8. WKBJ Theory

Question 1

What is WKBJ Theory?

Answer 1

-a tool for solving ODEs of the form:

ε² y’’ - q(x) y = 0, 0

Question 2

The Leading Order Solution

Form of y(x)

Answer 2

y(x) = A exp[Σ δn(ε) Sn(x)]
= A exp[δ0(ε)S0(x) + δ1(ε)S1(x) + ...]
= AE(x)
-sum from n=0 to n=∞
-with {δn(ε)} and {Sn(x)} to be found and δ0>>δ1>>δ2>>... as ε->∞

Question 3

The Leading Order Solution

Steps

Answer 3

• taking the WKBJ form of y, find y’ and y’’ then sub in to the original equation
• find dominant balance including q(x) to find δ0 and S0
• find the next balance to get δ1 and S1

Question 4

The Leading Order Solution

Eikonal Equation

Answer 4

-the equation for So’

So’ = ± √[q(x)]

Question 5

The Leading Order Solution

Transport Equation

Answer 5

-equation for S1’

S1’ = - So’‘/2So’

Question 6

The Leading Order Solution

Solution

Answer 6

y(x) = A E(x)
-where:
E(x) = exp[1/ε (±∫√[q(x)]dx) + 1 (log|q|^(-1/4) + c) + O(ε)]
-integral from x0 to x where x0 is chosen based on the region of interest for the problem

Question 7

The Leading Order Solution

General Solution

Answer 7

y(x) ~ A+ / |q(x)|^(1/4) exp[1/ε ∫√[q(x)]dx] + A- / |q(x)|^(1/4) exp[-1/ε ∫√[q(x)]dx]

Question 8

The Leading Order Solution

-when is the two term approximation the exact solution?

Answer 8

-when q is constant

Question 9

The Leading Order Solution

q(x)<0

Answer 9

-the exp terms correspond to oscillations with length scale ε which are modulated on the length scale of q(x)

Question 10

WKBJ Theory

Boundary Value Problems

Answer 10

-write in the form of a WKBJ problem:
ε² y’’ - q(x) y = 0,
-find general solution as normal
-use boundary conditions to find constants

Question 11

WKBJ Theory

Initial Value Problem: y’’ + fy

Answer 11

• follow the same steps as for the BVP
• when using the initial conditions, for the y’ condition there will be terms that can be ignored due to their relative size, so check order of terms

Question 12

WKBJ Theory

Initial Value Problem: y’‘+fy’+gy

Answer 12

• make the substitution T=εt
• seek a WKBJ solution x(T)=AE(T)
• sub in to initial equation
• consider O(1) terms to find So
• consider O(ε) terms to find S1
• sub in to y form for WKBJ to get general solution
• find unknown constants
• convert back to t from T

Question 13

WKBJ Theory

Eigenvalue Problems

Answer 13

• an eigenvalue problem:
• y’’ = λ w(x) y, y(a)=0, y(b)=0
• with w(x)>0 on [a,b] so λ>0
• in WKBJ form, ε=1/λ & q(x)=-w(x)<0 and suppose that λ»1
• sub in to general solution with x0=a
• solution will contain a constant n with some constraints corresponding to the fact that λ in the original equation can only take on certain values