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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
ψ (x,t) = 2 2 ψ (x,t)
2
∂x
v ∂t
Example: Non dispersive wave equation
(a second order linear partial differential equation)
∂2
1 ∂2
ψ (x,t) = 2 2 ψ (x,t)
2
∂x
v ∂t
Demonstrate that both standing and traveling
waves satisfy this equation (HW)
PROVIDED ω/k = v .
Thus we recognize that v represents the wave
velocity. (What does “velocity” mean for a
standing wave?)
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Example: Non dispersive wave equation
∂2
1 ∂2
ψ (x,t) = 2 2 ψ (x,t)
2
∂x
v ∂t
Separation of variables:
Then
With v = ω/k,
ψ (x,t) = X(x)T(t)
ψ (x,t) = Acos kx cosωt
+ Bcos kx sin ωt
+ C sin kx cosωt
+ Dsin kx sin ωt
A, B, C, D arbitrary constants
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Separation of variables:
Assume solution
ψ (x,t) = X(x)T(t)
Plug in and find
1 d 2 X(x) 1 1 d 2T (t)
= 2
2
2
X(x) dx
v T (t) dt
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 is
the same k as in the wavevector.
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Separation of variables:
Then
And
d 2 X(x)
2
= −k X(x)
2
dx
X(x) = B' p cos kx + B'q sin kx
2
d T (t)
2 2
2
= −v k T (t) = −ω T (t)
2
dt
T (t) = Bp cos ω t + Bq sin ω t
ψ (x,t) = X(x)T(t)
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Example: Non dispersive wave equation
Specifying initial conditions determines the (so far arbitrary)
coefficients:
ψ (x, 0);
∂ψ (x,t)
∂t
t =0
In-class worksheet had different examples of initial
conditions
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A
1
2
3
4
B
C
D
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T=mg
Generate standing waves in this system and measure values of
ω (or f) and k (or λ). Plot ω vs. k - this is the DISPERSION RELATION.
Determine phase velocity for system.
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The dispersion relation is an important concept for a system
in which waves propagate. It tells us the velocity at which
waves of particular frequency propagate (phase velocity) and
also the group velocity, or the velocity of a wave packet
(superposition of waves). The group velocity is the one we
associate with transfer of information (more in our QM
discussion).
Non dispersive: waves of different frequency have the same
velocity (e.g. electromagnetic waves in vacuum)
Dispersive: waves of different frequency have different
velocity (e.g. electromagnetic waves in medium; water
waves; QM)
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dispersion relation y = 32.714x + 23.338
omega (rad/sec)
R2 = 0.9736
450
400
350
300
250
200
150
100
50
0
w/k
Linear (w/k)
0
2
4
6
k (1/m)
8
10
12
L=2m
tension=5N
µ=0.005 kg/m
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Waves in a rope: Application of Newton’s law results in the
non dispersive wave equation! (for small displacements
from equilibrium; cosθ ≈ 1)
θ2
T
θ1
ψ
T
x
x+Δx
x
Given system parameters: T = tension in rope; µ = mass per unit length of rope
θ2
T
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θ1
ψ
T
x+Δx
x
Fperp = ma perp
Fperp = T sin θ 2 − T sin θ1
= T ( tan θ 2 cosθ 2 − tan θ1 cosθ1 )
Fperp
⎛ ∂ψ
=T⎜
⎝ ∂x
x
ma perp
∂ 2ψ
= µΔx 2
∂t
2
⎞
∂ψ
∂ψ
−
= T 2 Δx
⎟
∂x x ⎠
∂x
x+Δx
∂ 2ψ T ∂ 2ψ
=
2
2
∂t
µ ∂x
Non disp wave eqn with v2=T/µ
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THE NON-DISPERSIVE WAVE EQUATION
-REVIEW
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(non-dispersive) wave equation
Separation of variables
initial conditions, dispersion, dispersion relation
superposition, traveling wave <-> standing wave, (non-dispersive) wave equation
Mathematical representations of the above