ME4213 D Alembert's Solution of the Wave Equation
Transcript of ME4213 D Alembert's Solution of the Wave Equation
![Page 1: ME4213 D Alembert's Solution of the Wave Equation](https://reader031.fdocuments.in/reader031/viewer/2022020718/53fd3621dab5ca94038b4d3e/html5/thumbnails/1.jpg)
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
![Page 2: ME4213 D Alembert's Solution of the Wave Equation](https://reader031.fdocuments.in/reader031/viewer/2022020718/53fd3621dab5ca94038b4d3e/html5/thumbnails/2.jpg)
ME4213/4213E 2
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
![Page 3: ME4213 D Alembert's Solution of the Wave Equation](https://reader031.fdocuments.in/reader031/viewer/2022020718/53fd3621dab5ca94038b4d3e/html5/thumbnails/3.jpg)
ME4213/4213E 3
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
![Page 4: ME4213 D Alembert's Solution of the Wave Equation](https://reader031.fdocuments.in/reader031/viewer/2022020718/53fd3621dab5ca94038b4d3e/html5/thumbnails/4.jpg)
ME4213/4213E 4
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
![Page 5: ME4213 D Alembert's Solution of the Wave Equation](https://reader031.fdocuments.in/reader031/viewer/2022020718/53fd3621dab5ca94038b4d3e/html5/thumbnails/5.jpg)
ME4213/4213E 5
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
![Page 6: ME4213 D Alembert's Solution of the Wave Equation](https://reader031.fdocuments.in/reader031/viewer/2022020718/53fd3621dab5ca94038b4d3e/html5/thumbnails/6.jpg)
ME4213/4213E 6
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
![Page 7: ME4213 D Alembert's Solution of the Wave Equation](https://reader031.fdocuments.in/reader031/viewer/2022020718/53fd3621dab5ca94038b4d3e/html5/thumbnails/7.jpg)
ME4213/4213E 7
D’Alembert’s solution
)z()v()z(dv)v(hu
)ctx()ctx()t,x(u
![Page 8: ME4213 D Alembert's Solution of the Wave Equation](https://reader031.fdocuments.in/reader031/viewer/2022020718/53fd3621dab5ca94038b4d3e/html5/thumbnails/8.jpg)
ME4213/4213E 8
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)
![Page 9: ME4213 D Alembert's Solution of the Wave Equation](https://reader031.fdocuments.in/reader031/viewer/2022020718/53fd3621dab5ca94038b4d3e/html5/thumbnails/9.jpg)
ME4213/4213E 9
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
![Page 10: ME4213 D Alembert's Solution of the Wave Equation](https://reader031.fdocuments.in/reader031/viewer/2022020718/53fd3621dab5ca94038b4d3e/html5/thumbnails/10.jpg)
ME4213/4213E 10
Graphical interpretation of travelling wave
![Page 11: ME4213 D Alembert's Solution of the Wave Equation](https://reader031.fdocuments.in/reader031/viewer/2022020718/53fd3621dab5ca94038b4d3e/html5/thumbnails/11.jpg)
ME4213/4213E 11
If the initial shape is a triangular..
![Page 12: ME4213 D Alembert's Solution of the Wave Equation](https://reader031.fdocuments.in/reader031/viewer/2022020718/53fd3621dab5ca94038b4d3e/html5/thumbnails/12.jpg)
ME4213/4213E 12