chapter v wave theory 5.1 derivation of one dimensional wave equation v- wave theory.pdf · 5.1...
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71
CHAPTER V
WAVE THEORY
5.1 DERIVATION OF ONE DIMENSIONAL WAVE EQUATION
The wave equation in the one dimensional case can be derived from Hooke's law in
the following way: Imagine an array of little weights of mass m are interconnected
with mass less springs of length h and the springs have a stiffness of k .
Here ( )u x measures the distance from the equilibrium of the mass situated at x . The
forces exerted on the mass m at the location x h are
2
Newton 2. ( ) . ( , )F m a t m u x h t
t
(5.1)
Hooke 2 [ ( 2 , ) ( , )] [ ( , ) ( , )]x h xF F F k u x h t u x h t k u x t u x h t (5.2)
The equation of motion for the weight at the location x h is given by equating these
two forces.
2
2
( , )[ ( 2 , ) ( , ) ( , ) ( , )]
u x h tm k u x h t u x h t u x t u x h t
t
(5.3)
where the time-dependence of ( )u x has been made explicit.
If the array of weights consists of N weights spaced evenly over the length .L N h
of total mass .M N m , and the total stiffness of the array K k N then we can write
the above equation as:
2 2
2 2
( , ) [ ( 2 , ) 2 ( , ) ( , )]u x h t KL u x h t u x h t u x t
t M h
(5.4)
Taking the limit , 0N h (and assuming smoothness) one gets
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2 2 2
2 2
( , ) ( , )u x t KL u x t
t M x
(5.5)
2KL
Mis the square of the propagation speed in this particular case. Taking 2c
2KL
Mwe
have the one dimensional wave equation as
2 2
2 2 2
( , ) 1 ( , )u x t u x t
x c t
(5.6)
with some initial and boundary conditions.
5.2 SOLUTION OF ONE DIMENSIONAL WAVE EQUATION
The one-dimensional wave equation can be solved exactly by D'Alembert's solution,
Fourier transform method, or via separation of variables. D'Alembert devised his
solution in 1746, and Euler subsequently expanded the method in 1748. Let us
consider andx ct x ct . (5.7)
Applying the chain rule to obtain
. .u u u u u
x x x
(5.8)
. .u u u u u
ct t t
(5.9)
2 2 2 2
2 2 22
u u u u u u u u
x x
(5.10)
2 2 2 22 2
2 2 22
u u u u u u u uc c c
t t
(5.11)
Substituting (5.10) and (5.11) in (5.6) we have the wave equation of the following
form
2
0u
(5.12)
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Therefore, the solution of the wave equation takes the form
( , ) ( ) ( )u F G (5.13)
( , ) ( ) ( )u x t F x ct G x ct (5.14)
where F and G are arbitrary functions, with F representing a right-travelling wave
and G represents the left-travelling wave. The function F and G can be found
explicitly by using the boundary condition and the initial conditions.
5.2.1 SEMI-INFINITE STRING WITH FIXED END
For Semi-infinite string with fixed end the wave equation takes the form
2 2
2 2 2
( , ) 1 ( , ), 0, 0
u x t u x tt x
x c t
(5.15)
with boundary condition (0, ) 0u t (5.16)
and the initial condition ( ,0) ( ) ; ( ,0) ( )tu x f x u x g x (5.17)
Clearly the solution of the Eq. 5.15 will take the form
( , ) ( ) ( )u x t F x ct G x ct (5.18)
Applying the boundary condition 5.17 we have
( ) ( ) ( )F x G x f x (5.19)
0
1 1( ) ( ) ( ) ( ) ( ) ( )
x
F x G x g x F x G x g x dxc c
(5.20)
Solving Eqs. 5.19 and 5.20 we have
0 0
1 1 1 1( ) ( ) ( ) and ( ) ( ) ( )
2 2 2 2
x x
F x f x g x G x f x g xc c
(5.21)
Substituting 5.21 in 5.18 we have
0 0
1 1 1 1( , ) ( ) ( ) ( ) ( )
2 2 2 2
x ct x ct
u x t f x ct g x dx f x ct g x dxc c
0
0
1 1 1( , ) ( ) ( ) ( ) ( )
2 2 2
x ct
x ct
u x t f x ct f x ct g x dx g x dxc c
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1 1
( , ) ( ) ( ) ( )2 2
x ct
x ct
u x t f x ct f x ct g x dxc
(5.22)
Now we introducing a function 0
( ) ( )
z
S z g x dx
Therefore, 1 1
( , ) ( ) ( ) ( ) ( )2 2
u x t f x ct f x ct S x ct S x ctc
Applying the boundary condition 5.16 we have
( ) ( ) and ( ) ( )f ct f ct S ct S ct
Therefore, we rewrite the solution as follows
1 1
( , ) ( ) ( ) ( ) ( )2 2
u x t Y x ct Y x ct H x ct H x ctc
(5.23)
where
( ) ( ) 0 ( ) ( ) 0
( ) 0 ( ) 0
Y z f z z H z S z z
f z z S z z
,0
( ) ( )
z
S z g x dx (5.24)
Eqs. 5.22 is implemented in the MATLAB function „Semi_Infinite_String_Calc‟and
is given in Table 5.1.
Table 5.1: Semi_Infinite_String_Calc
%******************************************************************** % Wave Equation For Semi Infinite String Calculation
%******************************************************************** function [X,U]= Semi_Infinite_String_Calc(c,L,t,fx,gx) %c ----> Wave Speed.
%L ----> Max Length upto which you want to view the motion.
%t ----> Time.
%fx ----> Initial displacement function.
%gx ----> Initial velocity function.
%Use MATLAB syntax with mpower concept. Suppose f(x)= e-(x-5)^2. So you
have to write fx= @(x) exp(-(x-5).^2). X=0:0.01:L;
val1= X+c*t;val2= X-c*t; for i=1:1: length(val2) if t== 0 U=gx(X); end if t~=0 Y1= fx(val1(i)); if val2(i)>=0 Y2= fx(val2(i)); end if val2(i)< 0 Y2= -fx(-val2(i)); end YY= (Y1+Y2)*0.5; R1= 0:val1(i)/100:val1(i); vel1= gx(R1);
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H1= trapz(vel1)*val1(i)/100; if val2(i)>=0 R2=0:val2(i)/100:val2(i); end if val2(i)< 0 R2=0:val2(i)/100:-val2(i); end vel2= gx(R2); H2= trapz(vel2)*val2(i)/100; HH= (H1-H2)*0.5/c; U(i)= YY+HH; end end end %-------------------------End of the Function------------------------
Consider a semi infinite string with wave speed 2 m/s where the initial condition are
2( ) exp( ( 5) ) and ( ) 0f x x g x then comparison in the position of the string for
different time are tabulated in the Table 5.2.
Table 5.2 Comparison of the Position of a semi infinite string
Time = 1 sec Time = 2 sec
Time = 3 sec Time = 4 sec
Time = 5 sec Time = 6 sec
Time 7 sec Time 10 sec
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5.2.2 FINITE STRING WITH FIXED END
For finite string with fixed end the wave equation takes the form
2 2
2 2 2
( , ) 1 ( , )u x t u x t
x c t
(5.25)
with boundary condition (0, ) 0 and ( , ) 0u t u L t (5.26)
and the initial condition ( ,0) ( ) and ( ,0) ( )tu x f x u x g x (5.27)
Clearly the solution of the Eq. 5.15 will take the form
( , ) ( ) ( )u x t F x ct G x ct (5.28)
Applying the boundary condition 5.27 we have
1 1
( , ) ( ) ( ) ( )2 2
x ct
x ct
u x t f x ct f x ct g x dxc
Now we introducing a function 0
( ) ( )
z
S z g x dx
Therefore, 1 1
( , ) ( ) ( ) ( ) ( )2 2
u x t f x ct f x ct S x ct S x ctc
Applying the boundary condition 5.26 we have
( ) ( ) and ( ) ( )f ct f ct S ct S ct
( ) ( ) and ( ) ( )f L ct f L ct S L ct S L ct
Setting ct and L ct we have
( ) ( ) and ( ) ( )f f S S
( ) (2 ) and (2 ) ( )f f L S L S
Combining above two we have
( ) ( ( 2 )) ( ) ( ( 2 ))
( 2 ) ( 2 )
f f L S S L
f L S L
Setting 2 wehave ( ) ( 2 ) and ( ) ( 2 ).z L f z f z L H z H z L
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Therefore, both the function are defined for all values of z and also periodic in z ,
repeating itself at intervals of 2L all along its length. Therefore, we rewrite the
solution as follows
1 1
( , ) ( ) ( ) ( ) ( )2 2
u x t Y x ct Y x ct H x ct H x ctc
(5.29)
where
( ) 0
( ) 0
( ) (2 ) 2
( 2 ) 2 3
................
f z L z
f z z L
Y z f L z L z L
f z L L z L
(5.30)
( ) 0
( ) 0
( ) (2 ) 2
( 2 ) 2 3
................
S z L z
S z z L
H z S L z L z L
S z L L z L
, 0
( ) ( )
z
S z g x dx (5.31)
Eqs. 5.29, 5.30 and 5.31 is implemented in the MATLAB function „ Finite _ String _
Calc‟ is given in Table 5.3.
Table 5.3: Finite_String_Calc
%****************************************************************** % Wave Equation For Finite String Calculation %****************************************************************** function [X,U]= Finite_String_Calc(c,L,t,fx,gx) %c ----> Wave Speed. %L ----> String Length %t ----> Time. %fx ----> Initial displacement function. %gx ----> Initial velocity function. %Use MATLAB syntax with mpower concept. Suppose f(x)= e-(x-5)^2. %So you have to write fx= @(x) exp(-(x-5).^2). X= 0:0.1:L; for i=1:1:length(X) Z = [X(i) - c*t, X(i) + c*t]; for m = 1: 1: length(Z) z1=Z(m) ;n1 = z1/L;n2= n1+ 1; if (round (n1)<= n1), n= round(n2); end if (round (n1)> n1), n= round(n1); end if mod(n, 2)== 0 x=n*L- z1; A(m)= - fx(x);
limit = 0: x/100 : x;
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vel= gx(limit); B(m)= trapz(vel)*x/100; end if mod (n, 2)== 1 x=z1- (n-1)*L; A(m)= fx(x); limit = 0: x/100 : x; vel= gx(limit);B(m)= trapz(vel)*x/100; end end U(i) = 0.5*(A(1) + A(2))-0.5*c*(B(1)-B(2))/c; end end %-------------------------End of the Function----------------------
Consider a Finite string of length 10 m with wave speed 2 m/s where the initial
condition are 2( ) exp( 40( 1) ) and ( ) 0f x x g x then comparison in the position of
the string for different time are tabulated in the Table 5.4.
Table 5.4 Comparison of the Position of a finite string
Time = 1 sec Time = 2 sec
Time = 3 sec Time = 4 sec
Time = 5 sec Time = 6 sec
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Time 7 sec Time 8 sec
Time 9 sec Time 10 sec
5.3 NON-LINEAR WAVE EQUATION
The wave equation in an ideal fluid can be derived from hydrodynamics and the
adiabatic relation between pressure and density. The equation for conservation of
mass, Euler‟s equation (Newton‟s 2nd Law), and the adiabatic equation of state are
respectively (Jensen et al, 2000)
.( ) 0vt
(5.32)
( ) . 0v
p v vt
(5.33)
22
0 2
1( ) ...........
2S S
p pp p
(5.34)
and for convenience we define the quantity
2
S
pc
(5.35)
where c will turn out to be the speed of sound in an ideal fluid. In the above
equations, is the density, v
the particle velocity, p the pressure, and the subscript
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S denotes that the thermodynamic partial derivatives are taken at constant entropy.
The ambient quantities of the quiescent (time independent) medium are identified by
the subscript 0. We use small perturbations for the pressure and density,
0 ( )p p p t (5.36)
0 ( )t (5.37)
and note that ( )v t
is also a small quantity; that is, the particle velocity which results
from density and pressure perturbations is much smaller than the speed of sound.
Retaining higher-order terms in Eq. 5.32 and 5.33 yields a non-linear wave equation.
The non linear effects we include are contained in the quadratic density term in the
equation of state, Eq. 5.34, and the quadratic velocity term (the convection term) in
Euler‟s equation, Eq. 5.33. First multiply Eq. 5.34 by and take its divergence; next,
take the partial derivative of Eq. 5.32 with respect to time. Substituting one into the
other yields
22
2( )i j i jp v v
t
(5.38)
Here the indices ,i j = 1, 2, 3 indicate , ,x y z -components, respectively. Tensor
notation is used; repeated indices signify a summation (e.g., .i iv v ).
The first term on the right side of Eq. 5.38 can be rewritten using Eq. 5.34 and 5.35 as
2 2 2 20( )1( )
cp c
c
(5.39)
The convection term on the right side of the Eq. 5.38 is more difficult to evaluate, but
we can obtain an expression for it in the limit of small propagation angles with
respect to the main direction of propagation, e.g., the x -direction. Then we may
estimate iv using the linear impedance relation, to be later derived as Eq. 5.53,
together with the equation of state 5.34
81
2
, ( ) ( )0
i i x
cv O O
(5.40)
where ,i x is the Kroenecker delta symbol, so that
2 2 2
0
1( ) ( )i j i jv v c
(5.41)
Substituting Eq. 5.39 and 5.41 into Eq. 5.38, we obtain the nonlinear wave equation
22 2 2 2 3
2( ) ( , ) ....c O
t
(5.42)
where 0
is the density ratio and 0( )
1c
c
the nonlinear parameter of
the medium.
5.4 LINEAR WAVE EQUATION
The linear approximations which led to the acoustic wave equation, involve retaining
only the first order terms in the hydrodynamic equations. Applying the assumption
5.36 and 5.37 to the hydrodynamic equations Eq. 5.32 to 5.34 and then eliminating
second the order terms (products of small quantities), we have linearized equations as
0 . 0vt
(5.43)
0 ( ) 0v
pt
(5.44)
2p c (5.45)
5.4.1 WAVE EQUATION FOR PRESSURE
Considering that the time scale of oceanographic changes is much longer than the time
scale of acoustic propagation, we will assume that the material properties 0 and 2c are
independent of time t .
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Now, we can eliminate v
from Eqs. 5.43 and 5.44 to obtained the wave equation for
pressure and to do so we take the partial derivative of Eq. 5.33 with respect to time t
and then the divergence of Eq.5.44 and combining this two relation we have
0 0. .v
v pt t t
2
0 02. . . ( )
vv p
t t t
2
0 02. . . ( )
v vp
t t t
22
20p
t
Using the Eq. 5.45 we have
22
2 2
10
pp
c t
22
2 2
1. . 0
pi e p
c t
(5.46)
We have omitted the primes using the pressure and density perturbations. This is the
most fundamental equation of acoustics.
It describes the properties of a sound field in space and time and how those properties
evolve. It is quite unlike the incompressible flow equations to which you may be
accustomed because it describes very weak processes which happen over large
distances. The most fundamental obvious property of the wave equation is that it is
linear. This means that the sum of two solutions of the wave equation is also itself a
solution.
If the density is not in-depended of t then the wave equation takes the form
2
2 2
1 1. 0
pp
c t
(5.47)
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5.4.2 WAVE EQUATION FOR PARTICLE VELOCITY
Alternatively, we can take the divergence of Eq. 5.43 and the time derivative of Eq.
5.44, and combine the two using Eq. 5.55 to arrive at the wave equation for the
particle velocity,
22
2
1( . ) 0
vc v
t
(5.48)
and if the density is in-depended of t then the wave equation takes the form
22
2( . ) 0
vc v
t
(5.49)
This form of the wave equation is a vector equation coupling the three spatial
components of the particle velocity. It involves spatial derivatives of both density and
sound speed, and is therefore rarely used, except for uni-axial propagation problems.
5.4.3 WAVE EQUATION FOR VELOCITY POTENTIAL
22
2
1( . ) 0
vc v
t
If the density is constant or slowly varying, the vector equation 5.48 can be
transformed into a simple scalar wave equation by introducing the velocity potential,
defined by
v
(5.50)
Substituting Eq. 5.50 together with the constant density condition 0
, into Eq.
5.48, the latter takes the form
22 2
2( ) 0c
t
(5.51)
This equation is clearly satisfied if satisfies the simple wave equation
22
2 2
10
c t
(5.52)
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which is identical to the pressure wave equation 5.46. Both equations are valid for
varying sound speed, but for constant density only.
We note that there is simple relationship between velocity and pressure for plane-
wave solutions to the wave equation. This impedance relation is easily found using
the velocity potential form of the wave equation with the solution, ( )f x ct .
From Eq. 5.50, ( )xv f x ctx
and from the linearized Euler‟s equation 5.44,
0 ( )p cf x ctx
, where f denotes a derivative with respect to the
argument of the function f . Comparing the pressure and velocity expressions yields
the plane wave impedance relation as,
0
x
pC
v (5.53)
5.4.4 WAVE EQUATION FOR DISPLACEMENT POTENTIAL
By using the kinematics relation between velocity and displacementu
vt
, it is
easily shown that the displacement potential defined by
u (5.54)
and it is as well governed by the following simple wave equation,
22
2 2
10
c t
(5.55)
As was the case for other wave equations 5.46 and 5.52, 5.55 is also valid only for
media with constant density. However, discrete changes in density can be handled
through appropriate boundary conditions between regions of constant density. For
such problems the boundary conditions require continuity of pressure and
displacement (or velocity), and the potentials become discontinuous.
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From the kinematic relations between displacements and velocities, Eq. 5.43, 5.45,
and 5.54, we obtain the following expression for the acoustic pressure in terms of the
displacement potential,
2p K (5.56)
with K being the bulk modulus, when
2K c (5.57)
Eq. 5.56 denotes the constitutive relation for an ideal, linearly elastic fluid (Hook‟s
Law). Combination of Eq. 5.55 to 5.57, yields the alternative expression for the
acoustic pressure
2
2p
t
(5.58)
5.5 SOURCE REPRESENTATION
Underwater sound is produced by natural or artificial phenomena through forced mass
injection. Such forcing terms were neglected in the mass conservation equation 5.43,
and therefore also in the derived wave equations. However, such terms are easily
included, leading to inhomogeneous wave equations, e.g., for the displacement
potential
22
2 2
1( , )f r t
c t
(5.59)
where ( , )f r t represents the volume injection as a function of space and time. Similar
inhomogeneous forms of the wave equations for pressure or velocity are easily
derived.
5.6 THE HELMHOLTZ EQUATION
Since the coefficients to the two differential operators in Eq. 5.59 are independent of
time, the dimension of the wave equation can be reduced to three by use of the
frequency-time Fourier transform pair,
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1( ) ( )
2
i tf t f e d
(5.60)
1( ) ( )
2
i tf f t e dt
(5.61)
leading to the frequency-domain counterpart of the wave equation, or the Helmholtz
equation,
2 2[ ( )] ( , ) ( , )k r r f r (5.62)
where ( )k r is the medium wave number at radial frequency , and is given by
( )( )
k rc r
(5.63)
It should be pointed out that although the Helmholtz equation 5.62, due to the
reduction in the dimension of this PDE, is simpler to solve than the full wave
equation, Eq. 5.59, this simplification is achieved at the cost of having to evaluate the
inverse Fourier transform, Eq. 5.60. However, many ocean acoustic applications are
of narrow-band nature. The Helmholtz equation, rather than the wave equation,
therefore forms the theoretical basis for the most important numerical methods,
including the Wave number Integration (WI), Normal Mode (NM) and Parabolic
Equation (PE) approaches (Jensen et al, 2000).
It is important to stress the difference between narrow-band processing in Ocean
Acoustics and wide band processing in seismic. The latter approach is viable because
the length scale of the environmental features addressed in seismic experiments is of
the same order of magnitude as the seismic wavelengths, and the time scales of the
experiments are such cross-spectral coherence can be assumed. In other words,
seismic experiments are characterized by very few interactions with any single
boundary, whereas a typical ocean acoustic experiment can have hundreds or
thousands of interactions. This is basically the reason why time-domain approaches
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such as FDM and FEM have never gained widespread popularity in ocean acoustics
whereas they are very important numerical analysis tools in seismic community.
There is, however, much virtue to time-domain solutions in terms of physical
understanding, and time-domain solutions are produced routinely for exactly that
purpose, both by Fourier synthesis and by direct time-domain solutions of the wave
equation.
5.7 THE COORDINATE SYSTEMS
In a homogeneous medium, the homogeneous Helmholtz equation, Eq. 5.62, is easily
solved, with a choice of coordinate system being imposed by the source and boundary
geometry. Thus, if plane wave propagation is considered, a Cartesian coordinate
system r ( , , )x y z is the natural choice, with the Lap lace operator,
2 2 22
2 2 2x y z
(5.64)
yielding plane wave solutions of the form
( , , )r
r
ik
ik
Aex y z
Be
(5.65)
where ( , , )x y zk k k k is the wave vector and A and B are arbitrary amplitudes and r is
the radial coordinate. For a single plane wave component, the coordinate system can
be aligned with the propagation direction, e.g., with , 0y zk ,k yielding the simple
solution ( , , )x
x
ik
ik
Aex y z
Be
(5.66)
which corresponds to a forward and a backward propagating plane wave solution with
time dependence i te .
Similarly, the field produced by an infinite, homogeneous line source is conveniently
described in a cylindrical coordinate system r ( r, ,z ) , for which case given by
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2 22
2 2 2
1 1( )r
r r r r z
(5.67)
where r denotes the radial coordinates, the circumferential coordinate and z the
axial coordinate.
For a uniform line source, the field only varies with range r , reducing the Helmholtz
equation to the Bessel equation,
21( ) ( ) 0r k r
r r r
(5.68)
with the solution
0 0( ) J ( ) or Y ( )r A kr B kr (5.69)
where 0 0J and Y denote the Bessel functions of first and second kinds.
In terms of the Hankel functions, the solution in Eq. (5.66) may be written as
0
(1)
0 0 0
(2)
0 0
CH ( ) C[J ( ) Y ( )]
( )
DH ( ) D[J ( ) Y ( )]
kr kr i kr
r
kr kr i kr
(5.70)
The solution in Eq. (5.67) represents diverging and converging cylindrical and waves
for r , as is clear from the asymptotic form of the Hankel functions for kr ,
( )(1) 40
2H ( )
i kr
kr ekr
(5.71)
( )(2) 40
2H ( )
i kr
kr ekr
(5.72)
These asymptotic also show that the cylindrically symmetric field produced by a line
source decays in amplitude proportionally to 1/ 2r . Approaching the source, the line
source field exhibits a logarithmic singularity (Jensen et al, 2000).
89
In case of an Omni-directional point source, the field only depends on the range from
the source, and the solution is conveniently described in a spherical coordinate
system, with the reduced Helmholtz equation being
2 21( ) 0r k r
r r r
(5.73)
which has the solutions A B
( ) orikr ikrr e er r
(5.74)
Again the solution in Eq. 5.74 correspond to diverging and converging spherical
waves with amplitude decaying proportional to 1r in spherical range. The term
geometrical spreading loss refers to these amplitude decays. Thus, cylindrical
spreading loss is proportional to 1r and spherical spreading loss is 2r .
5.8 CIRCULAR MEMBRANE
When we studied the one-dimensional wave equation we found that the method of
separation of variables resulted in two Simple Harmonic Oscillator. Here we are
interested in the next level of complexity, when the ODEs which arise upon
separation may be different from the familiar SHO equation. This complexity arises
when non-cartesian coordinate systems are used.
We choose polar coordinate system and therefore the wave equation reduces to
2 2 2
2 2 2 2 2
( , , ) 1 ( , , ) 1 ( , , ) 1 ( , , )u r t u r t u r t u r t
r r r r c t
(5.75)
where ( , , )u r t is the vertical vertical displacement of a point on the membrane at
position ( , )r and time t .
For the sake of the solution of this problem we choose separation of variables and so
substituting ( , , ) ( ) ( ) ( )u r t R r T t (5.76)
in Eq. 5.75 we have
90
2 2 2
2 2 2 2 2
1 1 1( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
R R TT t T t T t R r R r
r r r r c t
2 2 2
2 2 2 22
2
1 1
1
( ) ( ) ( )
R R T
r r r r t kR r c T t
2 22 2 2
22 22 2 2
2and 0
( ) ( )
R Rr r k r R
Tr r m m c TR r t
22 2 2 2
2
22
2
22 2
2
( ) 0
0
0
R Rr r k r m R
r r
m
Tk c T
t
(5.77)
The solution of the above equations 5.77 are well established and the solutions are
( ) ( ) ( ) ;
( ) cos( ) sin( )
( ) cos( ) sin( )
m mR r AJ kr BY kr
C m D m
T t E kct F kct
(5.78)
We see that in this problem of circular membrane the function ( )mY kr are not allowed
as they go to as 0r therefore only the functions ( )mJ kr are permitted as they
are finite at 0r . The profile of the disturbance across the circular membrane will
then be described by functions of the form
( ) ( )
( ) cos( ) sin( )
( ) cos( ) sin( )
mR r AJ kr
B m C m
T t D kct E kct
(5.79)
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Boundary conditions:
The boundary condition is that the edge of the circular membrane is fixed so that it
cannot be displaced. This means that if where R be the radius of the membrane then
( ) 0mJ kR (5.80)
This determines the allowed values of k since kR must correspond to a zero of the
function ( )mJ kR .In other words, if mn are the n th positive roots of ( ) 0mJ s then
mnmnk
R
, where 0,1, 2,........and 1, 2,........m n . (5.81)
Therefore from the relation 5.76 we have the complete solution as
0
1
( , , ) cos( ) sin( ) cos( ) sin( ) ( )mn mn mn mn m m m mn
mn
u r t a k ct b k ct c m d m J k r
(5.82)
Initial Conditions:
For practical purpose we assume that the membrane is at rest i.e. ( , ,0) 0tu r and
( ,0, ) 0u r t .
Hence, applying this in Eq. 5.82 it is clear that the term sin( )mnk ct and sin( )m is no
longer needed. Therefore,
01
( , , ) cos( )cos( ) ( )mn mn m mn
mn
u r t a k ct m J k r
(5.83)
With another appropriate initial condition we can easily find mna and using the
equation 5.83 we will have the complete solution.
According to our assumption that the ( , ,0) 0tu r and ( ,0, ) 0u r t the (m, n) th
mode shape will be of the following form
( , ) cos( )cos( ) ( )mn m mnm n k ct m J k r (5.84)
where mnmnk
R
, mn is the n th positive roots of the ( ) 0mJ x . (5.85)
92
Eq. 5.84 is implemented in the MATLAB function “ModeShape_Circular_Membrane
_Calc‟ and mnk are calculated by the MATLAB function “Find_nth_bessel_root”. is
given in Table 5.5 and 5.6 and the comparison of the mode shape for different values
of t is shown in the Table 5.7.
Table 5.5: ModeShape_Circular_Membrane_Calc
%**************************************************************
% Mode Shape Calculation
%**************************************************************
function [X,Y,U] =
ModeShape_Circular_Membrane_Calc(c,radius,m,n,t) %c ---> Wave Propagation Speed. %radius ---> Radius of the Circular Membrane. %m ---> m th asimuthal number. %n ---> n th root of the bessel function. Nth_root=Find_nth_bessel_root(m, n);
Q= Nth_root /radius;N=20; R1=linspace(0, radius, N);Theta1=linspace(0, 2*pi, N); [R, Theta]=meshgrid(R1, Theta1);[X,Y] = pol2cart(Theta,R); U=cos(c*Q*t)*besselj(m, Q*R).*cos(m*Theta); end
%-------------------------- END -------------------------------
Table 5.6: Find_nth_bessel_root
%**************************************************************
% Finding the n th root of the Bessel function
%**************************************************************
function [root] = Find_nth_bessel_root(k, p) X=0.5:0.5:(10*p+1); Y = besselj(k, X); tol=1e-5; [a, b] = Position_nth_root(X, Y, p); if a~= b check = abs(a-b);i=1; while check>= tol XX=a:(b-a)/100:b; YY = besselj(k, XX); [a, b] = Position_nth_root(XX, YY, 1); i=i+1;check = abs(a-b); end end root=(a+b)/2; end function [a, b] = Position_nth_root(X, Y, n) l=0; m=length(X); for i=1:1:(m-1) if ( Y(i)*Y(i+1) <= 0 ) l=l+1;end; if (l== n) a=X(i); b=X(i+1);return; end end end %-------------------------- END -------------------------------
93
Table 5.7: Comparison of mode shape Circular Membrane
Time
sec (0,1) (1,1) (1,2)
0
1
2
3
4
5
Time
sec (0,2) (2,1) (2,2)
0
1
2
3
4
94
5.8 RECTANGULAR MEMBRANE
To find the motion of a rectangular membrane with sides of length xL and yL (in
absence of gravity) use the two-dimensional wave equation of the form
2 2 2
2 2 2 2
( , , ) ( , , ) 1 ( , , )u x y t u x y t u x y t
x y c t
(5.86)
where ( , , )u x y t is the vertical displacement of the membrane at position ( , )x y and
time t .For the sake of the solution of this problem we choose separation of variables
and so substituting ( , , ) ( ) ( ) ( )u x y t X x Y y T t (5.87)
in Eq. 5.86 we have
5
Time
sec (0,3) (3,2) (0,4)
0
1
2
3
4
5
95
2 2 2
2 2 2 2
1( ) ( ) ( ) ( ) ( ) ( )
X Y TY y T t X x T t X x Y y
x y c t
2 2 2 2 22
2 2 2
1
( ) ( ) ( )
c X c Y Tk
X x x Y y y T t t
2 2 2 22 2
2 2 2 2
1 1and 0x
X Y k Tk k T
X x Y y c t
2 2 22 2 2
2 2 20 ; 0 and 0x y
X X Tk X k X k T
x x t
(5.88)
where 2
2 2
2x y
kk k
c (5.89)
The solution of the above equations 5.88 are well established and the solutions are
( ) cos( ) sin( )
( ) cos( ) sin( )
( ) cos( ) sin( )
x x
y y
X x A k x B k x
Y y C k y D k y
T t E kt F kt
(5.90)
Boundary Condition:
The rectangular membrane fixed at all the boundaries ( 0, 0, , )x yx y x L y L that
means (0, , ) 0 ( ,0, )u y t u x t and ( ,0, ) 0 (0, , )x yu L t u L t .
Applying first two boundary condition we have 0A C and the second set of
boundary condition implies (for non trivial solution) sin( ) 0 sin( )x yk x k y that is
,x y
x y
m nk k
L L
, where 1, 2,........and 1, 2,........m n . (5.91)
Therefore from the relation 5.87 we have the complete solution as
11
( , , ) cos( ) sin( ) sin( )sin( )mn mn x y
mn
u x y t a kt b kt k x k y
(5.92)
where , andx yk k k are given in the Eq. 5.89 and 5.91.
96
Initial Condition:
For practical purpose we assume that the membrane is at rest i.e. ( , ,0) 0tu x y
Hence, 11
( , , ) cos( )sin( )sin( )mn x y
mn
u x y t a kt k x k y
(5.93)
With another appropriate initial condition we can easily find mna and using the
equation 5.83 we will have the complete solution. According to our assumption that
the (0, , ) 0 ( ,0, )u y t u x t and ( ,0, ) 0 (0, , )x yu L t u L t the (m, n) th mode shape will
be of the following form ( , ) cos( )sin( )sin( )x ym n kt k x k y (5.94)
where 2 2, ,x y x y
x y
m nk k k c k k
L L
(5.95)
Eq. 5.94 and 5.95 is implemented in the MATLAB function
“ModeShape_Rectangular_Membrane_Calc‟ and given in the Table 5.8. The
comparison of the mode shape for different values of t is shown in the Table 5.9
where length and breadth of the membrane are 10 m each and the wave speed 1 m/s.
Table 5.8 : ModeShape_Rectangular_Membrane_Calc
%*****************************************************************
% Mode Shape for Rectangular Membrane
%*****************************************************************
function [X,Y,U] = MoedeShape_Rectangular_Calc(Lx,Ly,c,t,m,n)
%Lx ----> Length of the Membrane %Ly ----> Breadth of the Membrane %c ----> Wave Speed %m ----> Azhimuthal number for X %n ----> Azhimuthal number for Y %t ----> Time
kx= m*pi/Lx;
ky= n*pi/Ly;
k= c*sqrt(kx^2 + ky^2); [X,Y]=meshgrid( (0:0.01:1)*Lx, (0:0.01:1)*Ly); U = 5*cos(k*t)*sin(kx*X).*sin(ky*Y);
end
%---------------------- END ---------------------------
97
Table 5.9: Comparison of mode shape Rectangular Membrane
Time (1,1) (1,2) (1,3)
0
1
2
3
4
5
Time (2,1) (2,2) (2,3)
0
1
2
3
4
98
5
Time (3,1) (3,2) (3,3)
0
1
2
3
4
5