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**Unformatted text preview: **’ = 0. Integrate both sides of the equation from x=-ε to x=+ ε , proceed to ε-> 0, and in so doing obtain the jump, if any, in u x . You may assume that u y is at worst piecewise discontinuous across x=0? Is u x continuous across x=0? 2. Now write down the eigenvalue equation, simplifying S ’ as suggested above. Integrate both sides of this equation from x=-ε to x=+ ε and in so doing obtain the jump in u x ’ across x=0. 3. Solve the eigenvalue equation for u x separately for x > 0 and for x < 0, assuming that u x (x) -> 0 as |x| -> infinity. There will be 2 arbitrary constants when you do this. 4. Apply the 2 jump conditions from 1 and 2 above. Thus, obtain the dispersion relation and so find ω 2 as a function of given k. 5. Compare with the short wavelength limit as done in class (for p ’ small). Make a sketch of | ω | vs k, for kL <<1 and kL >> 1, with a reasonable interpolation thru kL ~ 1....

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