Lecture 2: Acoustics - Columbia UniversityMany musical instruments-guitar (plucked)- ... lip...
33
E6820 SAPR - Dan Ellis L02 - Acoustics - 2006-01-26 - 1 EE E6820: Speech & Audio Processing & Recognition Lecture 2: Acoustics The wave equation Acoustic tubes: reflections & resonance Oscillations & musical acoustics Spherical waves & room acoustics Dan Ellis <[email protected]> http://www.ee.columbia.edu/~dpwe/e6820/ Columbia University Dept. of Electrical Engineering Spring 2006 1 2 3 4
Transcript of Lecture 2: Acoustics - Columbia UniversityMany musical instruments-guitar (plucked)- ... lip...
E68
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Dan
Elli
sL0
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s -
2006
-01-
26 -
1
EE
E68
20: S
peec
h &
Aud
io P
roce
ssin
g &
Rec
ogni
tion
Lec
ture
2:
Aco
ust
ics
Th
e w
ave
equ
atio
n
Aco
ust
ic t
ub
es: r
eflec
tio
ns
& r
eso
nan
ce
Osc
illat
ion
s &
mu
sica
l aco
ust
ics
Sp
her
ical
wav
es &
ro
om
aco
ust
ics
Dan
Elli
s <
dpw
e@ee
.col
umbi
a.ed
u>ht
tp://
ww
w.e
e.co
lum
bia.
edu/
~dp
we/
e682
0/
Col
umbi
a U
nive
rsity
Dep
t. of
Ele
ctric
al E
ngin
eerin
gS
prin
g 20
06
1 2 3 4
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Aco
ust
ics
& s
ou
nd
•A
cou
stic
s is
th
e st
ud
y o
f p
hysi
cal w
aves
•(A
cou
stic
) w
aves
tra
nsm
it e
ner
gy
wit
ho
ut
per
man
entl
y d
isp
laci
ng
mat
ter
(e.g
. oce
an w
aves
)
•S
ame
mat
h r
ecu
rs in
man
y d
om
ain
s
•In
tuit
ion
: pu
lse
go
ing
do
wn
a r
op
e
1
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Th
e w
ave
equ
atio
n
•C
on
sid
er a
sm
all s
ecti
on
of
the
rop
e:
•d
isp
lace
men
t
y
(
x
)
, ten
sio
n
S
, mas
s
ε
·
dx
→
late
ral f
orc
e is
...
φ 1
φ 2
S
Sx
y
ε
Fy
Sφ 2(
)si
n⋅
Sφ 1(
)si
n⋅
–=
Sx22
∂∂y
dx
≈
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Wav
e eq
uat
ion
(2)
•N
ewto
n’s
law
:
•C
all
(t
ensi
on
to
mas
s-p
er-l
eng
th)
hen
ce:
th
e W
ave
Eq
uat
ion
:
.. p
arti
al D
E r
elat
ing
cu
rvat
ure
an
d a
ccel
erat
ion
Fm
a=
Sx22
∂∂y
xd⋅
⋅ε
xdt22
∂∂y
⋅=
c2SεÚ
=
c2
x22
∂∂y
⋅t22
∂∂y
=
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So
luti
on
to
th
e w
ave
equ
atio
n
•If
(
any
f
(·)
)
then
also
wo
rks
for
Hen
ce, g
ener
al s
olu
tio
n:
yx
t,(
)f
xct
–(
)=
x∂∂y
f'x
ct–
()
=
x22
∂∂y
f''
xct
–(
)=
t∂∂yc–
f'x
ct–
()
⋅=
t22
∂∂y
c2f'
'x
ct–
()
⋅=
yx
t,(
)f
xct
+(
)=
c2
x22
∂∂y
⋅t22
∂∂y
=
y
xt,
()
y+x
ct–
()
y–x
ct+
()
+=
⇒
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So
luti
on
to
th
e w
ave
equ
atio
n (
2)
• a
nd
a
re t
rave
llin
g w
aves
-sh
ape
stay
s co
nst
ant
but
chan
ges
po
siti
on
:
•c
is t
rave
llin
g w
ave
velo
city
( ∆
x / ∆
t )
•y+
mov
es r
igh
t, y–
mov
es le
ft
•re
sult
ant
y(x)
is s
um
of
the
two
wav
es
y+x
ct–
()
y–x
ct+
()
time
0:
x x
y
y+ y-
time
T:
y+ y-
∆x
= c
·T
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Wav
e eq
uat
ion
so
luti
on
s (3
)
•W
hat
is t
he
form
of
y+ , y
– ?
-an
y do
ubly
-diff
eren
tiabl
e fu
nctio
n w
ill s
atis
fy
wav
e eq
uatio
n
•A
ctu
al w
aves
hap
es d
icta
ted
by
bo
un
dar
y co
nd
itio
ns
-e.
g. y
(x)
at t
= 0
-pl
us c
onst
rain
ts o
n y
at p
artic
ular
x’s
e.g.
inp
ut m
otio
n y(
0, t)
= m
(t)
rig
id te
rmin
atio
n y(
L, t
) =
0
x
y
y(0,
t) =
m(t
)
y+(x
,t)y(
L,t)
= 0
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Term
inat
ion
s an
d r
eflec
tio
ns
•S
yste
m c
on
stra
ints
:-
initi
al y
(x, 0
) =
0 (
flat r
ope)
-in
put y
(0, t
) =
m(t
) (
at a
gent
’s h
and)
(→
y+)
-te
rmin
atio
n y(
L, t
) =
0 (
fixed
end
)
-w
ave
equa
tion
y(x,
t) =
y+(x
- c
t) +
y– (x
+ c
t)
•A
t te
rmin
atio
n:
y(L
, t)
= 0
→
y+(L
- c
t) =
– y
– (L
+ c
t)i.e
. y+ a
nd y
– ar
e m
irror
ed in
tim
e an
d am
plitu
dear
ound
x =
L→
inve
rted
refl
ectio
n at
term
inat
ion
sim
ulat
ion
[trav
el1.
m]
x =
L
y(x,
t)=
y+
+ y
–
y+
y–
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Aco
ust
ic t
ub
es
•S
ou
nd
wav
es t
rave
l do
wn
aco
ust
ic t
ub
es:
-1-
dim
ensi
onal
; ver
y si
mila
r to
str
ings
•C
om
mo
n s
itu
atio
n:
-w
ind
inst
rum
ent b
ores
-ea
r ca
nal
-vo
cal t
ract
2
x
pressure
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Pre
ssu
re a
nd
vel
oci
ty
•C
on
sid
er a
ir p
arti
cle
dis
pla
cem
ent
:
•P
arti
cle
velo
city
henc
e vo
lum
e ve
loci
ty
•(R
elat
ive)
air
pre
ssu
re
ξx
t,(
)
x
ξ(x)
vx
t,(
)t∂∂ξ
=
ux
t,(
)A
vx
t,(
)⋅
=
px
t,(
)1 κ---
–x∂∂ξ ⋅
=
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Wav
e eq
uat
ion
for
a tu
be
•C
on
sid
er e
lem
enta
l vo
lum
e:
•N
ewto
n’s
law
:
•H
ence
x
Are
a dA
For
ce p
·dA
For
ce (
p+∂p
/ ∂x·
dx)·
dAV
olum
e dA
·dx
Mas
s ρ·
dA·d
x
Fm
a=
x∂∂
p–
dx
dA
⋅⋅
ρd
Ad
xt∂∂v ⋅
=
x∂∂
p⇒
ρt∂∂v
–=
c2
x22
∂∂ξ
⋅t22
∂∂ξ
=c
1 ρκ
--------
---=
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Aco
ust
ic t
ub
e tr
avel
ing
wav
es
•Tr
avel
ing
wav
es in
par
ticl
e d
isp
lace
men
t:
•C
all
•T
hen
vo
lum
e ve
loci
ty:
•A
nd
pre
ssu
re:
•(S
cale
d)
sum
& d
iff.
of
trav
elin
g w
aves
ξx
t,(
)ξ+
xct
–(
)ξ-
xct
+(
)+
=
u+α(
)cA
α∂∂ξ+
α()
–= Z
0ρ
c A------
=
ux
t,(
)A
t∂∂ξ ⋅u+
xct
–(
)u-
xct
+(
)–
==
px
t,(
)1 κ---–
x∂∂ξ ⋅
Z0
u+x
ct–
()
u-x
ct+
()
+[
]⋅
==
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Aco
ust
ic t
ub
e tr
avel
ing
wav
es (
2)
•D
iffe
ren
t re
sult
ants
for
pre
ssu
re a
nd
vo
lum
e ve
loci
ty:
x
u+ u- u(x,
t)=
u+
- u
-
p(x,
t)=
Z0[
u+ +
u- ]
c
c
Aco
ustic
tube
Volu
me
velo
city
Pre
ssu
re
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Term
inat
ion
s in
tu
bes
•E
qu
ival
ent
of
fixe
d p
oin
t fo
r tu
bes
?
•O
pen
en
d is
like
fixe
d p
oin
t fo
r ro
pe:
refl
ects
wav
e b
ack
inve
rted
•U
nlik
e fi
xed
po
int,
solid
wal
l refl
ects
tra
velin
g
wav
e w
ith
ou
t in
vers
ion
u 0(t)
(Vol
ume
velo
city
inpu
t)
Sol
id w
all f
orce
s u(
x,t)
= 0
henc
e u+
= u
-
henc
e u+
= -
u-
Ope
n en
d fo
rces
p(x,
t) =
0
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Sta
nd
ing
wav
es
•C
on
sid
er (
com
ple
x) s
inu
soid
al in
pu
t:
•P
ress
ure
/vo
lum
e m
ust
hav
e fo
rm
•H
ence
tra
velin
g w
aves
:
wh
ere
(sp
atia
l fre
qu
ency
, rad
/m)
(wav
elen
gth
)
•P
ress
ure
, vo
l. ve
loc.
res
ult
ants
sh
ow
st
atio
nar
y p
atte
rn: s
tan
din
g w
aves
-ev
en w
hen
| A| ≠
|B|
→si
mul
atio
n [s
intw
avem
ov.m
]
u 0t()
U0
ejω
t⋅
=
Ke
jω
tφ
+(
)
u+x
ct–
()
Ae
jkx
–ω
tφ
A+
+(
)=
u-x
ct+
()
Be
jkx
ωt
φB
++
()
=k
ωcÚ
=λ
cfÚ
2π
cωÚ
==
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Sta
nd
ing
wav
es (
2)
•F
or
loss
less
ter
min
atio
n (
|u+
| = |u
- |),
hav
e tr
ue
no
des
& a
nti
no
des
•P
ress
ure
an
d v
ol.
velo
c. a
re p
has
e sh
ifte
d-
in s
pace
and
in ti
me
∗
U0 ejω
t
kx =
π�
x =
λ /
2
pres
sure
= 0
(no
de)
vol.v
eloc
. = m
ax
(
antin
ode)
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17
Tran
sfer
fu
nct
ion
•C
on
sid
er t
ub
e ex
cite
d b
y :
-si
nuso
idal
trav
elin
g w
aves
mus
t sat
isfy
te
rmin
atio
n ‘b
ound
ary
cond
ition
s’-
satis
fied
by c
ompl
ex c
onst
ants
A a
nd B
in
-st
andi
ng w
ave
patte
rn w
ill s
cale
with
inpu
t m
agni
tude
-po
int o
f exc
itatio
n m
akes
a b
ig d
iffer
ence
...
u 0t()
U0
ejω
t⋅
=
ux
t,(
)u+
xct
–(
)u-
xct
+(
)+
=
Ae
jkx
–ω
t+
()
Be
jkx
ωt
+(
)+
=
ejω
tA
ejk
x–
Be
jkx
+(
)⋅
=
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Tran
sfer
fu
nct
ion
(2)
•F
or
op
en-e
nd
ed tu
be
of l
eng
th L
exc
ited
at x
= 0
by
:
(mat
ches
at x
= 0
, max
imum
at x
= L
)
•i.e
. sta
nd
ing
wav
e p
atte
rne.
g. v
aryi
ng L
for
a gi
ven ω
(an
d he
nce
k):
mag
nitu
de o
f UL d
epen
ds o
n L
(an
d ω
)...
U0e
jωt
ux
t,(
)U
0e
jωt
kL
x–
()
cos
kLco
s----
--------
--------
---------
⋅=
kω c----
=
U0 ejω
t U0 ejω
t
U0
U0
UL
UL
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Tran
sfer
fu
nct
ion
(3)
•V
aryi
ng
ω fo
r a
giv
en L
, i.e
. at
x =
L :
•O
utp
ut
vol.
velo
c. a
lway
s la
rger
th
an in
pu
t
•U
nb
ou
nd
ed fo
r
i.e. r
eso
nan
ce (
amp
litu
de
gro
ws
w/o
bo
un
d)
UL
U0
-------
uL
t,(
)u
0t,
()
--------
--------
1 kLco
s----
--------
---1 ω
LcÚ
()
cos
--------
--------
--------
----=
==
L
u(L
)�u(
0)
u(L
)�u(
0)�∞ a
t ω
L/c
= (
2r+
1)π/
2, r
= 0
,1,2
...
L2
r1
+(
)πc
2ω
-------
2r
1+
()λ 4---
==
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Res
on
ant
mo
des
•F
or
loss
less
tu
be
wit
h
, m o
dd
, λ w
avel
eng
th,
is u
nb
ou
nd
ed, m
ean
ing
:
-tr
ansf
er fu
nctio
n ha
s po
le o
n fr
eque
ncy
axis
-en
ergy
at t
hat f
requ
ency
sus
tain
s in
defin
itely
-co
mpa
re to
tim
e do
mai
n...
•e.
g 1
7.5
cm v
oca
l tra
ct, c
= 3
50 m
/s→
ω0
= 2π
· 50
0 H
z (t
hen
150
0, 2
500
...)
Lm
λ 4--- ⋅=
uL(
)u
0()
--------
---
L =
λ0 /
4
L =
3 · λ 1
/ 4
→ ω
1 =
3ω
0
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21
Sca
tter
ing
jun
ctio
ns
•S
olv
e e.
g. f
or
u- k
and
u+ k+
1: (
gen
eral
ized
ter
m.)
Are
a A
kA
rea
Ak+
1
At a
brup
t cha
nge
in a
rea:
• p
ress
ure
mus
t be
cont
inuo
us
pk(
x, t)
= p
k+1(
x, t)
• v
ol. v
eloc
. mus
t be
cont
inuo
us
uk(
x, t)
= u
k+1(
x, t)
• tr
avel
ing
wav
es
u+
k, u
- k, u
+k+
1, u
- k+1
w
ill b
e di
ffere
nt
u+k
u-k
u+k+
1
u-k+
1
+
+u+
k
u-k
u+k+
1
u-k+
1
2r 1 +
r
1 -
r1
+ r
2r
+ 1
r -
1r
+ 1
r =
Ak+
1
Ak
“Are
a ra
tio”
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Co
nca
ten
ated
tu
be
mo
del
•Vo
cal t
ract
act
s as
a w
aveg
uid
e
•D
iscr
ete
app
rox.
as
vary
ing
-dia
met
er t
ub
e:
Glo
ttis�
u 0(t)
Glo
ttis
Lips
�u L
(t)
Lips
x =
0
x =
L
x
Ak,
Lk Ak+
1, L
k+1
E68
20 S
AP
R -
Dan
Elli
sL0
2 -
Aco
ustic
s -
2006
-01-
26 -
23
Co
nca
ten
ated
tu
be
reso
nan
ces
•C
on
cate
nat
ed t
ub
es →
sca
tter
ing
jun
ctio
ns
→ la
ttic
e fi
lter
•C
an s
olv
e fo
r tr
ansf
er f
un
ctio
n -
all-
po
le
•A
pp
roxi
mat
e vo
wel
syn
thes
is f
rom
res
on
ance
sso
un
d e
xam
ple
: ah
ee
oo
+
+u+
k
u-k
e-jωτ 1
e-jωτ 1
+
+e-jω
τ 2
e-jωτ 2
+
+
+ +
e-jω2τ
1
+ +
+ +
e-jω2τ
2
-1-0
.50
0.5
1
-1
-0.50
0.51
Imaginary part
E68
20 S
AP
R -
Dan
Elli
sL0
2 -
Aco
ustic
s -
2006
-01-
26 -
24
Osc
illat
ion
s &
mu
sica
l aco
ust
ics
•P
itch
(p
erio
dic
ity)
is e
ssen
ce o
f m
usi
c:
-w
hy?
why
mus
ic?
•D
iffe
ren
t ki
nd
s o
f o
scill
ato
rs:
-si
mpl
e ha
rmon
ic m
otio
n (t
unin
g fo
rk)
-re
laxa
tion
osci
llato
r (v
oice
)-
strin
g tr
avel
ing
wav
e (p
luck
ed/s
truc
k/bo
wed
)-
air
colu
mn
(non
linea
r en
ergy
ele
men
t)
3
E68
20 S
AP
R -
Dan
Elli
sL0
2 -
Aco
ustic
s -
2006
-01-
26 -
25
Sim
ple
har
mo
nic
mo
tio
n
•B
asic
mec
han
ical
osc
illat
ion
:
•S
pri
ng
+ m
ass
( +
dam
per
)
•e.
g. t
un
ing
fork
•N
ot
gre
at fo
r m
usi
c:-
fund
amen
tal (
cosω
t) o
nly
-re
lativ
ely
low
ene
rgy
xω
2x
–=
xA
ωt
ϕ+
()
cos
=
m x
F =
kx
ζω
2 =
k� m
E68
20 S
AP
R -
Dan
Elli
sL0
2 -
Aco
ustic
s -
2006
-01-
26 -
26
Rel
axat
ion
osc
illat
or
•M
ult
i-st
ate
pro
cess
:-
one
stat
e bu
ilds
up p
oten
tial (
e.g.
pre
ssur
e)-
switc
h to
sec
ond
(rel
ease
) st
ate
-re
vert
to fi
rst s
tate
etc
.
•e.
g. v
oca
l fo
lds:
( ht
tp://
ww
w.m
edic
ine.
uiow
a.ed
u/ot
olar
yngo
logy
/cas
es/n
orm
al/n
orm
al2.
htm
)
•O
scill
atio
n p
erio
d d
epen
ds
on
forc
e (t
ensi
on
)-
easy
to c
hang
e-
hard
to k
eep
stab
le→
less
use
d in
mus
ic
pu
E68
20 S
AP
R -
Dan
Elli
sL0
2 -
Aco
ustic
s -
2006
-01-
26 -
27
Rin
gin
g s
trin
g
•e.
g. o
ur
ori
gin
al ‘r
op
e’ e
xam
ple
•M
any
mu
sica
l in
stru
men
ts-
guita
r (p
luck
ed)
-pi
ano
(str
uck)
-vi
olin
(bo
wed
)
•C
on
tro
l per
iod
(p
itch
):-
chan
ge le
ngth
(fr
ettin
g)-
chan
ge te
nsio
n (t
unin
g pi
ano)
-ch
ange
mas
s (p
iano
str
ings
)
•In
flu
ence
of
exci
tati
on
... [
plu
ck1a
.m]
L
tens
ion
S�m
ass/
leng
th ε
ω2
= π
2 S�
L2 ε
E68
20 S
AP
R -
Dan
Elli
sL0
2 -
Aco
ustic
s -
2006
-01-
26 -
28
Win
d t
ub
e
•R
eso
nan
t tu
be
+ en
erg
y in
pu
t
•e.
g. c
lari
net
-lip
pre
ssur
e ke
eps
reed
clo
sed
-re
flect
ed p
ress
ure
wav
e op
ens
reed
-re
info
rced
pre
ssur
e w
ave
pass
es th
roug
h
• F
ing
er h
ole
s d
eter
min
e fi
rst
refl
ecti
on
→ e
ffec
tive
wav
egu
ide
len
gth
ener
gy
nonl
inea
rel
emen
tsc
atte
ring
junc
tion
(ton
ehol
e)
(qua
rter
wav
elen
gth)
acou
stic
wav
egui
de
ω =
π
c�2
L
E68
20 S
AP
R -
Dan
Elli
sL0
2 -
Aco
ustic
s -
2006
-01-
26 -
29
Ro
om
aco
ust
ics
•S
ou
nd
in f
ree
air
exp
and
s sp
her
ical
ly:
•S
ph
eric
al w
ave
equ
atio
n:
solv
ed b
y
•In
ten
sity
fal
ls a
s
4
radi
us r
r22
∂∂p
2 r---r∂∂p
⋅+
1 c2-----
t22
∂∂p
⋅=
pr
t,(
)P
0 r------
ejω
tkr
–(
)⋅
=
p2
∝1 r2-----
E68
20 S
AP
R -
Dan
Elli
sL0
2 -
Aco
ustic
s -
2006
-01-
26 -
30
Eff
ect
of
roo
ms
(1):
Imag
es
•Id
eal r
eflec
tio
ns
are
like
mu
ltip
le s
ou
rces
:
•‘E
arly
ech
oes
’ in
ro
om
imp
uls
e re
spo
nse
:
-ac
tual
refl
ectio
ns m
ay b
e h r
(t),
not
δ(t
)
sour
celis
tene
r
virt
ual (
imag
e) s
ourc
esre
flect
edpa
th
t
h roo
m(t
)
dire
ct p
ath
early
ech
os
E68
20 S
AP
R -
Dan
Elli
sL0
2 -
Aco
ustic
s -
2006
-01-
26 -
31
Eff
ect
of
roo
ms
(2):
mo
des
•R
egu
larl
y-sp
aced
ech
oes
beh
ave
like
aco
ust
ic t
ub
es:
•R
eal r
oo
ms
hav
e lo
ts o
f m
od
es!
-de
nse,
sus
tain
ed e
choe
s in
impu
lse
resp
onse
-co
mpl
ex p
atte
rn o
f pea
ks in
freq
uenc
y re
spon
se
E68
20 S
AP
R -
Dan
Elli
sL0
2 -
Aco
ustic
s -
2006
-01-
26 -
32
Rev
erb
erat
ion
•E
xpo
nen
tial
dec
ay o
f re
flec
tio
ns:
•F
req
uen
cy-d
epen
den
t-
grea
ter
abso
rptio
n at
hig
h fr
eque
ncie
s→
fast
er d
ecay
•S
ize-
dep
end
ent
-la
rger
roo
ms →
long
er d
elay
s →
slo
wer
dec
ay
•S
abin
e’s
equ
atio
n:
•T
ime
con
stan
t va
ries
wit
h s
ize,
ab
sorp
tio
n
t
h roo
m(t
)~
e-t / T
RT
600.
049
VSα
--------
--------
-=
E68
20 S
AP
R -
Dan
Elli
sL0
2 -
Aco
ustic
s -
2006
-01-
26 -
33
Su
mm
ary
•Tr
avel
ling
wav
es
•A
cou
stic
tu
bes
& r
eso
nan
ce
•M
usi
cal a
cou
stic
s &
per
iod
icit
y
•R
oo
m a
cou
stic
s &
rev
erb
erat
ion
Par
tin
g T
ho
ug
ht:
•M
usi
cal b
ott
les
