me4213 d alembert's solution of the wave equation
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ME4213/4213E
ME4213/4213E D’Alembert’s Solution of
the Wave Equation
H.P. LEE Department of Mechanical Engineering
EA-05-20 Email: [email protected]
Semester 2 2011/2012
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The Wave Equation The wave equation governing the transverse vibration of
strings, the longitudinal vibration of rod and the torsional
vibration of bars can be expressed in the following
general form
The partial differential equation has been solved by the
method using separation of variables.
2
22
2
2
x
uc
t
u
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D’Alembert’s solution The wave equation can also be solved using
D’Alembert’s approach.
Introduce the new independent variables
We have effectively changed the variables from x and t to u and v.
ctxv
ctxz
)z,v(u)t,x(u
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D’Alembert’s solution
zv uuz
u
v
u
x
z
z
u
x
v
v
u
x
u
x
z)
z
u
v
u(
zx
v)
z
u
v
u(
vx
u
x
zzvzvv uu2uz
u
zz
u
v2
v
u
v
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D’Alembert’s solution Similarly
zv cucuz
uc
v
uc
t
z
z
u
t
v
v
u
t
u
t
z)
z
uc
v
uc(
zt
v)
z
uc
v
uc(
vt
u
t
)uu2u(cz
u
zc
z
u
vc2
v
u
vc zzvzvv
2222
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D’Alembert’s solution
2
22
2
2
x
uc
t
u
)uu2u(c)uu2u(c zzvzvv
2
zzvzvv
2
0vz
uu
2
vz
0vz
uu
2
vz
)v(hv
u
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D’Alembert’s solution
)z()v()z(dv)v(hu
)ctx()ctx()t,x(u
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Points to note
The solution consists of two terms, the first term
is a wave travelling to the left whereas the
second term is a wave travelling to the right.
The functions are to be determined from the
initial conditions (initial shape and velocity)
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Points to note
Consider the wave shown below. If the argument
of F1 are the same, the function has the same
value.
If x0 – ct0 = x1 – ct1, or (x1 – x0) = c(t1 – t0), the
function F1 remains the same. Hence, F1
represents a right travelling wave
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Graphical interpretation of travelling wave
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If the initial shape is a triangular..
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