More Useful: find Normal Mode solutions:

1
Continua Continued
More Useful find Normal Mode solutions
good solution if
normal modes have a sinusoidal shape.
therefore if
2
Set f p/2 for the moment and apply
product-to-sum
Rearrange to f(x-vt)
Normal mode two counter-propagating traveling
waves
3
node
antinode
4
Infinite strings give infinite solutions. Try a
bound string
f(0)0 and f(L)0
L, m, T
Wave equation still applies.
Normal mode solutions still apply.
But now we can apply boundary conditions
Boundary conditions reveal specific normal mode
frequencies! (for the clamped string)
5
(No Transcript)
6
These are the normal mode solutions, NOT the
solution to some driving force or initial
condition!!
7
If we shake one end.
With these Boundary conditions
y
0
L
x
is this a good solution?
Boundary conditions reveal normal mode
wavelengths frequencies! (for the clamped
string)
8
Apply boundary condition
Sine is zero if
Apply other boundary condition
9
n 1 B 0.1 L 10 v 1 w 1.05 (npv/L)
10
n 2 B 0.1 L 10 v 1 w 1.05 (npv/L)
11
n 5 B 0.1 L 10 v 1 w 1.05 (npv/L)
12
n 5 B 0.1 L 10 v 1 w 1.01 (npv/L)
13
n 5 B 0.1 L 10 v 1 w 1.001 (npv/L)
14
Wavenumber
French is the only book still in print that
defines k 1/l !!!
The equation of motion for a stretched string (a
continuum) is the wave equation. It has
traveling wave and normal mode solutions (which
are really the same thing).