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



Future Work

Introduction

Goals:

1 Lower frequencies than expected2 Applications3 We want do do this simply




Future Work

Introduction

Goals:1 Lower frequencies than expected

2 Applications3 We want do do this simply




Future Work

Introduction

Goals:1 Lower frequencies than expected2 Applications

3 We want do do this simply




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



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.




Future Work


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




Future Work



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.




Future Work



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.




Future Work



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)




Future Work


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.




Future Work


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.




Future Work


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.




Future Work


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)




Future Work


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




Future Work


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.




Future Work


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.




Future Work

The Cylindrical Wave EquationSeparation of VariablesProblems With This Method

The Cylindrical Wave Equation




Future Work


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.




Future Work


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 .




Future Work


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.




Future Work


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)




Future Work


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.




Future Work


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.




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.




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.




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




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.




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


http://www.feynmanlectures.caltech.edu/I_47.html

http://www.oocities.org/geoy0703/physics/waves/AP-wave.html

http://www.oocities.org/geoy0703/physics/waves/AP-wave.html

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