ME4213/4213E

ME4213/4213E D’Alembert’s Solution of

the Wave Equation

H.P. LEE Department of Mechanical Engineering

EA-05-20 Email: mpeleehp@nus.edu.sg

Semester 2 2011/2012

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

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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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