Ljud i byggnad och samhälle

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Ljud i byggnad och samhälle

(VTAF01) – Wave propagation (II)

MATHIAS BARBAGALLO

DIVISION OF ENGINEERING ACOUSTICS, LUND UNIVERSITY

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… recap from last lecture (I) – waves in solid media

• Longitudinal waves (∞ medium ≈ beams)

– Quasi-longitudunal waves (finite ≈ plates)

• Shear waves

• Bending waves (dispersive)

c

_L

= E ρ

c

_qL

= 𝐸

^′

ρ = E(1 − υ

²

) ρ

c

_sh

= G

ρ = E

2(1 + υ)ρ

c

_B

= ω

⁴

𝐵 B 𝜕 ⁴ v _y 𝑚

𝜕x ⁴ + m 𝜕 ² v _y

𝜕t ² = 0 G 𝜕 ² v _y

𝜕x ² − ρ 𝜕 ² v _y

𝜕t ² = 0 E ^′ 𝜕 ² v _x

𝜕x ² − ρ 𝜕 ² v _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 address here

B

_beam

= E bh

³

12

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• 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

… recap from last lecture (II) – waves in solid media

• One will always find a frequency where the wavelengths of bending waves in a structure and of longitudinal waves in air will match and,

therefore, effective

sound radiation occurs

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• General approach to derive equations of motion:

1. Newton’s law – dynamic equilibrium

2. Constitutive relations – forces, stresses and strains

• Förhållanden mellan två fysiska kvantiteter i ett visst material.

a. Force – stress b. Stress – strain

3. Strain – displacement relation

… recap from last lecture (III) – waves in solid media

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… recap from last lecture (IV) – Wave equation solution

• 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 t ± x c

f x ± vt = f(x ± vt + T)

f

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… recap from last lecture (V) – Wave equation solution

Travelling waves:

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• Bending waves (dispersive)

• More complex solutions including near-field terms

c

_B(ω)

= ω

⁴

𝐵 B 𝜕 ⁴ u _y 𝑚

𝜕x ⁴ + m 𝜕 ² u _y

𝜕t ² = 0

… recap from last lecture (VI) – waves in solid media

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• More complex solutions including near-field terms

… recap from last lecture (VII) – waves in solid media

Source: Sound and Vibration, Wallin, Carlsson, Åbom, Bodén, Glav

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• More complex solutions including near-field terms

… recap from last lecture (VIII) – waves in solid media

Source: Sound and Vibration, Wallin, Carlsson, Åbom, Bodén, Glav

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Outline

Introduction

Wave propagation in fluids Wave phenomena

Summary

Musical instruments

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Learning outcomes

• Wave propagation in fluid media

• Wave phenomena – Interference – Standing waves

» Resonances

» Eigenmodes

• How musical instruments work

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Outline

Introduction

Wave propagation in fluids Wave phenomena

Summary

Musical instruments

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Waves in fluid media

• 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

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Outline

Introduction

Wave propagation in fluids Wave phenomena

Summary

Musical instruments

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Summation of noise (I)

• Types of sources

‒ Correlated (or coherent)

» Constant phase difference, same frequency

» Interferences (constructive/destructive)

‒ Uncorrelated (or uncoherent)

L _p,tot = 20 log ෍

n=1 N

10 ^L ²⁰

^p,n

L _p,tot = 10 log ෍

n=1 N

10 ^L ¹⁰

^p,n

For uncorrelated sources, the 3

^rd

term vanishes

• The total RMS pressure:

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Summation of noise (I)

• Uncorrelated sources – constructive interference

• If they two sources have same sound pressure:

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Summation of noise (I)

• Correlated sources – constructive interference

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Summation of noise (I)

• Interference at a wall

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Wave phenomena (II)

• Interferences: constructive / destructive

• Standing waves (coherent source)

y ₋ x, t = ොy cos(ωt − kx)

y ₊ x, t = ොy cos(ωt + kx) y x, t = y ₋ x, t + y ₊ x, t = 2ොy sin(kx)cos ωt

Position-dependent amplitude oscillating according to cos(ωt) Two travelling waves of same frequency, type

and fixed phase relation propagating in opposite directions

y ₁ x, t = ොy cos(ωt − kx)

y ₂ x, t = ොy cos(ωt − kx + θ) y x, t = y ₁ x, t + y ₂ x, t = 2ොycos θ

2 sin(ωt − kx + θ)

Constructive/destructive depending on Ф

Source: Dan Russell

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Standing waves in a string: resonances & eigenmodes

λ=2L f

₁

=v/2L

Fundamental eigenfrequency / 1

^st

harmonic

λ=L f

₂

=2f

₁

Second eigenfrequency / 2

^nd

harmonic

λ=(2/3)L f

₃

=3f

₁

Third eigenfrequency / 3

^rd

harmonic

In general:

λ=2L/n f _n =n·v/2L

Eigenmode: different ways a string (structure in general) can vibrate generating standing waves

Examples: 1 / 2 / 3 / 4.

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Standing waves and higher harmonics (I)

Any motion = sum of motion of all the harmonics

Source: J. Hetricks

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Standing waves and higher harmonics (II)

Any motion = sum of motion of all the harmonics

Source: http://signalysis.com

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Outline

Introduction

Wave propagation in fluids Wave phenomena

Summary

Musical instruments

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Music instruments

• Systems’ eigenfrequencies are derived without external forces – the homogeneous solution.

• Interesting phenomena happens on the other hand when external forces with their own driving frequencies interact with systems’ eigenfrequencies – i.e.

resonance phenomena happen – the particular solution.

• Systems eigenfrequencies, and accordingly systems’ response to sound and

vibrations, will therefore sustain, maintain and add character to external driving frequencies. In musical acoustics one speaks of loudness, quality, timbre.

• Think about musical instrument, concert rooms.

• When this interaction is not properly managed though problems will occur

(collapsing bridges due to external excitation is an extreme example).

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Music instruments: string (e.g. violin)

λ=2L/n

f _n =n·v/2L v=(tension/mass-length) ^1/2

When ones plays  Changes L

Knobs (tune)  Vary tension

To discuss: Piano?

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Music instruments: wood-wind

Open-open / Closed-closed:

λ=2L/n f _n =n·v/2L

NOTE: Open-closed vary

v=(temperature/molecular weight) ^1/2

Change of v (molecular Cover holes  Vary L

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Music instruments: soundboards

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Standing waves and timbre (I)

• Characteristics of sound:

‒ Loudness (amplitude)

‒ Pitch (frequency)

‒ Quality or Timbre

» “Cocktail” characteristic of every instrument/source

» Different combination of higher harmonics

» What makes us distinguish one instrument from another

• To discuss: helium & voice

‒ Change of molecular weight (i.e. v)  natural frequency goes up

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Standing waves and timbre (II)

Source: Quora

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Outline

Introduction

Wave propagation in fluids Wave phenomena

Summary

Musical instruments

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Summary

• Wave propagation in fluid media

• Wave phenomena

– Interference (constructive/destructive) – Standing waves

» Resonances

» Eigenmodes

• How musical instruments work

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Ljud i byggnad och samhälle

Ljud i byggnad och samhälle

(VTAF01) – Wave propagation (II)

MATHIAS BARBAGALLO

… recap from last lecture (I) – waves in solid media

• Longitudinal waves (∞ medium ≈ beams)

– Quasi-longitudunal waves (finite ≈ plates)

• Shear waves

• Bending waves (dispersive)

c

= E ρ

c

= 𝐸

ρ = E(1 − υ

) ρ

c

= G

ρ = E

2(1 + υ)ρ

c

= ω

𝐵 B 𝜕 4 v y 𝑚

𝜕x 4 + m 𝜕 2 v y

𝜕t 2 = 0 G 𝜕 2 v y

𝜕x 2 − ρ 𝜕 2 v y

𝜕t 2 = 0 E ′ 𝜕 2 v x

𝜕x 2 − ρ 𝜕 2 v x

𝜕t 2 = 0

x y

Plate: E, G, ρ, υ, h

m = ρh

B

= Eh

12(1 − υ

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

B

= E bh

12

• Bending waves (dispersive)

• Dispersion relations – frequency dependance of wave speed

c

= ω

𝐵 B 𝜕 4 u y 𝑚

𝜕x 4 + m 𝜕 2 u y

𝜕t 2 = 0

… recap from last lecture (II) – waves in solid media

• One will always find a frequency where the wavelengths of bending waves in a structure and of longitudinal waves in air will match and,

therefore, effective

sound radiation occurs

• General approach to derive equations of motion:

1. Newton’s law – dynamic equilibrium

2. Constitutive relations – forces, stresses and strains

• Förhållanden mellan två fysiska kvantiteter i ett visst material.

a. Force – stress b. Stress – strain

3. Strain – displacement relation

… recap from last lecture (III) – waves in solid media

… recap from last lecture (IV) – Wave equation solution

• 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 t ± x c

f x ± vt = f(x ± vt + T)

f

… recap from last lecture (V) – Wave equation solution

Travelling waves:

• Bending waves (dispersive)

• More complex solutions including near-field terms

c

= ω

𝐵 B 𝜕 4 u y 𝑚

𝜕x 4 + m 𝜕 2 u y

𝜕t 2 = 0

… recap from last lecture (VI) – waves in solid media

𝐵 B 𝜕 ⁴ v _y 𝑚

𝜕x ⁴ + m 𝜕 ² v _y

𝜕t ² = 0 G 𝜕 ² v _y

𝜕x ² − ρ 𝜕 ² v _y

𝜕t ² = 0 E ^′ 𝜕 ² v _x

𝜕x ² − ρ 𝜕 ² v _x

𝜕t ² = 0

𝐵 B 𝜕 ⁴ u _y 𝑚

𝜕x ⁴ + m 𝜕 ² u _y

𝜕t ² = 0

𝐵 B 𝜕 ⁴ u _y 𝑚

𝜕x ⁴ + m 𝜕 ² u _y

𝜕t ² = 0

𝜕 ² p

𝜕x ² − 1 c ²

𝜕 ² p

𝜕t ² = 0

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

L _p,tot = 20 log ෍

10 ^L ²⁰

L _p,tot = 10 log ෍

10 ^L ¹⁰