– Wave propagation (I)

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Ljud i byggnad och samhälle (VTAF01) – Wave propagation (I)

MATHIAS BARBAGALLO

DIVISION OF ENGINEERING ACOUSTICS, LUND UNIVERSITY

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M. Barbagallo, D. Bard / Ljud i byggnad och samhälle / VTAF01 / 1 April 2019

… recap from last lecture (I)

• SDOF – mass, spring, damper

– Basic system yet very powerful and with broad applicability – Introduces concepts of:

» Equation of motion + initial conditions define a specific motion

» Eigenfrequency – own frequency of the system, belongs to it

» Resonance – frequency of external driving force matches eigenfrequency

– Transfer functions: ratio between a force and a response i.e.

output/input.

» Valid for linear systems, harmonic motions, independent of outer conditions

» Full with useful info – eigenfrequencies, phase shifts, eigenmodes.

NOTE: Degrees of freedom (DOF): number of independent displacement components to define exact position of a system

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… recap from last lecture (II)

• Undamped SDOF

Mü(t) + Ku(t) = F(t)

Inertial force

Elastic force

Applied force

F(t) = F

_driv

·cos(t)

) cos(

1 ) 1

sin(

) cos(

)

(

2

0 0

0 0 0

0

t

K t F t v

u t

u

_total ^driv



 

  

 





 





 

 



 







Homogeneous

Particular

M

 K



0

𝑢 𝑡 = 0 = 𝑢

₀

; ሶ𝑢 𝑡 = 0 = 𝑣

₀

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… recap from last lecture (III)

• Damped SDOF

Mü(t) + Rů(t) + Ku(t) = F(t)

Inertial force

Elastic force Damping

force

Applied force

F(t) = F

_driv

·cos(t)

u(t)

0

; ( 0 )

) 0

( t u u t v

u     

 sin( ) cos( )  sin( ) cos( )

)

( t e

² ⁰

B

₁

t B

₂

t D

₁

t D

₂

t

u

_total



^^^^t



_d

 

_d

   

Homogeneous Particular

2

0

1 



_d

 

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… recap from last lecture (IV)

• Complex representation of a damped SDOF

Mü(t) + Rů(t) + Ku(t) = F(t)

F(t) = F

_driv

·cos(t)

u(t)

෤u(ω) = F

_driv

K − Mω

²

+ Riω

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J. Negreira / Ljud i byggnad och samhälle / VTAF01 / 26 March 2018

SDOF – Driving frequencies

• Ex:

– Without damping – With damping

• Different driving freqs

 0

 

 0

    

₀

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SDOF – Frequency response functions (FRF) – (I)

• In general, FRF = transfer function, i.e.:

‒ Contains system information

‒ Independent of outer conditions

𝐻

_𝑖𝑗

𝜔 = 𝑠 ෥

_𝑖

(𝜔)

෥

𝑠

_𝑗

(𝜔) = output input

Source: https://community.plm.automation.siemens.com/t5/Testing-Knowledge-Base/The-FRF-and-its-Many-Forms

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SDOF – Frequency response functions (FRF) – (II)

Source: https://community.plm.automation.siemens.com/t5/Testing-Knowledge-Base/The-FRF-and-its-Many-Forms

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SDOF – Frequency response functions (FRF) – (IV)

• Real and imaginary parts – the imaginary part has interesting information

Source: https://community.plm.automation.siemens.com/t5/Testing-Knowledge-Base/The-FRF-and-its-Many-Forms

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Helmholtz resonator (II)

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Helmholtz resonator (IV)

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Outline

Wave propagation in solid media Introduction

Wave propagation in fluids

Summary

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Learning outcomes

• Wave propagation in solid media

– Longitudinal/quasi-longitudinal waves – Shear waves

– Bending waves

• Wave propagation in fluid media

• Wave equation solution

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Types of waves – classification

• Depending on propagation media

‒ Mechanical waves (solids and fluids)

‒ Electromagnetical waves (vacuum)

• Propagation direction – 1D, 2D and 3D

• Based on periodicity

– Periodics and non-periodics

• Based on particles’ movement in relation with propagation direction:

‒ Longitudinal waves (solids and fluids)

‒ Transverse waves (solids)

• … NOTE: waves do not transport mass, just energy

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Outline

Wave propagation in solid media Introduction

Wave propagation in fluids

Summary

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Types of waves in solid media (I)

• Longitudinal waves

• Shear waves

• Torsional waves

• Bending waves

• Rayleigh waves

• Lamb waves

• …

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Types of waves in solid media (II)

• Longitudinal waves (∞ medium ≈ beams) – Quasi-longitudunal waves (finite ≈ plates)

• Longitudinell våg, en våg där punkterna i vågmediet svänger i vågens

utbredningsriktning. Härvid komprimeras mediet, och den återställande kraften ges av tryck. Ett exempel är en ljudvåg eller en vanlig fjäder.

• Motsatsen är en transversell våg där punkterna i vågmediet svänger vinkelrätt mot utbredningsriktningen. Exempel: stränginstrument, vattnet i en

damm och elektromagnetisk strålning.

c

_L

= E ρ

c

_qL

= 𝐸

^′

ρ = E

ρ(1 − υ

²

)

𝜕

²

u

_x

𝜕x

²

− ρ E

^′

𝜕

²

u

_x

𝜕t

²

= 0

Plate: E, G, ρ, υ, h

Source: Wikipedia.se

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Types of waves in solid media (III)

• Longitudinal waves (∞ medium ≈ beams) – Quasi-longitudunal waves (finite ≈ plates)

– Poissons konstant, Poissons tal eller tvärkontraktionstalet är en materialkonstant som anger hur ett material reagerar på tryck- och dragkrafter. När ett material (blått) töjs ut i en riktning dras det ihop i andra riktningar (grönt).

Plate: E, G, ρ, υ, h

c

_L

= E ρ

c

_qL

= 𝐸

^′

ρ = E

ρ(1 − υ

²

)

Source: Wikipedia.se

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Types of waves in solid media (IV)

• Shear waves

Skjuvning, eller skjuvtöjning, är en deformation utan volymändring. Den definieras som vinkeländringen skapad av deformationen.

Plate: E, G, ρ, υ, h

𝜕

²

u

_y

𝜕x

²

− ρ 𝐺

𝜕

²

u

_y

𝜕t

²

= 0 ^c

^sh

⁼

G

ρ = E

2(1 + υ)ρ

Source: Wikipedia.se

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Types of waves in solid media (V)

• Bending waves (dispersive)

• Böjvågen är en mekanisk vågtyp som finns i tunna fasta strukturer som platta och balkar i form av böjning. Böjvågor är speciellt viktiga i akustiken då de är bra på att stråla ut ljud till omgivningen.

c

_B(ω)

= ω

⁴

𝐵

B 𝜕

⁴

u

_y

𝑚

𝜕x

⁴

+ m 𝜕

²

u

_y

𝜕t

²

= 0

x y

Plate: E, G, ρ, υ, h

m = ρh

B

_plate

= Eh

³

12(1 − υ

²

)

B

_beam

= E bh

³

12

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Types of waves in solid media (VI)

• Bending waves (dispersive)

• Dispersion relations – frequency dependance of wave speed

c

_B(ω)

= ω

⁴

𝐵

B 𝜕

⁴

u

_y

𝑚

𝜕x

⁴

+ m 𝜕

²

u

_y

𝜕t

²

= 0

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Types of waves in solid media (VII)

• Longitudinal waves (∞ medium ≈ beams) – Quasi-longitudunal waves (finite ≈ plates)

• Shear waves

• Bending waves (dispersive)

c

_L

= E ρ

c

_qL

= 𝐸

^′

ρ = E

ρ(1 − υ

²

)

c

_sh

= G

ρ = E

2(1 + υ)ρ

c

_B(ω)

= ω

4

𝐵

B 𝜕

⁴

u

_y

𝑚

𝜕x

⁴

+ m 𝜕

²

u

_y

𝜕t

²

= 0

𝜕

²

u

_y

𝜕x

²

− ρ 𝐺

𝜕

²

u

_y

𝜕t

²

= 0

𝜕

²

u

_x

𝜕x

²

− ρ E

^′

𝜕

²

u

_x

𝜕t

²

= 0

x y

Plate: E, G, ρ, υ, h

m = ρh

B

_plate

= Eh

³

12(1 − υ

²

) NOTE: torsional waves (beams and columns) are not addressed here

B

_beam

= E bh

³

12

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Derivation of longitudinal wave equations (I)

• General approach to derive equations of motion:

1. Newton’s law – dynamic equilibrium

2. Constitutive relations – forces, stresses and strains

• Relations between two physical quantities in a material

a. Force – stress b. Stress – strain

3. Strain – displacement relation (definition)

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Derivation of longitudinal wave equations (II)

• Stress: Dragspänning eller normalspänning definieras som den negativa spänning som uppstår i en enaxligt belastad stång utsatt för en dragkraft. Krafterna normeras med planets yta, så att dessa spänningar har enheten för tryck. Vanligtvis betecknas

dragspänning med den grekiska bokstaven sigma.

• Strain: Töjning (elongation) är ett enhetslöst, geometrioberoende mått

på deformationsgraden och betecknas ε (epsilon). Töjning kan anta både positiva och negativa värden beroende på om objektet utsatts för drag- eller tryckspänning.

• Constitutive relations: Förhållanden mellan två fysiska kvantiteter i ett visst material.

Source: Wikipedia.se

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Wave equation solution (I)

• Travelling waves:

• Alternative forms:

• Note:

• Periodic functions:

• Harmonic functions: is a sinus or cosinus

y = f(x ± vt)

Space Time

Propagation speed

y = f x ± ω

k t = f kx ± ωt

k = f kx ± ωt

f x ± vt = f(x ± vt + T)

f

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Wave equation solution (II)

Travelling waves – d’Alambert’s solution:

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Wave equation solution (III)

Time and position dependency: ^{u x, t = ෞ} ^u

⁺

cos ωt − kx = ෞ u

₊

e

^{−i(ωt−kx)}

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Other types…

• In reality, combinations of aforementioned waves can exist, e.g.

• Surface waves

Water waves (long+transverse waves)

Particles in clockwise circles. The radius of the circles decreases increasing depth

Pure shear waves don’t exist in fluids

• Body waves

Rayleigh waves (long+transverse waves)

Particles in elliptical paths. Ellipses width decreases with increasing depth

Change from depth>1/5 of λ

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Outline

Wave propagation in solid media Introduction

Wave propagation in fluids

Summary

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Waves in fluid media (I)

• Sound waves: longitudinal waves

‒ Pressure as field variable

‒ Velocity as field variable

Comparing both equations: (acoustic impedance)

𝜕 ² p

𝜕x ² − 1 c ²

𝜕 ² p

𝜕t ² = 0

c

_air

= γP

₀

ρ(T = 0°C) 1 + T

2 ∙ 273 = 331.4 1 + T 2 ∙ 273 , c

_medium

= D

ρ ,

p x, t = ෞ p _± cos(ωt ± kx) = ෞ p _± e ^{−i(ωt±kx)}

𝜕 ² v

𝜕t ² = c ² 𝜕 ² v

𝜕x ² v x, t = 1

ρc p ෞ _± e ^{−i(ωt±kx)}

Z ≡ p _±

v _± = ±ρc

k = 2π

λ

𝑇 = 20℃; 𝑐

_𝑎𝑖𝑟

= 343 𝑚/𝑠

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Outline

Wave propagation in solid media Introduction

Wave propagation in fluids

Summary

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Summary

• Wave propagation in solid media

• Wave propagation in fluid media

REFERENCES: Animations retrieved from Dan Russell’s website

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