Consider a Pekeris waveguide with the speed of sound c1 = 1500 m/s and density p1 = 1000 kg/m3 in the water column, and with c2 = 1800 m/s and _2 = 2000 kg/m3 in the bottom. The water depth is 100 m. A line source at depth zs is generating a plane acoustic field in the waveguide.

a. Defining the slowness of the mth normal mode as

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where kxm is the horizontal wavenumber of the mode, state the upper and lower limit of pm for modes propagating in the positive x-direction.

b. For a source frequency exciting 3 modes, make a sketch of the mode functions for pressure and for the particle velocity potential. Discuss the differences.

c. Derive the expression for the vertical wavelength of the modes.

d. Using the results from questions (a) and (c), state the lower limit for the vertical wavelength of a mode at angular frequency !.

e. Use the result from (d) to determine how many modes you have at frequency

f D 30 Hz.