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					                                                  College of Engineering and Computer Science
                                                            Mechanical Engineering Department
                                                                  Engineering Analysis Notes
                                                                   Larry Caretto          May 31, 2011


                         Solutions to the Wave Equation
The wave equation
The one-dimensional wave equation, shown below, describes the propagation of a disturbance,
u, over space and time. For example, u might be the amplitude of a vibrating string which varies
with space and time.

                                               2u      2u
                                                    c2 2                                              [1]
                                               2t     x

The D’Alambert solution and its proof
The D’Alambert solution to equation [1] is written in terms of coordinates ξ and η, defined as
follows:

                               x  ct           and             x  ct                             [2]

The D’Alambert solution to the wave equation is written in terms of two arbitrary functions, F(ξ)
and G(η). This gives the following solution.

                            u  F ( )  G()  F ( x  ct )  G( x  ct )                             [3]

To show that this is a solution, we have to rewrite the wave equation in terms of these two
functions. This means that we have to transform the coordinates in the wave equation from (x,t)
to (ξ,η). To start the coordinate transformations we look at the first derivative terms. If we have u
as a function of  and , we can get the time derivatives with respect to t by the following
equation.

                                          u  u  u
                                                                                                     [4]
                                          t t  t 

The D’Alambert solution gives simple expressions for the ξ and η derivatives since F is a function
of ξ only and G is a function of η only.

                             u                   dF ()
                                F ()  G ()          F ' ( )                                   [5]
                                                d

Here we use the notation F’(ξ) for the first derivative of F with respect to ; we will subsequently
define a second derivative in a similar manner.

                                        dF ( )                         d 2 F ( )
                           F ' ( )                     F ' ' ( )                                   [6]
                                         d                               d 2


Jacaranda (Engineering) Room 3333                       Mail Code                    Phone: 818.677.6448
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Wave equation solutions                      L. S. Caretto, May 31, 2011                            Page 2


In a similar fashion we can write

                             u   
                                   F ()  G()  dG()  G' ()                                    [7]
                                                  d

Finally, from the definitions of ξ = x + ct and η = x - ct, we can write the following partial
derivatives.

                                                      
                                            c              c                                        [8]
                                         t             t
Combining this equation with equations [4], [5], and [7] gives the following result.

                                      u
                                          cF 'cG'  c( F 'G' )                                      [9]
                                      t
We find the second time derivative by taking the time derivative of this first derivative. We can
simplify the result using the equations above for coordinate transformation, the D’Alambert
solution that u = F + G, and the definitions of F’’ and G’’ as ordinary second derivatives.

   2u  u   u   u                             
                               c c( F 'G ' )  c    c( F 'G ' )  c 2 ( F ' 'G ' ' ) [10]
  t 2
         t t t  t t  t                      

We can repeat this process for the x-derivatives; the main difference is in the partial derivatives of
the new coordinates with respect to x.

                                                       
                                             1             1                                        [11]
                                          x             x
With these relationships, we can obtain the first derivative with respect to x as follows.

      u  u  u       [ F ( )  G ( )]       [ F ( )  G ( )]
                    (1)                      (1)                      F ' ( )  G ' ( ) [12]
      x x  x                                        

The second derivative is obtained by taking the derivative of the first derivative.

    2u  u   u   u                                
                                (1) ( F 'G ' )  (1)    ( F 'G ' )  ( F ' 'G ' ' ) [13]
   x 2
          x x x  x x  x                         

We can now show that the two expressions for the second derivatives in equations [10] and [13]
satisfy the original two-dimensional wave equation in [1].

               2u      2u             2u                               2u
                    c2 2                    c 2 F ' 'G ' '     c2         c 2 F ' 'G ' '     [14]
               2t     x              t2
                                                                         x 2



Thus, the D’Alambert solution, u = F(x + ct) + G(x – ct), where F and G are arbitrary functions
satisfies the differential equation. We now have to show how we can use this solution to satisfy
the differential equation and the initial conditions.
Wave equation solutions                                      L. S. Caretto, May 31, 2011                      Page 3


The D’Alambert solution with initial conditions
We want to be able to satisfy arbitrary initial conditions on the displacement, u, and its first
derivative, the velocity, at t = 0. These arbitrary initial conditions are written as follows.

                                                                             u
                                       u (0, x)  f ( x)          and                     g ( x)               [15]
                                                                             t   x 0


In this section we show that the solution to the wave equation [1] with the boundary conditions in
equation [15] can be written as follows.

                                                                                         x  ct
                              u (t , x)   f ( x  ct )  f ( x  ct ) 
                                         1                                 1
                                         2                                       g ( )d
                                                                           2c x  ct
                                                                                                                [16]


In this equation, we have used the D’Alambert solution for the first two terms where the arbitrary
functions F() and G() in the D’Alambert solution, are both the initial condition function f(). We say
that u(t=0,x) = f(x) and u(t,x) has a contribution f(x + ct) and f(x – ct). This means that we use the
same functional form, but the argument to the function f() at later times is computed as x + ct and
x – ct. For example if f(x) = 1 when x = 0 then f(x + ct) would equal 1 when x + ct = 0. (This
would occur at all points where x = -ct.

The final term in equation [16] is the integral of the initial derivative condition that is integrated
over the dummy variable, . The first two terms have just been shown to satisfy the differential
equation. (Recall that any function of x + ct or x – ct satisfied the differential equation.) In order
to show that the integral term in equation [16] satisfies the differential equation we need to use
the general formula for differentiation under the integral sign.

                                                b( y )
                                                                 b           a
                                           y      ( x)dx  y (b)  y (a)                                 [17]
                                                a( y)

We can apply this general result to compute the space and time derivatives for the wave
equation. First find the time derivatives. Using the general rule above, we find the following
result.

                              x  ct
                                                    ( x  ct )                ( x  ct )
                         t        g ()d             t
                                                                 g ( x  ct ) 
                                                                                    t
                                                                                            g ( x  ct )
                              x  ct
                                                                                                                [18]
                           cg ( x  ct )  (c) g ( x  ct )  cg ( x  ct )  g ( x  ct )

The second derivative of the integral is just the derivative of the first derivative, which becomes

                         x  ct                          x  ct
                  2                                                
                  t 2     g ( )d 
                         x  ct
                                                        ctg ( )d  t cg ( x  ct )  cg ( x  ct )
                                                t t x 
                                                                                                                [19]
              ( x  ct ) g ( x  ct )  ( x  ct ) g ( x  ct ) 
          c                                                        c g ' ( x  ct )  g ' ( x  ct )
                                                                         2

             t             ( x  ct )      t        ( x  ct ) 
Wave equation solutions                                 L. S. Caretto, May 31, 2011                               Page 4


We have used the usual notation, g’, to indicate an ordinary first derivative.

                                                                    dg  
                                                       g '                                                      [20]
                                                                     d
In a similar fashion we can take the second-order space derivative of the integral by starting with
the first derivative.

                      x  ct
                                      ( x  ct )                ( x  ct )                g ( )
                       ctg ( )d  x g ( x  ct )  x g ( x  ct )  x
                  x x                                                                                             [21]
                                                            g ( )
                     (1) g ( x  ct )  (1) g ( x  ct )            g ( x  ct )  g ( x  ct )
                                                             x

We then take the second derivative from the result above.

                               x  ct                  x  ct
                    2                                            
                       2 
                    x x  ct
                             g ( )d            ctg ( )d  x g ( x  ct )  g ( x  ct )
                                          x x x                                                                  [22]
              ( x  ct ) g ( x  ct ) ( x  ct ) g ( x  ct )
                                                                  )  (1) g ' ( x  ct )  (1) g ' ( x  ct )
                  x        ( x  ct )      x         ( x  ct )

If we substitute equations [19] and [22] into the original two-dimensional wave equation in [1] we
see that the integral term satisfies the differential equation.

                            x  ct                                                             x  ct
                     2                                                                 2
                                g ( )d  c g ' ( x  ct )  g ' ( x  ct )  c                g ( )d
                                               2                                    2
                                                                                                                    [23]
                     t 2   x  ct
                                                                                        x 2   x  ct


We can now show that the integral is added to the solution to satisfy the initial conditions. First
consider the condition that u(0,x) = f(x). Setting t = 0 in the proposed solution gives.

                                                                x0
        u (0, x)   f ( x  0)  f ( x  0)      0g ( )d  2  f ( x)  f ( x)  0  f ( x)
                  1                             1                1
                                                                                                                    [24]
                  2                             2c x 

Here we use the result that the value of a definite integral is zero when both the lower and upper
limits are the same. Taking the time derivative of the solution and using equation [18] for the first
derivative of the integral in the solution gives the following result.

                                                                                         x  ct
                        u ( x, t ) 1  f ( x  ct ) f ( x  ct )  1 
                           t
                                    
                                    2       t
                                                     
                                                           t         2c t  g ( )d
                                                                            x  ct


            1   ( x  ct ) f ( x  ct )  ( x  ct ) f ( x  ct )  1
                                                                         cg ( x  ct )  cg ( x  ct )
                                                                                                                    [25]
                                         
            2  t
                             ( x  ct )       t       ( x  ct )  2c
                                                                      
        
            1
              cf ' ( x  ct )  (c) f ' ( x  ct )  1 cg ( x  ct )  cg ( x  ct )
            2                                           2c
Setting t = 0 in this equation shows that g(x) is the initial velocity.
Wave equation solutions                             L. S. Caretto, May 31, 2011                      Page 5


                      u ( x, t )
                                   cf ' ( x)  (c) f ' ( x)  cg ( x)  cg ( x)  g ( x)
                                   1                             1
                                                                                                         [26]
                         t t  0 2                              2c

Thus the solution proposed in equation [16] satisfies the wave equation and the initial conditions.

Propagation of initial conditions
Consider the following initial conditions giving g = 0 and u a simple triangular profile initially.

                                                                     x  1    1  x  0
                                                                     
                        g ( x)  0        all x             f ( x)   1  x    0  x 1                 [27]
                                                                     0
                                                                              otherwise

With g(t) everywhere zero, equation [16] tells us that u(t,x) = [f(x+ct) + f(x-ct)]/2. This solution
results in the following profiles of the initial conditions at later times as shown in the figure below.




                                                                                                      t=0
                                                                                                      ct=1
                                                                                                      ct = 2
  time, t




                                                                                                      ct = 3
                                                                                                      x+ct
                                                                                                      x+ct
                                                                                                      x-ct
                                                                                                      x-ct




            -5   -4        -3        -2       -1        0        1       2     3        4        5
                                                   distance, x


The solutions at later times are computed from equation [16] in the following way. The argument
is easier to understand if we talk about the initial condition as f(ξ) or f(η) where ξ = x + ct and η =
x – ct. Of course, f is a single function, as defined in equation [27], such that f(ξ) or f(η) is zero
except where its argument lies between -1 and +1. Between these points this function has the
triangular shape defined in equation [27] and shown as the initial conditions in the figure above.

At a point (t,x) where the time, t, = a/c, the value of ξ = x + ct = x + a. Since the original initial
condition is zero at all points except where -1 ≤ ξ ≤ 1, f(x + a) will be zero except for the region
where -1 ≤ x + a ≤ 1, that is the region where -1-a ≤ x ≤ 1-a. For example, when a = ct = 2, f(x +
ct) will be zero only in the region -3 ≤ x ≤ -1. For ct = 2, then, the f(x+ct) term will occur in this
region. Similarly, when ct = a, the solution for f(η) = f(x – ct) will be zero at all points outside the
region -1 ≤ η ≤ 1, which corresponds to -1 ≤ x – a ≤ 1 or a–1 ≤ x ≤ 1+a. Thus when a = ct = 2 the
u(x,t) term resulting from f(x – ct) will reflect the initial condition between x = 1 and x = 3.
Wave equation solutions                        L. S. Caretto, May 31, 2011                        Page 6


Because the solution u(t,x) = [f(x+ct) + f(x-ct)]/2, the magnitude of the initial condition will be cut in
half in each component.

Solution by separation of variables
Consider the wave equation with the following set of initial and boundary conditions.

                     2u 1  2u
                                  0 0  x  L, t  0
                    t 2 c 2 x 2                                                                     [28]
                                                                     u
                   u(0, t )  u( L, t )  0     u(0, t )  f ( x )      (0, t )  g ( x )
                                                                     t
Here, the boundary conditions give u = 0 at x = 0, x = L. The initial conditions on u and its first
derivative with respect to time are both known functions of x. These functions are typically called
the initial displacement and the initial velocity, respectively.

The solution of the wave equation by separation of variables proceeds in a manner similar to the
solution of other partial differential equations. We postulate a solution that is the product of two
functions, X(x) a function of x only and T(t) a function of time only. With this assumption, our
solution becomes.

                                              u(x,t) = X(x)T(t)                                       [29]

We do not know, in advance, if this solution will work. However, we assume that it will and we
substitute it for u in equation [28]. Since X(x) is a function of x only and T(t) is a function of t only,
we obtain the following result when we substitute equation [29] into equation [28].

     1  2u  2u 1  2 X ( x )T (t )  2 X ( x )T (t ) X ( x )  2T (t )          2 X ( x)
              2  2                                      2                 T (t )            0 [30]
     c 2 t 2 x  c       t 2               x 2           c        t 2               x 2

If we divide the final equation through by the product X(x)T(t), and move the x derivative to the
other side of the equal sign, we obtain the following result.

                                      1  2 X ( x)     1  2T ( t )
                                                    2                                                [31]
                                    X ( x ) x 2    c T (t ) t 2

The left hand side of equation [31] is a function of x only; the right hand side is a function of y
only. The only way that this can be correct is if both sides equal a constant. This also shows that
the separation of variables solution works. In order to simply the solution, we choose the
         1                 2
constant to be equal to . This gives us two ordinary differential equations to solve.

                                  1  2 X ( x)     1  2T (t )
                                                2               2                                 [32]
                                X ( x ) x 2
                                                c T (t ) t 2




  The choice of – for the constant as opposed to just plain  comes from experience. Choosing
1                  2

the constant to have this form now gives a more convenient result later. If we chose the constant
to be simply , we would obtain the same result, but the expression of the constant would be
awkward.
Wave equation solutions                      L. S. Caretto, May 31, 2011                           Page 7


Equation [32] shows that we have two separate differential equations, each of which has a known
                                                                               2
general solution. These equations and their general solutions are shown below.

            d 2 X ( x)
                  2
                        2 X ( x)  0                 X ( x)  A sin( x)  B cos(x)              [33]
               dx

                d 2T (t )
                     2
                           2T (t )  0            T (t )  C sin(ct )  D cos(ct )              [34]
                  dt

From the solutions in equations [33] and [34], we can write the general solution for u(x,y) =
X(x)T(t) as follows.

                    u( x, t )  Asin(x )  B cos(x )C sin(ct )  D cos(ct )                  [35]

We now apply the boundary conditions shown with the original equation [28] to evaluate the
constants A, B, C, and D. If we substitute the boundary condition that u = 0 at x = 0 into equation
[35], get the following result.

                 u(0, t )  0  Asin( 0)  B cos( 0)C sin(ct )  D cos(ct )                  [36]

Because sin(0) = 0 and cos(0) = 1, equation [36] will be satisfied for all y only if B = 0. Thus, we
set B = 0. Next we apply the solution in equation [35] (with B = 0) to the boundary condition at x
= L.

                         u( L, t )  0  Asin(L)C sin(ct )  D cos(ct )                         [37]

Equation [37] can only be satisfied if the sine term is zero. This will be true only if L is an
integral times . If n denotes an integer, we must have

                                                                 n
                                    L  n        or                                              [38]
                                                                  L
Since any integral value of n gives a solution to the original differential equations, with the
boundary conditions that u = 0 at the two boundaries considered so far, the most general solution
is one that is a sum of all possible solutions, each multiplied by a different constant. In the
general solution for one value of n, which we can now write as Asin(nx)[Csin(nct) + Dcos(nct)],
with n = nx/L, we can write the product of two constants, AC, as the single constant, An., which
may be different for each value of n. Similarly we can write the product AD as the constant B n.
Again, this constant can be different for different values of n. The general solution which is a sum
of all solutions with different values of n is written as follows

                                    
                        u( x, t )   An sin(n ct )  Bn cos(n ct )sin(n x )                    [39]
                                   n 1


(We start with n = 1 since n = 0 gives zero for the eigenfunction.) We can use eigenfunction
expansions to get the initial conditions on displacement and velocity.


2
 As usual, you can confirm that this solution satisfies the differential equation by substituting the
solution into the differential equation.
Wave equation solutions                            L. S. Caretto, May 31, 2011                Page 8


                                                                                 
          u( x,0)  f ( x )   An sin(n c0)  Bn cos(n c0)sin(n x )   Bn sin(n x )      [40]
                                 n 1                                            n 1


We can obtain an equation for Bn by using the orthogonality relationships for integrals of the sine.
If we multiply both sides by sin(mx/L), where m is another integer, and integrate from a lower
limit of zero to an upper limit of L, we get the following result.

                                      mx          
                                                             mx   nx 
                      L                              L

                         f ( x ) sin      dx    Bn sin      sin  dx
                      0               L         0 n 1     L   L 
                                                                                                 [41]
                           L
                                   mx   nx              2  mx 
                                                                     L
                         Bn sin      sin  dx  Bm  sin       dx
                        n 1 0     L   L             0       L 

In the second row of equation [41] we can reverse the order of summation and integration
because these operations commute. We then recognize that the integrals in the summation all
vanish unless m = n, leaving only this integral to evaluate. Solving for Bm and evaluating the last
        3
integral in equation [41] gives the following result.

                                                  mx 
                                  L

                                   f ( x ) sin
                                                   L 
                                                        dx
                                                             2
                                                               L
                                                                            mx 
                          Bm     0
                                                              f ( x ) sin     dx             [42]
                                             2  m x                      L 
                                        L
                                                             L0
                                         sin  L dx
                                        0            

For any initial displacement, then, we can perform the integral on the right hand side of equation
[42] to compute the values of Bm and substitute the result into equation [39]. We find the value of
the constants Am in a similar manner by fitting the condition for the initial velocity. First we take
the time derivative of our solution for u.

                      u( x, t ) 
                                  n cAn cos(n ct )  Bn sin(n ct )sin(n x )              [43]
                        t        n 1


Using this equation for the initial condition on velocity as g(x) gives.

     u                                                                      
        ( x,0)  g ( x )   n cAn cos(n c0)  Bn sin(n c0)sin(n x )   n cAn sin(n x ) [44]
     t                    n 1                                              n 1


The eigenvalue expansion to obtain the coefficients A n proceeds in exactly the same manner as
the expansion used to find Bn. The only differences are in the use of g(x) rather than f(x) and the
presence of the factor of λnc multiplying An. Accounting for these differences, we can infer the
equation for An from equation [42] for Bm.



3
  Using a standard integral table, and the fact that the sine of zero is zero and the sine of m is
zero for integer m, we find the following result:
                                                     L
     2  mx   x            2mx               2mL  L
L
                       L                L    L
 sin  L dx   2  4m sin L  0  2  4m sin L   2
0                                                  
Wave equation solutions                          L. S. Caretto, May 31, 2011                    Page 9


                                    L
                             L              nx 
                            nc     
                                g ( x) sin 
                                            L 
                                                 dx
                                                              2
                                                                  L
                                                                             nx 
                    An             0
                                    L
                                             nx 
                                                           
                                                             nc  
                                                                 g ( x) sin 
                                                                             L 
                                                                                  dx             [45]

                                     sin 2 
                                             L 
                                                  dx             0

                                    0

For any given initial conditions f(x) and g(x), we can find the values of A n and Bn by performing
the integrals in equations [42] and [45]. We can then substitute these results into equation [39],
rewritten below with the substitution of nπ/L for λ.

                                  
                                           nct           nct   nx 
                     u( x, t )    An sin        Bn cos       sin                       [46]
                                 n 1      L              L   L 

We can first consider the case where the initial velocity is zero at all points. According to
equation [45] this will give all values of An = 0 so that our equation becomes.

                                                 
                                                         nct   nx 
                                    u( x, t )   Bn cos       sin                            [47]
                                                n 1     L   L 

We can show that this form is equivalent to the D’Alembert solution by using the trigonometric
identities for the sine of the sum and difference of two angles. These formulae are written below.
(Note that we can obtain the second formula from the first one by setting y = –y in the first
equation and using the facts that sin(–y) = –sin(y) and cos(–y) = cos(y).)

                                     sin(x + y) = sin x cos y + sin y cos x                       [48]

                                     sin(x – y) = sin x cos y – sin y cos x                       [49]

Adding these two equations gives

                                     sin(x + y) + sin(x – y) = 2 sin x cos y                      [50]

We can apply this result to equation [47] writing the product of the two trigonometric functions as
follows.

                      nx   nct        nx nct        nx nct 
                2 sin      cos     sin           sin                                  [51]
                      L   L             L    L          L    L 

Simplifying the result and substituting it into equation [47] gives.

                                    1    n ( x  ct )   n ( x  ct ) 
                     u ( x, t )       Bn sin L   sin L 
                                    2 n 1                              
                                                                                                  [52]


Equation [52] gives us the form of the D’Alembert solution with the x + ct and x – ct dependence
of two functions.

In a similar fashion we can consider the case where the displacement is zero giving the following
equation.
Wave equation solutions                           L. S. Caretto, May 31, 2011                        Page 10


                                                   
                                                            nct   nx 
                                     u ( x, t )   An sin        sin                                 [53]
                                                  n 1      L   L 

As before we can manipulate the terms using trigonometric relations for the cosine of the sum
and difference of two angles. These are

                                      cos(x + y) = cos x cos y – sin y sin x                              [54]

                                      cos(x – y) = cos x cos y + sin y sin x                              [55]

Subtracting equation [54] from equation [55] gives

                                      cos(x – y) – cos(x + y) = 2 sin x sin y                             [56]

We can apply this result to equation [53] writing the product of the two trigonometric functions as
follows.

                                     1    n ( x  ct )   n ( x  ct ) 
                     u ( x, t )        An cos L   cos L 
                                     2 n 1                              
                                                                                                          [57]


We could have done the same operations with the general equation in which both f(x) and g(x)
are nonzero to give.

                                     1       n ( x  ct )   n ( x  ct ) 
                      u ( x, t )       An cos L   cos L 
                                     2 n1                                  
                                                                                                          [58]
                                  1      n ( x  ct )        n ( x  ct ) 
                                   Bn sin               sin               
                                  2 n1        L                    L        

Although the equation was solved for the region 0 ≤ x ≤ L, we know that the periodic functions of
sine and cosine extend beyond this region. In fact, they will produce periodic extensions of the
original solution for t = 0 which is shown below.

                     1    nx            nx  1    nx   nx 
         u( x,0)         L   L   2  Bn sin L   sin L 
                     2 n 1
                            An cos   cos
                                                    n 1           
                                                                                                          [59]


The cosine terms cancel in this equation leaving only the sine terms. In the equation below, we
recognize that the initial condition, u(x,0) is simply f(x).

                                      1    nx           nx          nx 
              u( x,0)  f ( x )         Bn sin L   sin L    Bn sin L 
                                      2 n 1                   n 1         
                                                                                                          [60]


                                       
                                                n x 
Equation [60] tells us that f ( x )   Bn sin        so that
                                      n 1      L 
                        
                                n ( x  ct )                                
                                                                                       n ( x  ct ) 
        f ( x  ct )   Bn sin                and           f ( x  ct )   Bn sin                  [61]
                       n 1          L                                      n 1          L        
Wave equation solutions                           L. S. Caretto, May 31, 2011                    Page 11


For the case where the initial velocity, g(x), is zero, which is given in equation [52], we see that
the separation of variables solution reduces to the D’Alembert solution that u(x,t) = [f(x + ct) – f( x
– ct)]/2.

Because the initial condition, f(x) is expressed in terms of sine eigenfunctions, we have the same
results as a Fourier series in terms of sines. A Fourier sine series gives an odd periodic
extension of the initial conditions in regions beyond the boundary. This is illustrated in the figure
below for a region 0 ≤ x ≤ 1, and an initial condition that is a triangular peak, with a height of one,
at the midpoint of the region.


                 1
           0.5                                                                                  Initial
   f(x)




                                                                                                Conditions
                 0                                                                              Periodic
                                                                                                Extension
          -0.5
             -1
                     -2               -1                  0                1                2
                                                          x

Although the initial conditions are defined only for 0 ≤ x ≤ 1, the sine expansion gives the odd
periodic extensions of the initial conditions beyond the actual region. These extensions are the
basis for the f(x + ct) and f(x – ct) terms in the region for large times. The chart below shows the
propagation of the initial conditions shown above at a time where ct = 0.4.

            1

          0.5
                                                                                                 f(x)
   f(x)




            0                                                                                    f(x+ct)/2
                                                                                                 f(x - ct)/2
          -0.5            ct = 0.4

           -1
                 -2        -1.5      -1    -0.5       0       0.5      1        1.5     2
                                                      x

At this point, the triangular initial conditions which are nonzero between x = 0.4 and x = 0.6 have
just reached the boundaries of the region 0 ≤ x ≤ 1. However, as the propagation of the actual
initial conditions is about the leave the region, the periodic extensions are about to enter the
region. Thus we see that the periodic extension centered at x = 1.5 has a component f(x + ct)
that is now centered at 1.1 and is about to propagate into the actual region from the right.
Similarly, the periodic extension at centered at x = –0.5 has a component f(x + ct) that is now
centered at x = –0.1, and is about to enter the region from the left.

When these periodic extensions enter the region, the two solution components will have positive
and negative components that will cancel. The effect of this is shown in the three charts below
for ct = 0.45. The top two charts show the plots of f(x+ct) and f(x-ct), respectively. In the bottom
chart, the expression for the wave form, u(x,t) = [f(x+ct) + f(x-ct)]/2, is plotted.

At the left side of the top chart for f(x+ct) we see the periodic extension of the initial condition that
was initially centered at x = 1.5, whose triangular base extended from 1.4 to 1.6. (Since t = 0
initially, the value of x + ct = 1.4 at the start of this triangle when t = 0. When ct = 0.45, x + ct =
Wave equation solutions                     L. S. Caretto, May 31, 2011                      Page 12


1.4 corresponds to x + 0.45 = 1.4 or x = 0.95. Thus, the start of the triangle which was at x = 1.4
when t = 0 will be located at 0.95 when ct = 0.45. A similar analysis will show that there is a
periodic extension of the initial condition, originally centered at x = –0.5, whose base extends
from –0.6 to –0.4. The part of the triangle that was at x – ct = –0.4 when t = 0 will be located at
an x coordinate given by the equation x – 0.45 = –0.05. This is where we see the start of the
triangle in the plot of f(x-ct).

              1


            0.5
 f(x+ct)




                                                           ct = 0.450
              0


            -0.5
                   0       0.2              0.4               0.6              0.8               1
                                                      x
               1

             0.5
  f(x-ct)




                                                            ct = 0.450
               0

            -0.5
                   0       0.2               0.4              0.6              0.8                1
                                                      x
               1

             0.5
  u(x,t)




                                                            ct = 0.450
               0

            -0.5
                   0       0.2               0.4              0.6              0.8                1
                                                      x


The net solution, u(x,t) = [f(x+ct) + f(x-ct)]/2 is zero between x = 0.15 and 0.85 where both f(x+ct)
and f(x-ct) are zero. Between x = 0.85 and x = 0.95, f(x+ct) is zero, and f(x-ct) is positive, so
u(x,t) is simply f(x-ct)/2. However, between x = 0.95 and x = 1, f(x+ct) is negative, and f(x-ct) is
positive, so u(x,t) = [f(x+ct) + f(x-ct)]/2 has both a positive and negative contribution, which makes
the slope of the wave form more strongly negative.

A similar effect occurs between x = 0.05 and x = 0.15, f(x-ct) is zero, and f(x+ct) is positive, so
u(x,t) is simply f(x+ct)/2. However, between x = 0 and x = 0.05, f(x-ct) is negative, and f(x+ct) is
positive, so u(x,t) = [f(x+ct) + f(x-ct)]/2 has both a positive and negative contribution, which makes
the slope of the wave form more strongly positive.
Wave equation solutions                     L. S. Caretto, May 31, 2011                       Page 13


In summary, the periodic extensions that enter the region from either side cancel part of the initial
conditions that have propagated from the center point of the region, giving a steeper slope to the
start and finish of the solution for u(x,t). When ct = 0.5, the periodic extensions will exactly cancel
the components propagating from x = 0.5 giving a zero solution in the entire region 0 ≤ x ≤ 1.

				
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