Ocean currents affect sound propagation. For simplicity onsider sound from a line source propagating in a laminar flow velocity V.z/ parallel to the ocean bottom and in the positive x-direction. Linearizing about the background state as in Sect. 2.1 one can derive the following convected wave equation:

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where and are the acoustic particle velocities in the x and z-directions, respectively.

a. Show that the normal modes of this equation satisfy

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Note that V.z = 0 gives the usual modal equation.

b. Ocean currents will satisfy a no-slip condition implying that the flow velocity vanishes at the bottom. Nevertheless, consider an ocean with uniform flow, uniform sound speed and with a perfectly rigid bottom. What is the dispersion relation? Plot representative curves for different modes and flow speeds. Include the asymptotes.