In electromagnetics, acoustics and seismology, lateral waves or head waves are interfacial waves that are launched at or near grazing incidence with respect to the interface of two different media with different physical properties, such as permittivity, acoustic impedance or slowness. While lateral waves are often classified as surface waves, they are distinct from conventional surface waves such as surface plasmon polaritons in optics or surface acoustic waves in acoustics. Lateral waves play a role in various different physical phenomena, such as radio propagation at large distances, extraordinary optical transmission and total internal reflection. In exploration seismology and geophysics, lateral waves constitute the basis of the seismic refraction method.
Mathematical formulation The mathematical physics governing lateral waves is analogous across electromagnetics, acoustics, and seismology, as the diffraction mechanisms and boundary conditions for wave coupling share a common theoretical framework; consequently, the following description utilizes an electromagnetic formulation as a representative example. Lateral waves emerge as a specific contribution during the asymptotic evaluation of the Sommerfeld integrals used to solve the inhomogeneous wave equation for a point or line source near a plane interface separating two homogeneous media with different wavenumbers, k 1 {\displaystyle k_{1}} and k 2 {\displaystyle k_{2}} . For a magnetic line source located at depth z ′ {\displaystyle z'} in Medium 1 ( z < 0 ) {\displaystyle (z<0)} , the resulting magnetic field H {\displaystyle H} in that medium can be represented as a Fourier integral:
H ( y , z ) = − ω ϵ 1 4 π ∫ − ∞ ∞ [ e i κ 1 | z − z ′ | + Γ ( η ) e − i κ 1 ( z + z ′ ) ] e i η y κ 1 d η {\displaystyle H(y,z)=-{\frac {\omega \epsilon _{1}}{4\pi }}\int _{-\infty }^{\infty }\left[e^{i\kappa _{1}|z-z'|}+\Gamma (\eta )e^{-i\kappa _{1}(z+z')}\right]{\frac {e^{i\eta y}}{\kappa _{1}}}d\eta }
where:
η {\displaystyle \eta } is the horizontal wavenumber.
κ j = k j 2 − η 2 {\displaystyle \kappa _{j}={\sqrt {k_{j}^{2}-\eta ^{2}}}} represents the vertical propagation constant in medium j {\displaystyle j} .
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