ljud i byggnad och samhälle (vtaf01) wave propagation (ii)

JNegreira_AkustikLecture25Nov(VTAF01) – Wave propagation (II)
MATHIAS BARBAGALLO DIVISION OF ENGINEERING ACOUSTICS, LUND UNIVERSITY
M. Barbagallo, D. Bard / Ljud i byggnad och samhälle / VTAF01 / 2 April 2019
… recap from last lecture (I) – waves in solid media
• Longitudinal waves (∞ medium ≈ beams)
– Quasi-longitudunal waves (finite ≈ plates)
m = ρh
Bplate = Eh3
12(1 − υ2)NOTE: torsional waves (beams and columns) are not address here
Bbeam = E bh3
12
• Bending waves (dispersive)
cB(ω) = ω 4
• One will always find a
frequency where the
wavelengths of bending
of longitudinal waves in
air will match and,
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
… recap from last lecture (III) – waves in solid media
… recap from last lecture (IV) – Wave equation solution
• Travelling waves:
• Alternative forms:
y = f(x ± vt)
x
c
f
… recap from last lecture (V) – Wave equation solution
Travelling waves:
• Bending waves (dispersive)
cB(ω) = ω 4
… recap from last lecture (VI) – waves in solid media
• 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
• 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
Outline
Introduction
Musical instruments
Learning outcomes
• Wave phenomena
• How musical instruments work
Outline
Introduction
Musical instruments
Waves in fluid media
• Sound waves: longitudinal waves
Pressure as field variable
Velocity as field variable
2p
x2 −
1
p x, t = p± cos(ωt ± kx) = p±e −i(ωt±kx)
2v
λ
Outline
Introduction
Musical instruments
Summation of noise (I)
» Interferences (constructive/destructive)
• The total RMS pressure:
• Uncorrelated sources – constructive interference
• If they two sources have same sound pressure:
• Correlated sources – constructive interference
• Interference at a wall
Wave phenomena (II)
• Interferences: constructive / destructive
y− x, t = y cos(ωt − kx)
y+ x, t = y cos(ωt + kx) y x, t = y− x, t + y+ x, t = 2y 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
y1 x, t = y cos(ωt − kx)
y2 x, t = y cos(ωt − kx + θ) y x, t = y1 x, t + y2 x, t = 2ycos
θ
depending on
Source: Dan Russell
Standing waves in a string: resonances & eigenmodes
λ=2L
f1=v/2L
fn=n·v/2L
Eigenmode: different ways a string (structure in general) can vibrate generating standing waves
Examples: 1 / 2 / 3 / 4.
Standing waves and higher harmonics (I)
Any motion = sum of motion of all the harmonics
Source: J. Hetricks
Standing waves and higher harmonics (II)
Any motion = sum of motion of all the harmonics
Source: http://signalysis.com
Outline
Introduction
Musical instruments
Music instruments
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).
Music instruments: string (e.g. violin)
λ=2L/n
When ones plays Changes L
Knobs (tune) Vary tension
To discuss: Piano?
Music instruments: wood-wind
v=(temperature/molecular weight) 1/2
Change of v (molecular
Cover holes Vary L
Music instruments: soundboards
Standing waves and timbre (I)
• Characteristics of sound:
» What makes us distinguish one instrument from another
• To discuss: helium & voice
Change of molecular weight (i.e. v) natural frequency goes up
Standing waves and timbre (II)
Source: Quora
Outline
Introduction
Musical instruments
Summary
• Wave phenomena
– Interference (constructive/destructive)
– Standing waves
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ljud i byggnad och samhälle (vtaf01) wave propagation (ii)

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