Striking a Beat
Ashley Martin
PHY 495
Introduction
Striking a Beat
History
Outline
Cartesian
Coordinates
Ashley Martin
Solutions
Dynamics
Harmonics
Polar Coordinates
PHY 495
Solutions
Harmonics
Dynamics
Striking the Center
Spring 2012
References
Striking a Beat
Ashley Martin
PHY 495
Introduction
History
Outline
Cartesian
Coordinates
Solutions
Dynamics
Harmonics
Polar Coordinates
Solutions
Harmonics
Dynamics
Where is it optimal to strike a circular drum?
Striking the Center
References
Striking a Beat
Ashley Martin
PHY 495
Introduction
History
I
Daniel Bernoulli (1700-1782) - introduced concept of
Bessel functions
I
Leonhard Euler (1707-1783) - used Bessel funtions of
both zero and integral orders
I
Friedrich Bessel (1784-1846) - generalized the Bessel
function
Outline
Cartesian
Coordinates
Solutions
Dynamics
Harmonics
Polar Coordinates
Solutions
Harmonics
Dynamics
Striking the Center
References
Striking a Beat
Ashley Martin
PHY 495
Introduction
I wish to ascertain the difference one hears between striking
a drum at the center and off center.
I
Two Dimensions
I
I
I
Cartesian Coordinates
Polar Coordinates
Striking the Center vs. Striking Off Center
History
Outline
Cartesian
Coordinates
Solutions
Dynamics
Harmonics
Polar Coordinates
Solutions
Harmonics
Dynamics
Striking the Center
References
Striking a Beat
Rectangular Membrane
Ashley Martin
PHY 495
Introduction
The wave equation in Cartesian Coordinates
History
Outline
∂2u
= c2
∂t 2
∂2u ∂2u
+
; c=
∂x 2 ∂y 2
r
t > 0, 0 ≤ x ≤ L, 0 <≤ y ≤ H
T
ρ
Cartesian
Coordinates
Solutions
Dynamics
Harmonics
Polar Coordinates
Solutions
Harmonics
Dynamics
Striking the Center
References
Striking a Beat
Conditions
Ashley Martin
PHY 495
Introduction
History
The fixed boundary conditions are
u(0, y , t) = 0, u(L, y , t) = 0
u(x, 0, t) = 0, u(x, H, t) = 0
Outline
Cartesian
Coordinates
Solutions
Dynamics
Harmonics
Polar Coordinates
with initial conditions
u(x, y , 0) = f (x, y )
ut (x, y , 0) = g (x, y )
Solutions
Harmonics
Dynamics
Striking the Center
References
Striking a Beat
Separation of Variables
Ashley Martin
Let u(x, y , t) = F (x)G (y )T (t), then
PHY 495
F (x)G (y )T 00 (t) = c 2 (F 00 (x)G (y )T (t) + F (x)G 00 (y )T (t))
Introduction
History
Outline
which leads to
T 00
F 00
Cartesian
Coordinates
G 00
1
=
+
= −λ
c2 T
F
G
Solutions
Dynamics
Harmonics
Polar Coordinates
Solutions
Harmonics
Dynamics
thus obtaining two equations
Striking the Center
References
00
2
T + c λT = 0.
(1)
F 00 G 00
+
= −λ
F
G
(2)
Striking a Beat
Time Dependence
Ashley Martin
PHY 495
Introduction
History
Outline
√
T (t) = A cos ωt + B sin ωt, ω = c λ.
Frequency of oscillations for the harmonics are
ω
c √
ν=
=
λ.
2π
2π
Cartesian
Coordinates
Solutions
Dynamics
Harmonics
Polar Coordinates
Solutions
Harmonics
Dynamics
Striking the Center
References
Striking a Beat
Spatial Equations
Ashley Martin
PHY 495
Introduction
F 00
G 00
=−
− λ = −µ , −µ < 0.
F
G
History
Outline
Cartesian
Coordinates
This leads to two equations
Solutions
Dynamics
Harmonics
Polar Coordinates
00
F + µF = 0
(3)
Solutions
Harmonics
Dynamics
Striking the Center
References
G 00 + (λ − µ)G = 0.
(4)
Striking a Beat
Apply BCs
Ashley Martin
PHY 495
Introduction
F (0) = 0 =⇒ A = 0,
F (L) = 0 =⇒ B = 0 or sin λx
L =0
Fn (x) = Bn sin
nπx
,λ=
L
nπ 2
L
, n = 1, 2, . . ..
Similarly,
Cartesian
Coordinates
Solutions
Dynamics
Harmonics
Polar Coordinates
Solutions
Harmonics
Dynamics
Striking the Center
G (0) = 0 , G (H) = 0
Gm (y ) = Dm sin
History
Outline
mπ 2
mπy
,λ − µ =
, m = 1, 2, . . ..
H
H
References
Striking a Beat
Ashley Martin
Yields the product solutions
PHY 495
nπx
mπy
unm (x, y , t) = (Anm cos ωnm t + Bnm sin ωnm t) sin
sin
;
L
H
where,
History
Outline
λnm =
ωnm = c
nπ 2
L
+
r nπ 2
L
Cartesian
Coordinates
mπ 2
+
Solutions
Dynamics
Harmonics
H
mπ 2
H
Polar Coordinates
Solutions
Harmonics
Dynamics
.
Striking the Center
Thus the general solutions is
u(x, y , t) =
Introduction
∞ X
∞
X
(Anm cos ωnm t+Bnm sin ωnm t) sin
n=1 m=1
References
nπx
mπy
sin
.
L
H
Striking a Beat
Fourier Coefficients
Ashley Martin
PHY 495
Introduction
Initial displacement is u(x, y , 0) = f (x, y ), thus
f (x, y ) =
∞ X
∞
X
History
Outline
Anm sin
n=1 m=1
nπx
mπy
sin
.
L
H
Rewriting as a single sum gives
∞
X
nπx
f (x, y ) =
An (y ) sin
, where
L
n=1
∞
X
mπy
An (y ) =
Anm sin
.
H
m=1
Cartesian
Coordinates
Solutions
Dynamics
Harmonics
Polar Coordinates
Solutions
Harmonics
Dynamics
Striking the Center
References
Fourier Coefficients
Striking a Beat
Ashley Martin
PHY 495
Introduction
The coefficients of Fourier sine series
Z
2 L
nπx
An (y ) =
dx,
f (x, y ) sin
L 0
L
Z H
2
mπy
Anm =
An (y ) sin
dy .
H 0
H
Results in a double Fourier sine series,
Z HZ L
mπy
4
nπx
sin
dxdy .
Anm =
f (x, y ) sin
LH 0 0
L
H
History
Outline
Cartesian
Coordinates
Solutions
Dynamics
Harmonics
Polar Coordinates
Solutions
Harmonics
Dynamics
Striking the Center
References
Striking a Beat
Fourier Coefficients
Ashley Martin
PHY 495
Introduction
History
Outline
Initial velocity, ut (x, y , 0) = g (x, y ),
g (x, y ) =
∞ X
∞
X
n=1 m=1
Bnm =
4
ωnm LH
Z
H
Z
1
nπx
mπy
Bnm sin
sin
.
ωnm
L
H
L
g (x, y ) sin
0
0
Cartesian
Coordinates
nπx
mπy
sin
dxdy .
L
H
Solutions
Dynamics
Harmonics
Polar Coordinates
Solutions
Harmonics
Dynamics
Striking the Center
References
Striking a Beat
Ashley Martin
This completes the full solution of the vibrating rectangular
membrane problem:
PHY 495
Introduction
History
∞ X
∞
X
mπy
nπx
u(x, y , t) =
sin
,
(Anm cos ωnm t+Bnm sin ωnm t) sin
L
H
n=1 m=1
Anm =
Bnm
4
LH
H
Z
4
=
ωnm LH
Z
L
f (x, y ) sin
0
0
Z
ωnm
HZ
L
nπx
mπy
sin
dxdy
L
H
nπx
mπy
g (x, y ) sin
sin
dxdy
L
H
0
0
r nπ 2 mπ 2
=c
+
L
H
Outline
Cartesian
Coordinates
Solutions
Dynamics
Harmonics
Polar Coordinates
Solutions
Harmonics
Dynamics
Striking the Center
References
Striking a Beat
Harmonics
Ashley Martin
PHY 495
Introduction
History
Outline
Cartesian
Coordinates
nπx
mπy
ϕnm (x, y ) = sin
sin
L
H
Nodal lines occur when ϕnm (x, y ) = 0.
Solutions
Dynamics
Harmonics
Polar Coordinates
Solutions
Harmonics
Dynamics
Striking the Center
References
Striking a Beat
Ashley Martin
PHY 495
Introduction
History
Outline
Cartesian
Coordinates
Solutions
Dynamics
Harmonics
Polar Coordinates
Solutions
Harmonics
Dynamics
[3]
Striking the Center
References
n=m=1
sin
πx
= 0, x = 0, L;
L
sin
πy
= 0, y = 0, H
H
Striking a Beat
Ashley Martin
PHY 495
Introduction
History
Outline
Cartesian
Coordinates
Solutions
Dynamics
Harmonics
Polar Coordinates
Solutions
Harmonics
Dynamics
[3]
Striking the Center
References
n = 3, m = 2
sin
3πx
L 2L
= 0, x = 0, , , L;
L
3 3
sin
2πy
H
= 0, y = 0, , H
H
2
Striking a Beat
Circular Membrane
Ashley Martin
The wave equation in polar coordinates
2
∂2u
1 ∂u
1 ∂2u
2 ∂ u
=c
+
+ 2 2 ;
∂t 2
∂r 2
r ∂r
r ∂θ
PHY 495
Introduction
History
Outline
Cartesian
Coordinates
Solutions
Dynamics
Harmonics
t > 0, 0 ≤ r ≤ a, 0 ≤ θ ≤ 2π
Polar Coordinates
Solutions
Harmonics
Dynamics
Striking the Center
References
[3]
Striking a Beat
Conditions
Ashley Martin
PHY 495
The fixed boundary conditions are
Introduction
History
u(a, θ, t) = 0
t > 0, 0 ≤ θ ≤ 2π.
The periodic conditions are
Outline
Cartesian
Coordinates
Solutions
Dynamics
Harmonics
Polar Coordinates
u(r , θ, t) = u(r , θ + 2π, t)
uθ (r , θ, t) = uθ (r , θ + 2π, t),
with initial conditions
u(r , θ, 0) = f (r , θ)
ut (r , θ, 0) = g (r , θ)
Solutions
Harmonics
Dynamics
Striking the Center
References
Striking a Beat
Separation of Variables
Ashley Martin
PHY 495
Introduction
History
Outline
Let u(r , θ, t) = R(r )Θ(θ)T (t),
Cartesian
Coordinates
Solutions
Dynamics
Harmonics
Θ00 + n2 Θ = 0
00
(5)
2 2
T +c λ T =0
(6)
r 2 R 00 + rR 0 + (r 2 λ2 − n2 )R = 0.
(7)
Polar Coordinates
Solutions
Harmonics
Dynamics
Striking the Center
References
Striking a Beat
Simple Equations
Ashley Martin
PHY 495
Introduction
History
Outline
Cartesian
Coordinates
Solutions
Dynamics
Harmonics
The solutions to (5) and (6) are
Θn (θ) = An cos nθ + Bn sin nθ;
Polar Coordinates
√
T (t) = Cnm cos ωnm t + Dnm sin ωnm t, ωnm = c λnm .
Solutions
Harmonics
Dynamics
Striking the Center
References
Striking a Beat
Radial Equation
Ashley Martin
PHY 495
The solutions to
functions.
r 2 R 00
+
rR 0
+
(r 2 λ2
−
n2 )R
= 0, are Bessel
Introduction
History
Outline
Cartesian
Coordinates
Solutions
Dynamics
Harmonics
Polar Coordinates
Solutions
Harmonics
Dynamics
Striking the Center
References
Figure: A few Bessel Functions with their zeros, znm [7].
Apply BCs
Striking a Beat
Ashley Martin
PHY 495
Introduction
History
Outline
Cartesian
Coordinates
u(a, θ, t) = 0 for t > 0 and 0 ≤ θ ≤ 2π =⇒ R(a) = 0.
Since we expect solutions to be finite at the center, therefore
√
R(r ) = CJn ( λr ).
Solutions
Dynamics
Harmonics
Polar Coordinates
Solutions
Harmonics
Dynamics
Striking the Center
References
Striking a Beat
Apply BCs
Ashley Martin
PHY 495
Since, R(a) = 0,
Introduction
√
Jn ( λa) = 0.
History
Outline
Cartesian
Coordinates
Listing a few of the mth zeros of Jn
znm
m=1
m=2
m=3
m=4
n=0
2.4048
5.5201
8.6537
11.792
n=1
3.8317
7.0156
10.173
13.324
n=2
5.1356
8.1472
11.620
14.796
n=3
6.3802
9.7610
13.015
16.223
Table: Approximate location of the zeros of Bessel functions of
the first kind, Jn (znm ) = 0 [7].
Solutions
Dynamics
Harmonics
Polar Coordinates
Solutions
Harmonics
Dynamics
Striking the Center
References
Striking a Beat
Ashley Martin
PHY 495
Denoting
that znm is the mth zero of Jn (x), then,
√
Jn ( λa) = 0, tells us that
√
λa = znm
λnm =
z
nm
a
2
.
Substituting this into R(r ) gives
z
nm
R(r ) = Jn
r .
a
Introduction
History
Outline
Cartesian
Coordinates
Solutions
Dynamics
Harmonics
Polar Coordinates
Solutions
Harmonics
Dynamics
Striking the Center
References
General Solution
Striking a Beat
Ashley Martin
PHY 495
Introduction
History
Outline
Cartesian
Coordinates
u(r , θ, t) =
∞ X
∞
X
Jn (λnm r ) [Anm cos nθ + Bnm sin nθ] cos ωnm t
n=0 m=1
+Jn (λnm r ) [Cnm cos nθ + Dnm sin nθ] sin ωnm t.
Solutions
Dynamics
Harmonics
Polar Coordinates
Solutions
Harmonics
Dynamics
Striking the Center
References
Harmonics
Striking a Beat
Ashley Martin
The circular membrane harmonics are given by
!
znm r
Φ(r , θ) = cos(nθ)Jn
.
a
znm r
) = 0.
Nodal curves occur when cos nθ = 0 or Jn (
a
PHY 495
Introduction
History
Outline
Cartesian
Coordinates
Solutions
Dynamics
Harmonics
Polar Coordinates
Solutions
Harmonics
Dynamics
Striking the Center
References
Figure: Vibrational Modes of a Circular Membrane with dashed
nodal curves [3].
Striking a Beat
Nodal Circles
Ashley Martin
PHY 495
Introduction
History
znm r
We wish to find values of r such that
is a zero of the
a
Bessel function. Thus,
znm r
= znj , j ≤ m
a
r=
znj
a, r ≤ a.
znm
Outline
Cartesian
Coordinates
Solutions
Dynamics
Harmonics
Polar Coordinates
Solutions
Harmonics
Dynamics
Striking the Center
References
Striking a Beat
Ashley Martin
[3]
For m = 2, we have two circles, r = a and r = zzn1
for each
n2
n. We will need to calculate r for each n = 0, 1, 2 by using
the table.
PHY 495
Introduction
History
Outline
Cartesian
Coordinates
Solutions
Dynamics
Harmonics
For n = 0,
r=
2.4048
zn1
=
a ≈ 0.4356a.
zn2
5.5201
For n = 1,
Polar Coordinates
Solutions
Harmonics
Dynamics
Striking the Center
References
3.8317
r=
a ≈ 0.5462a,
7.0156
and for n = 2,
r=
5.1356
a ≈ 0.6304a.
8.1472
General Solution
Striking a Beat
Ashley Martin
PHY 495
Introduction
History
Outline
Cartesian
Coordinates
u(r , θ, t) =
∞ X
∞
X
Jn (λnm r ) [Anm cos nθ + Bnm sin nθ] cos ωnm t
n=0 m=1
+Jn (λnm r ) [Cnm cos nθ + Dnm sin nθ] sin ωnm t
Solutions
Dynamics
Harmonics
Polar Coordinates
Solutions
Harmonics
Dynamics
Striking the Center
References
Striking a Beat
Initial displacement, u(r , θ, 0) = f (r , θ),
u(r , θ, 0) =
∞ X
∞
X
Ashley Martin
PHY 495
Jnm (λnm r ) [Anm cos nθ + Bnm sin nθ] .
Introduction
n=0 m=1
Let an (r ) =
∞
P
History
Anm Jn (λnm r ) and bn (r ) =
∞
P
Outline
Bnm Jn (λnm r ).
m=1
m=1
Solutions
Dynamics
Harmonics
Rewriting the equation gives
u(r , θ, 0) =
∞
X
Cartesian
Coordinates
Polar Coordinates
[an (r ) cos nθ + bn (r ) sin nθ], where
n=0
an (r ) =
bn (r ) =
Solutions
Harmonics
Dynamics
Striking the Center
References
2π
1
π
Z
1
π
Z
f (r , θ) cos nθdθ,
0
2π
f (r , θ) sin nθdθ.
0
Striking a Beat
Ashley Martin
PHY 495
Since an (r ) =
∞
P
Introduction
Anm Jn (λnm r ),
History
m=1
Anm =
Outline
2
a2 [Jn+1 (zn )]2
a
Z
an (r )Jn (λnm r )dr .
0
Similarly,
Cartesian
Coordinates
Solutions
Dynamics
Harmonics
Polar Coordinates
Bnm =
2
a2 [Jn+1 (zn )]2
Z
a
bn (r )Jn (λnm r )dr
0
Solutions
Harmonics
Dynamics
Striking the Center
References
Striking a Beat
Ashley Martin
PHY 495
Introduction
History
Intitial velocity, ut (r , θ, 0) = g (r , θ),
Z a
2
Cnm =
cn (r )Jn (λnm r )dr ,
ωnm a2 [Jn+1 (zn )]2 0
Z a
2
dn (r )Jn (λnm r )dr ,
Dnm =
ωnm a2 [Jn+1 (zn )]2 0
Outline
Cartesian
Coordinates
Solutions
Dynamics
Harmonics
Polar Coordinates
Solutions
Harmonics
Dynamics
Striking the Center
References
Striking a Beat
Axisymmetric Solution
Ashley Martin
PHY 495
Introduction
History
The solution independent of θ,
Outline
Cartesian
Coordinates
u(r , t) =
∞
X
m=1
J0m
z
0m
a
r
[αm cos ω0m t + βm sin ω0m t]
.
Notice for n ≥ 1, each vibrational mode has nodal curves
that pass through the center so none of these modes can be
excited.
Solutions
Dynamics
Harmonics
Polar Coordinates
Solutions
Harmonics
Dynamics
Striking the Center
References
Striking a Beat
Conclusion
Ashley Martin
PHY 495
Introduction
History
Outline
Cartesian
Coordinates
Where is it optimal to strike a drum?
Off Center
Solutions
Dynamics
Harmonics
Polar Coordinates
Solutions
Harmonics
Dynamics
Striking the Center
References
Acknowledgements
Striking a Beat
Ashley Martin
PHY 495
Introduction
History
Outline
Cartesian
Coordinates
Solutions
Dynamics
Harmonics
Polar Coordinates
Solutions
Harmonics
Dynamics
Striking the Center
References
Dr. Russel Herman, Allison Martin, Navid Sharifian, and my
friends and family!
A. Bruder.
Two mathematical models for the tympanic membrane.
pages 157–166, 2005.
http://mtbi.asu.edu/files/Two_Mathematical_
Models_for_the_Tympanic_Membrane.pdf.
R. Herman.
An introduction to fourier and complex analysis with
applications to the spectral analysis of signals.
http://people.uncw.edu/hermanr/mat367/
FCABook/Book2010/FCA_Main.pdf, March 2012.
R. Herman.
An introduction to mathematical physics via oscillations.
http://people.uncw.edu/hermanr/phy311/
MathPhysBook/MathPhys_Main.pdf, March 2012.
M. Kac.
Can one hear the shape of a drum?
The American Mathematical Monthly, 73:1–23, 1966.
Striking a Beat
Ashley Martin
PHY 495
Introduction
History
Outline
Cartesian
Coordinates
Solutions
Dynamics
Harmonics
Polar Coordinates
Solutions
Harmonics
Dynamics
Striking the Center
References
http://mathdl.maa.org/images/upload_library/
22/Ford/MarkKac.pdf.
O. Shipilova.
Differential equations lecture vi, February 2008.
https://noppa.lut.fi/noppa/opintojakso/
bm20a2101/luennot/lecture_4.pdf.
A. Wazwaz.
Partial Differential Equation and Solitary Wave Theory.
Higher Education, Beijing, 2009.
D. Young.
An introduction to partial differential equations in the
undergraduate curriculum.
https://http:
//www.math.hmc.edu/~ajb/PCMI/lecture1.pdf.
[4] [6] [3] [2] [1] [5] [7]
Striking a Beat
Ashley Martin
PHY 495
Introduction
History
Outline
Cartesian
Coordinates
Solutions
Dynamics
Harmonics
Polar Coordinates
Solutions
Harmonics
Dynamics
Striking the Center
References