Wave motion and spectral methods

Phys 750 Lecture 13

Wave equation‣ Wave motion is described by a partial differential

equation in time and position

‣ For example: vibrations on a generic string described by a displacement field evolving according to u(x, t)

⇥⇤2u

⇤t2+R

⇤u

⇤t= T

⇤2u

⇤x2� �

⇤4u

⇤x4+ · · ·

⇥ =

R =

T =

� =

mass per unit lengthloss coefficientline tensionnonlinear correction

x

u(x, t)

Wave equation‣ In the limit of small amplitudes, weak spatial

modulation, and slow vibrations, the PDE is linear:

‣ The wave velocity is a characteristic speed related to the string’s tension and mass per unit length

‣ Advection factoring:

�2u

�t2= c2 �2u

�x2

c =p

T/�

✓�

�t+ c

�

�x

◆✓�

�t� c

�

�x

◆u(x, t) = 0

Wave equation

‣ For an infinite string, any right- or left-travelling wave is a solution

‣ Lossless propagation

‣ Since the PDE is linear, all linear superpositions of solutions are also solutions:

x

F (x� ct)

F (x + ct)

u(x, t) = aF (x ± ct) + bG(x ± ct) + · · ·

x

envelopefunction

Space-time mesh‣ Naïve approach is to apply usual discretization techniques

‣ PDE requires a double mesh:

‣ Numerical estimates of the time and space derivatives lead to a recurrence relationwith

u(n)i = u(i�x, n�t)

u(n+1)i + u(n�1)

i � 2u(n)i

(�t)2⇥ c2

�u(n)

i+1 + u(n)i�1 � 2u(n)

i

(�x)2

⇥

u(n+1)i = 2

�1� s2)u(n)

i � u(n�1)i + s2

�u(n)

i+1 + u(n)i�1

⇥

s = c�t/�x controls the stability of the recursion

Space-time mesh

x

t

nn+1

n−1

ii+1i−1

‣ How we assign the current field from its values at previous time steps — e.g., — is represented by a stencil:

u(n+1)i = 2

�1� s2)u(n)

i � u(n�1)i + s2

�u(n)i+1 + u(n)

i�1

�

Reflections‣ All finite systems have a

boundary, where the wave is contained by reflection

‣ A fixed end leads to reversal of motion and a phase shift

‣ Finite differences are troublesome there

�F (�x� c(t� tb))

F (x� ct)

⇡

Reflections

x

ct

L

L

2L

ct = x

ct = 2L� x

‣ Single reflection event leads to the superposed solution

‣ With the understanding that foru(x, t) = 0 x > L

u(x, t) = F (x� ct)

� F (2L� x� ct)

Reflections

x

ct

ct = �x

ct = x� 2L

ct = 2L� x

ct = x

ct = 2L� x

ct = 2L+ x

ct = 4L� x

u(x, t) =1X

n=�1

⇥F (2nL+ x� ct)

� F (2nL� x� ct)⇤

‣ String segment undergoes an infinite number of reflections:

‣ Vanishes at and

‣ Periodic extension of the envelope function

x = 0x = L

Reflections

�F (�x� ct)

F (x� ct) = ���F (�(�x)� ct)

⇥

‣ A wave contained in a finite interval has a stable solution consisting of all multiply-reflected contributions

‣ Constructive parts: u(x, t) ⇥ F (x� ct)� F (�x + ct)

Normal modes‣ Suppose that the wave is confined to

‣ Barring pathological examples, the periodic extension of has a discrete Fourier representation

‣ The components are arbitrary except that the resulting envelope function must be real ( )

x � [0, L]

F (x) =⇥�

n=�⇥Fnein�x/L

F

Fn

F ⇤n = F�n

Normal modes‣ Waves in motion have the form

‣ Behaviour of each mode is determined by a characteristic wave-vector and angular frequency

F (±x� ct) =⇤�

n=�⇤Fne±iknxe�i�nt

kn =n�

L, ⇥n = ckn

n

�⇥2u

⇥t2= T

⇥2u

⇥x2��⇥2

nu = �Tk2nu ⇥n = (T/�)1/2kn = ckn

Normal modes

x = 0 x = L

u(0, t) = u(L, t) = 0

‣ Solution:

‣ Boundary conditions are automatically satisfied for all time

‣ Each mode is orthogonal to the others, and there is no energy transfer between them ( )an = 0

u(x, t) = F (x� ct)� F (�x� ct) =��

n=1

an sin knx cos �nt

Normal modes‣ From the initial conditions determine components by overlap with each mode:

‣ Complete behaviour at all subsequent times:

u(x, 0) =1X

n=1

an sin knx

an =1

L

Z L

0dxu(x, 0) sin knx

u(x, t) =1X

n=1

an sin�nx

Lcos

�nct

L

Truncation errors‣ Make tractable by putting a bound on the mode sums:

‣ Spatial resolution is set by the wavelength of the highest mode

‣ Discarded modes should have negligible power

�x = 2L/nc

1X

n=1

!ncX

n=1

� = 2⇥/kn = 2L/n

E =1

2

Zdx

✓⇥u

⇥t

◆2

+ c2✓⇥u

⇥x

◆2�

=1X

n=1

�n|an|2 �ncX

n=1

n2|an|2 +O((nc + 1)|anc+1|)2

Spectral method‣ Alternative strategy for more general PDEs that support

wavelike motion, e.g.,

‣ We can start from the ansatz

‣ Solve the space-only conventional ODE for each mode:

u(x, t) = un(x)ei!nt��t

@

2u

@t

2+ b

@u

@t

= c

2 @2u

@x

2+ f

@

4u

@x

4+ · · ·

(i!n � �)2un(x) + b(i!n � �)un(x)

= c

2u

00n(x) + fu

0000n (x) + · · ·

Spectral method

‣ View as a (real-valued, ) eigenequation

‣ Imposition of boundary conditions leads to a discrete eigenspectrum, corresponding to the normal modes

� = b/2

✓c

2 @

2

@x

2+ f

@

4

@x

4

◆un(x) =

✓�!

2n +

1

4b

2

◆un(x)