In[1]:=
Integrate@ 2 ê L * Sin@n * Pi * x ê LD ^ 2, 8x, 0, L<D
Out[1]=
1-
In[2]:=
1-
Out[2]=
1-
In[3]:=
Sin@2 n pD
2np
Sin@2 n pD
2np
H*Proves these are orthonormal since Sin@2*n*PiD=0.*L
Sin@2 n pD
2np
Plot@82 x, 2 H1 - xL<, 8x, 0, 1<, PlotStyle Ø 8Red, Blue<D
2.0
1.5
Out[3]= 1.0
0.5
0.2
0.4
0.6
0.8
1.0
In[4]:=
H*Shows fHxL for h=1.*L
In[5]:=
Integrate@u * Sin@uD, 8u, 0, n * Pi ê 2<, Assumptions Ø 8n œ Integers<D
Out[5]=
In[6]:=
-
In[7]:=
In[25]:=
In[9]:=
2
n p CosB
np
2
F + SinB
np
2
F
Integrate@Hn * Pi - uL * Sin@uD, 8u, n * Pi ê 2, n * Pi<, Assumptions Ø 8n œ Integers<D
1
Out[6]=
1
2
n p CosB
np
2
F + SinB
np
2
F - Sin@n pD
H*Two integrals for Fourier coefficients in notes.*L
y@m_, x_, t_D := 8 ê Pi ^ 2 ê H2 * m + 1L ^ 2 * H- 1L ^ m * Sin@H2 * m + 1L * Pi * xD * Cos@H2 * m + 1L * Pi * tD
H*Taking h=L=v=1.*L
2
13.4.nb
In[27]:=
Plot@Sum@y@m, x, 0D, 8m, 0, 100<D, 8x, 0, 1<, PlotRange Ø 8- 1, 1<, AxesLabel Ø 8x, y<D
y
1.0
0.5
Out[27]=
0.2
0.4
0.6
0.8
1.0
x
-0.5
-1.0
H*This is the function y@x,0D which agrees with BC III the plucked string.*L
In[31]:=
Plot3D@Sum@y@m, x, tD, 8m, 0, 100<D, 8x, 0, 1<,
8t, 0, 10<, PlotRange Ø 8- 1, 1<, AxesLabel Ø 8x, t, y<D
Out[31]=
H*This shows the string oscillating up and down in time!*L
13.4.nb
In[29]:=
Animate@Plot@Sum@y@m, x, tD, 8m, 0, 100<D, 8x, 0, 1<, PlotRange Ø 8- 1, 1<, AxesLabel Ø 8x, y<D,
8t, 0, 10<, AnimationRunning Ø TrueD
t
y
1.0
0.5
Out[29]=
0.2
0.4
0.6
0.8
-0.5
-1.0
H*This is an animation of the string in time.*L
1.0
x
3