N3_WaveProp_13Nov19_VTAN01Acoustics (VTAN01) 3. Wave propagation NIKOLAS VARDAXIS DIVISION OF ENGINEERING ACOUSTICS, LUND UNIVERSITY
Outline
Introduction
Wave equation solution
– Longitudinal/quasi-longitudinal waves
– Shear waves
– Bending waves
Electromagnetical waves (vacuum)
Longitudinal waves (solids and fluids)
Transverse waves (solids)
Outline
Introduction
Wave equation solution
• Longitudinal waves (∞ medium ≈ beams)
– Quasi-longitudunal waves (finite ≈ plates)
m = ρh
Bplate = EhK
12(1− υ.)NOTE: torsional waves (beams and columns) are not addressed here
Bbeam   = E bhK
Wave equation solution
• Alternative forms:
y = f(x ± vt) Space Time
Propagation speed
kx ± ωt k
= f kx ±ωt
f
Wave equation solution (II) Time and position dependency: u x, t = uUV cos ωt− kx = uUVeYZ(6[Y\E)
Wave equation solution (II) Time and position: u x, t = uUV cos ωt− kx = uUVeYZ(6[Y\E)
Other types…
• 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
Particles in elliptical paths. Ellipses width decreases with increasing depth
Change from depth>1/5 of λ
Outline
Introduction
Wave equation solution
.p x.
2 d 273 = 331.4 1 + T
2 d 273 ,cijkZli = D ρ ,
p x, t = p±V cos(ωt± kx) = p±VeYZ(6[±\E)
.v t.
Z ≡ p± v±
• Sound intensity
Sound power (i.e. rate of energy) per unit area [W/m2]
» Instantaneous value:
» In a free field:
In decibels (SWL / LW / L∏)…
Ls = 10 log pu.
p^jv. = 20 log pu p^jv
pu = pu f ≡ RMS  pressure p^jv = 2·10Y{  Pa = 20  µμPa p][i = 101  300  Pa p[}[(t) = p][i ± p(t)
I = pv = 1 T p t v t dt
a
 ; I^jv = 10Y.W m.
W t = I x,t dS
L = 10 log W W^jv
 ; W^jv = 10Y.W
Wave equation solution
• Standing waves (coherent source)
yY x, t = y cos(ωt − kx) yU x, t = y cos(ωt + kx) y x, t = yY x, t + yU 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
y x, t = y cos(ωt − kx) y. x, t = y cos(ωt− kx + θ)
y x, t = y x, t + y. x, t = 2ycos θ 2 sin  (ωt − kx + θ)
Constructive/destructive depending on
Source: Dan Russell
λ=2L f1=v/2L Fundamental eigenfrequency / 1st harmonic
λ=L f2=2f1 Second eigenfrequency / 2nd harmonic
λ=(2/3)L f3=3f1 Third eigenfrequency / 3rd harmonic
In general:
λ=2L/n fn=n·v/2L
Eigenmode: different ways a string (structure in general) can vibrate generating standing waves
Standing waves and higher harmonics (I)
Any motion = sum of motion of all the harmonics
Source: J. Hetricks
Any motion = sum of motion of all the harmonics
Source: http://signalysis.com
Wave equation solution
v=(tension/mass-length) 1/2
Knobs (tune) à Vary tension
To discuss: Piano?
Music instruments: wood-wind
v=(temperature/molecular weight) 1/2
Change of v (molecular weight) Cover holes à Vary L
Music instruments: soundboards
Pitch (frequency)
» What makes us distinguish one instrument from another
• To discuss: helium & voice
Change of molecular weight (i.e. v) à natural frequency goes up
Standing waves and timbre (II)
Source: Quora
Wave equation solution
• Wave equation solution
• Wave phenomena
– Interference (constructive/destructive)
Thank you for your attention! nikolas.vardaxis@construction.lth.se