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 Incoming links: Elastic wave polarization

Past exam of the elasticity and plasticity course of the University of Oxford / 2011 / 2 / e / Solution 2026-10-31

For the

b_{1}

root the elastic wave polarization is parallel to the complex derivative vector

(ik, - b_{1})

, giving

B_{1} = \frac{i b _{1}}{k} A_{1} .

(1)

For the

b_{2}

root the field has zero divergence,

ik A_{2} - b_{2} B_{2} = 0

, giving

B_{2} = \frac{ik}{b _{2}} A_{2} .

(2)

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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