Template:Incident potential for two dimensions

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Incident potential

To create meaningful solutions of the velocity potential [math]\phi[/math] in the specified domains we add an incident wave term to the expansion for the domain of [math]x \lt 0[/math] above. The incident potential is a wave of amplitude [math]A[/math] in displacement travelling in the positive [math]x[/math]-direction. We would only see this in the time domain [math]\Phi(x,z,t)[/math] however, in the frequency domain the incident potential can be written as

[math] \phi_{\mathrm{I}}(x,z) =e^{-k_{0}x}\chi_{0}\left( z\right). [/math]

The total velocity (scattered) potential now becomes [math]\phi = \phi_{\mathrm{I}} + \phi_{\mathrm{D}}[/math] for the domain of [math]x \lt 0[/math].

The first term in the expansion of the diffracted potential for the domain [math]x \lt 0[/math] is given by

[math] a_{0}e^{k_{0}x}\chi_{0}\left( z\right) [/math]

which represents the reflected wave.

In any scattering problem [math]|R|^2 + |T|^2 = 1[/math] where [math]R[/math] and [math]T[/math] are the reflection and transmission coefficients respectively. In our case of the semi-infinite dock [math]|a_{0}| = |R| = 1[/math] and [math]|T| = 0[/math] as there are no transmitted waves in the region under the dock.