1
THE NON-DISPERSIVE WAVE EQUATION
Reading:
Main 9.1.1
GEM 9.1.1
Taylor 16.1, 16.2, 16.3
(Thornton 13.4, 13.6, 13.7)
¶2
1 ¶2
y (x,t) = 2 2 y (x,t)
2
¶x
v ¶t
Example: Non dispersive wave equation
(a second order linear partial differential equation
¶
1 ¶
y (x,t) = 2 2 y (x,t)
2
¶x
v ¶t
2
2
Demonstrate that both standing and traveling
waves satisfy this equation (HW)
PROVIDED w/k = v .
Thus we recognize that v represents the wave
velocity. (What does “velocity” mean for a
standing wave?)
2
3
SEPARATION OF VARIABLES:
Assume solution
y(x,t) = X(x)T(t)
Plug in and find
1 d X(x) 1 1 d T (t)
= 2
2
X(x) dx
v T (t) dt 2
2
2
Argue both sides must equal constant and set to a
constant. For the moment, we'll say the constant
must be negative, and we'll call it -k2 and this
will be the same k as in the wave vector (you’ll
see!).
4
Separation of variables:
nd order
which
is
a
linear,
2
d X(x)
2
= -k X(x) ODE with solution
2
dx
X(x) = B' p cos kx + B'q sin kx
2
Then
d T (t)
2 2
2
= -v k T (t) = -w T (t)
2
dt
T (t) = Bp cos w t + Bq sin w t
2
And
Thus ….
y(x,t) = X(x)T(t)
linear, 2nd order
ODE with solution
Example: Non dispersive wave equation
(a second order linear partial differential equation)
¶2
1 ¶2
y (x,t) = 2 2 y (x,t)
2
¶x
v ¶t
y(x,t) = X(x)T(t)
y (x,t) = Acos kx coswt
+ Bcos kx sin wt
+ C sin kx coswt
+ Dsin kx sin wt
Separation of variables:
Then
With v = w/k,
A, B, C, D arbitrary constants
5
6
Activity: Non dispersive wave equation
Specifying initial conditions determines the (so far arbitrary)
coefficients:
¶y (x,t)
y (x,0);
¶t t = 0
In-class worksheet has different examples of initial conditions
7
A
1
2
3
4
B
C
D
8
THE NON-DISPERSIVE WAVE EQUATION
-REVIEW
•
•
•
•
(non-dispersive) wave equation
separation of variables makes PDE -> ODE
initial conditions determine arbitrary constants,
mathematical representations of the above