~∇P∣∣∣z=0,L
= 0∂P
∂t
∣∣∣∣t=0
= 0,
we find that
Plm(r, z, t) = AlmJ0 (krlr) cos(mπz
L
)cos (ωlt) . (9)
Tyler Markham Modeling the Speed of Sound Through a Corrugated Tube
IntroductionDerivation of 1D Wave Equation for Variable Cross-Sectional Area
Analytical Approach in Cylindrical Coordinates2D Numerical Approach in Cartesian Coordinates
Future Work
The Cylindrical Wave EquationSeparation of VariablesProblems With This Method
Problems With This Method
If our radial boundary is constant then
n̂r · ~∇P∣∣∣r=r0
=∂P
∂r
∣∣∣∣r=r0
= 0.
It then follows that
P (r, z, t) =
∞∑l=1
∞∑m=1
∞∑n=1
AlmnJ0
(j1,n
(r
r0
))cos(mπz
L
)cos (ωlt) ,
(10)where j1,n is the nth root of the first order bessel function of thefirst kind.
Tyler Markham Modeling the Speed of Sound Through a Corrugated Tube
IntroductionDerivation of 1D Wave Equation for Variable Cross-Sectional Area
Analytical Approach in Cylindrical Coordinates2D Numerical Approach in Cartesian Coordinates
Future Work
The Cylindrical Wave EquationSeparation of VariablesProblems With This Method
Problems With This Mehtod
However, if our radial boundary is nonconstant:
n̂r → n̂r(z)⇒ n̂r(z) · ~∇P∣∣∣b(z)
= 0.
An analytical solution may exist, but cannot be done simply.
We will need to approach this numerically.
Tyler Markham Modeling the Speed of Sound Through a Corrugated Tube
IntroductionDerivation of 1D Wave Equation for Variable Cross-Sectional Area
Analytical Approach in Cylindrical Coordinates2D Numerical Approach in Cartesian Coordinates
Future Work
The Cylindrical Wave EquationSeparation of VariablesProblems With This Method
Problems With This Mehtod
However, if our radial boundary is nonconstant:
n̂r → n̂r(z)⇒ n̂r(z) · ~∇P∣∣∣b(z)
= 0.
An analytical solution may exist, but cannot be done simply.
We will need to approach this numerically.
Tyler Markham Modeling the Speed of Sound Through a Corrugated Tube
IntroductionDerivation of 1D Wave Equation for Variable Cross-Sectional Area
Analytical Approach in Cylindrical Coordinates2D Numerical Approach in Cartesian Coordinates
Future Work
The Cylindrical Wave EquationSeparation of VariablesProblems With This Method
Problems With This Mehtod
However, if our radial boundary is nonconstant:
n̂r → n̂r(z)⇒ n̂r(z) · ~∇P∣∣∣b(z)
= 0.
An analytical solution may exist, but cannot be done simply.
We will need to approach this numerically.
Tyler Markham Modeling the Speed of Sound Through a Corrugated Tube
IntroductionDerivation of 1D Wave Equation for Variable Cross-Sectional Area
Analytical Approach in Cylindrical Coordinates2D Numerical Approach in Cartesian Coordinates
Future Work
Finite Difference Method in 2D
Let the spatial step-size be h and let the time-step be τ . For afinite difference method we utilize the approximations
∂2P
∂x2=P (xi+1, yj , tk)− 2P (xi, yj , tk) + P (xi−1, yj , tk)
h2
∂2P
∂y2=P (xi, yj+1, tk)− 2P (xi, yj , tk) + P (xi, yj−1, tk)
h2
∂2P
∂t2=P (xi, yj , tk+1)− 2P (xi, yj , tk) + P (xi, yj , tk−1)
τ2
where xi = ih, yj = jh, tk = kτ , and i ∈ Z.
Tyler Markham Modeling the Speed of Sound Through a Corrugated Tube
IntroductionDerivation of 1D Wave Equation for Variable Cross-Sectional Area
Analytical Approach in Cylindrical Coordinates2D Numerical Approach in Cartesian Coordinates
Future Work
Recursion Relation for (k + 1)st Pressure Entry
Inserting this into the 2D wave equation in Cartesian coordinates,we have the following recursion relation:
Pi,j,k+1 =(2− λ2
)Pi,j,k + λ2 (Pi+1,j,k + Pi−1,j,k)
+ λ2 (Pi,j+1,k + Pi,j−1,k)− Pi,j,k−1,
where λ is defined by
λ =vsτ
h.
Tyler Markham Modeling the Speed of Sound Through a Corrugated Tube
IntroductionDerivation of 1D Wave Equation for Variable Cross-Sectional Area
Analytical Approach in Cylindrical Coordinates2D Numerical Approach in Cartesian Coordinates
Future Work
Next Spring
Goals for next spring:
1 Remove boundaries on sides
2 Apply this to the 3D cylindrical case
3 Include damping effects
4 Find the group velocity as a function of the corrugationparameters
Tyler Markham Modeling the Speed of Sound Through a Corrugated Tube
IntroductionDerivation of 1D Wave Equation for Variable Cross-Sectional Area
Analytical Approach in Cylindrical Coordinates2D Numerical Approach in Cartesian Coordinates
Future Work
Next Spring
Goals for next spring:
1 Remove boundaries on sides
2 Apply this to the 3D cylindrical case
3 Include damping effects
4 Find the group velocity as a function of the corrugationparameters
Tyler Markham Modeling the Speed of Sound Through a Corrugated Tube
IntroductionDerivation of 1D Wave Equation for Variable Cross-Sectional Area
Analytical Approach in Cylindrical Coordinates2D Numerical Approach in Cartesian Coordinates
Future Work
Next Spring
Goals for next spring:
1 Remove boundaries on sides
2 Apply this to the 3D cylindrical case
3 Include damping effects
4 Find the group velocity as a function of the corrugationparameters
Tyler Markham Modeling the Speed of Sound Through a Corrugated Tube
IntroductionDerivation of 1D Wave Equation for Variable Cross-Sectional Area
Analytical Approach in Cylindrical Coordinates2D Numerical Approach in Cartesian Coordinates
Future Work
Next Spring
Goals for next spring:
1 Remove boundaries on sides
2 Apply this to the 3D cylindrical case
3 Include damping effects
4 Find the group velocity as a function of the corrugationparameters
Tyler Markham Modeling the Speed of Sound Through a Corrugated Tube
IntroductionDerivation of 1D Wave Equation for Variable Cross-Sectional Area
Analytical Approach in Cylindrical Coordinates2D Numerical Approach in Cartesian Coordinates
Future Work
Next Spring
Goals for next spring:
1 Remove boundaries on sides
2 Apply this to the 3D cylindrical case
3 Include damping effects
4 Find the group velocity as a function of the corrugationparameters
Tyler Markham Modeling the Speed of Sound Through a Corrugated Tube
IntroductionDerivation of 1D Wave Equation for Variable Cross-Sectional Area
Analytical Approach in Cylindrical Coordinates2D Numerical Approach in Cartesian Coordinates
Future Work
Acknowledgements
I would like to thank Dr. Perkins for the time he took to guide methrough my research, Dr. Ruiz for providing this opportunity, theUNCA Undergraduate Research Department, and the physicsdepartment as a whole.
Tyler Markham Modeling the Speed of Sound Through a Corrugated Tube
IntroductionDerivation of 1D Wave Equation for Variable Cross-Sectional Area
Analytical Approach in Cylindrical Coordinates2D Numerical Approach in Cartesian Coordinates
Future Work
Resources
http://www.feynmanlectures.caltech.edu/I_47.html
http://www.oocities.org/geoy0703/physics/waves/
AP-wave.html
The Science and Applications of Acoustics Ed 2 by Daniel R.Raichel
Tyler Markham Modeling the Speed of Sound Through a Corrugated Tube