Evanescent Field due to Total Internal Reflection

Note that if $c_1 < c_2 \vert\sin(\theta_1^+)\vert$, the horizontal component of the wavenumber in medium 2 becomes imaginary:

• Acoustic field in medium 2 is ``evanescent''
• Wave in medium 1 undergoes
  ``total internal reflection''
• No power travels from medium 1 to medium 2
• Evanescent field decays exponentially to the right
• ``Tunneling'' possible given medium 3 in which wave propagation resumes
• Reference: Vibrations and Waves in Physics by I.G. Main, Cambridge University Press, 1978.

But what does it mean?

$$\begin{eqnarray*} p(t,\underline{x}) &=& \cos\left(\omega t - \underline{k}^T\... ...{-k_x x}\right\}}}\\ [5pt] &=& e^{-k_x x} \cos(\omega t - k_y y) \end{eqnarray*}$$

• An imaginary wavenumber corresponds to an exponential decay
• Sign of $\kappa_x$ is chosen to match boundary conditions at plane
• Time dependence applies to all points to the right of the boundary (no ``propagation'')