The Wave Equation

♦ The method of characteristics

♦ Inclusion of boundary conditions

♦ Traveling waves and stationary waves

TRANSVERSE OSCILLATIONS OF AN ELASTIC STRING

small displacements y

θ2

θ

x

1

T2

T1

yuniform linear density ρ

• T1 cos θ1 = T2 cos θ2for small θ, cos θ ≃ 1 ⇒ T1 = T2 = T

• ρ δx∂2y

∂t2= T sin θ2 − T sin θ1

sin θ ≃ tan θ ≃∂y

∂x=⇒ ρ δx

∂2y

∂t2= T

[

(∂y

∂x)2 − (

∂y

∂x)1

]

︸ ︷︷ ︸

(∂2y/∂x2)δx+...

Thus ρ∂2y

∂t2= T

∂2y

∂x2

i.e. ,∂2y

∂x2=

1

c2∂2y

∂t2, c2 ≡

T

ρwave equation

2nd-order linear PDEs

A(x, y)uxx+2B(x, y)uxy+C(x, y)uyy+D(x, y)ux+E(x, y)uy+F (x, y)u = R(x, y)

• B2 − AC < 0 elliptic. Ex.: Laplace eqn. uxx + uyy = 0

• B2 −AC = 0 parabolic. Ex.: diffusion eqn. ut − αuxx = 0

• B2 −AC > 0 hyperbolic. Ex.: wave eqn. utt − c2uxx = 0

♦ General solutions of PDEs depend on arbitrary functions

(analogous to solutions of ODEs depending on arbitrary constants)

−→ boundary conditions to determine such functions

• Cauchy boundary conditions:

assign function u and normal derivative ∂u/∂n on given curve γ in xy plane

(relevant to hyperbolic PDEs)

D’Alembert’s solution

!2y

!x2=

1

c2

!2y

!t2

u = x ! ct

v = x + ct

Change variables

!

!x=

!u

!x

!

!u+!v

!x

!

!v=

!

!u+

!

!v

!

!t=

!u

!t

!

!u+!v

!t

!

!v= " c

!

!u+ c

!

!v

!2y

!x2=

1

c2

!2y

!t2

"

!!u

+!!v

"#$

%&'

2

y =1

c2

(c!!u

+ c!!v

"#$

%&'

2

y =!!u

(!!v

"#$

%&'

2

y

!"2y

"u"v= 0

y u,v( ) = f u( ) + g v( ) i.e. y x,t( ) = f x ! ct( ) + g x + ct( )General solution

f, g arbitrary functions

• The curves in the xt plane

x− ct = const.

x+ ct = const.

are called the characteristics of the wave equation.

x−ct=const.

x

t

x+ct=const.

CHARACTERISTICS

Consider the PDE Aytt + 2Bytx + Cyxx +Dyt + Eyx + Fy = R

Q =

(

A B

B C

)

matrix of 2nd− order coefficients

• Characteristics of the above PDE are defined as the curves

χ(t, x) = const.

such that their normal n is rotated by 90◦ by Q or is annihilated by Q, i.e.,

n ·Q n = 0.

• Since n ∝ ∇χ, for characteristics ∇χ ·Q ∇χ = 0.

⊲ Example. Wave eqn. : −1

c2ytt + yxx = 0 ⇒ Q =

(

−1/c2 0

0 1

)

The curves χ∓(t, x) = x∓ ct =const. are the characteristics of the wave equation because

∇χ∓ =

(

∓c

1

)

⇒ ∇χ ·Q ∇χ = (∓c 1 )

(

−1/c2 0

0 1

)(

∓c

1

)

= (∓c 1 )

(

±1/c

1

)

= 0

• The condition ∇χ ·Q ∇χ = 0 implies, using ∇χ = (χt, χx), that

(χt χx )

(A B

B C

)(χt

χx

)

= 0 ,

i.e., Aχ2t + 2Bχtχx + Cχ2

x = 0 .

Expressing the derivatives in terms of x′(t) = −χt/χx,

A[x′(t)]2 − 2Bx′(t) + C = 0 .

♠ Hyperbolic eqns. (B2 −AC > 0) have 2 families of characteristics

♠ Parabolic eqns. (B2 −AC = 0) have 1 (Q ∇χ = 0)

♠ Elliptic eqns. (B2 −AC < 0) have none

Uses of characteristics

• Characteristics χ∓(t, x) = const. can be used to solve hyperbolic equations

by means of the transformation of variables

u = χ−(t, x)

v = χ+(t, x)

⊲ Example: D’Alembert solution of the wave equation

• Characteristics serve to analyze whether boundary value problems

for PDEs are well posed.

⊲ Example: Cauchy conditions on curve γ well-defined

provided γ is not a characteristic

[Cauchy-Kovalevska theorem]

Cauchy boundary conditions and characteristics

Consider the PDE Aytt + 2Bytx + Cyxx = H(yt, yx, y, t, x)

• Cauchy conditions: Suppose y and the normal derivative ynare assigned on the curve γ specified by

G(t, x) = 0

⊲ The normal and tangential directions to γ are n ∝ ∇G = (Gt, Gx), τ ∝ (−Gx, Gt).

⊲ Given y on γ, we can compute tangential derivative yτ . From yn and yτ we can get yt and yx.

• Can we determine higher derivatives as well?

∂

∂τyt = τ · ∇ yt ∝ (−Gx Gt )

(ytt

yxt

)

= −Gxytt +Gtyxt

∂

∂τyx = τ · ∇ yx ∝ (−Gx Gt )

(ytx

yxx

)

= −Gxytx +Gtyxx

⇒ 3 linear equations in ytt, ytx, yxx, with unique solution if det 6= 0

det

A 2B C

−Gx Gt 0

0 −Gx Gt

6= 0

⇒ AG2t + 2BGtGx + CG2

x 6= 0

That is, Cauchy conditions on curve γ are well-defined

provided γ is not a characteristic

! y x,t( ) = f x " ct( ) + g x + ct( )!

2y

!x2=1

c2

!2y

!t2

!"#$%&'%()"*+*,$-.,/+

f and g are determined by the initial conditions:

Suppose at time t = 0, the wave has an initial displacement U(x) and an initial velocity V (x)

y x,0( ) = f x( ) + g x( ) =U x( )

!y x,0( )!t

= "c #f x( ) + c #g x( ) =V x( ) $ f x( ) " g x( ) = "1

cV x '( )dx

b

x

% '

f x( ) =1

2U x( ) !

1

2cV x '( )dx

b

x

" '

g x( ) =1

2U x( ) +

1

2cV x '( )dx '

b

x

!

y x,t( ) =1

2U x ! ct( ) +U x + ct( )"#

$% +

1

2cV x( )dx ! V x( )dx

b

x!ct

&b

x+ct

&"

#

'''

$

%

(((

=1

2U x ! ct( ) +U x + ct( )"#

$% +

1

2cV x( )dx

x!ct

x+ct

&

!"#$%&'($)*+,$*-*.&/$0(1+&-23/&0$4*56/&1(7(-+$0(/(&5(4$8097$0(5+:$$

y x,t( ) =1

2U x ! ct( ) +U x + ct( )"#

$%

V (x) = 0

y x,t( ) =1

2U x ! ct( ) +U x + ct( )"# $% +

1

2cV x( )dx

x!ct

x+ct

&'

(

))

*

+

,,

U x( )

y x,t( ) = f x ! ct( ) + g x + ct( )

What is the form of f (x), g(x)?

If time dependence is cos ! t( ) the full x,t( ) dependence is given by

y x,t( ) = f x ! ct( ) + g x + ct( )

!"#$%&#%'('

Speed of wave c =

!

k•

• Frequency f =1

!=

"

2#

t

! =2"

#y(x,t)

x

! =2"

ky(x,t)• Wavelength ! =

2"

k

k is "wavenumber"

y(x,t)=Acos kx ! ! t( ) + Bcos kx "! t( )

We can write the equation of a travelling wave in a number of analogous forms:

Velocity Wavelength Period Angular

frequency

( )sinA kx t!" / k! 2 / k# 2 /# ! !

( )sinA k x vt" v 2 / k# 2 / vk# vk

sin 2x t

A #$ %

& '( )"* +, -. /0 1

/$ % $ % 2 /# %

( )sin 2 /A x vt# $"& '0 1 v $ / v$ 2 /v# $

N.B. Can include phase most easily by putting

( ) ( ), Re expy x t A i kx t!& '= "& '0 10 1

where A is complex.

N.B.2 Sometimes more convenient to switch x and t, i.e.

( ) ( ), siny x t A t kx!= "

This is still a travelling wave moving to the right.

For non-sinusoidal wave moving to right with speed v, can always write as ( )f x vt! .

!"#$%&#'()*#+,-)

y = Asin kx !"t( ) + Asin kx +"t( )= 2Asin kxcos"t

y = Asin kx !"t + 2#1

( ) + Asin kx +"t + 2#2

( )= 2Asin kx + #

1+ #

2( )cos "t + #

1! #

2( )

.%',)/,&,'#00()

y t + !t( )

y t( )

1-"#$%&#'()*#+,2)

!"#$%&'()*+#,$-.)/#0-

!.E!"= 0

!.B!"= 0

! " E!"= #

$B!"

$t

! " B!"=1

c2

$E!"

$t

!"#$%&&'()%*+",-.()/01%%)(2"3%4)

!" #$B!"

$t= !

$" # B!"

$t= !

1

c2

$2E

!"

$t2

!2E

!"=1

c2

"2E

!"

"t2

EM plane wave E!"= E!"z( )

!2E

!"

!z2=1

c2

!2E!"

!t2

!E!"

!z= 0" E

z= 0

51".(6%1(%)$"6%)

! "! " E

!"= #!

2E

!"+! !.E

!"

( )

0

†

†

!"#$%&'$(")*

Ex = Asin kx !"t( )

Ey = Bsin kx !"t + #( )

Ex

Ey