For an acoustic field in a waveguide, how would you take advantage of the orthogonality of normal modes to construct a modal, rather than plane-wave beamformer? Use a vertical array.

1. Write down an expression in terms of discrete normal modes for the field of a point source received on a vertical array. Take the complex conjugate of this result and use the result to represent a distribution of point sources. With this source distribution and the known Green’s function of this problem, propagate the resulting field outward. (With this range-independent geometry, outward is the same as

a. What happens at the position of the original point source?

b. How does the vertical array geometry affect the results?

c. This method is called backpropagation. Is there a difference between this method

and Bartlett matched field processing?