 Solution

These relations also follow by inserting each decay root into either row of the matrix. The complete decaying displacement field has complex components

U (Y) = A_{1} e^{- b_{1} Y} + A_{2} e^{- b_{2} Y}, V (Y) = \frac{i b _{1}}{k} A_{1} e^{- b_{1} Y} + \frac{ik}{b _{2}} A_{2} e^{- b_{2} Y},

(3)

multiplied by

e^{ik (X - c t)}

, followed by the real part. The first constituent is irrotational and the second is divergence-free; both are needed to cancel the two surface tractions.

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