Show that the measured pressure in the far field is approximately p(r, t) = zf(t − c−1r)/r2.

An acoustic dipole can be modeled by two point sources (monopoles) residing close together but vibrating exactly out of phase. Suppose point source 1 is located at (0, 0, −d) and point source 2 is located at (0, 0, d), where d is very small. Further suppose that source 1 is generating the narrow-band signal f(t), leading to a spherical pressure wave given by p(r1,t) = f(t − c−1r1)/r1 and source 2 is generating the narrow-band signal −f(t), leading to a spherical pressure wave given by p(r2,t) = −f(t − c−1r2)/r2, where the two radii r1 and r1 are measured from their respective sources.

(a) Show that the measured pressure in the far field is approximately p(r, t) = zf(t − c−1r)/r2.

(b) Sketch isopressure lines in the x-z plane for the above field pattern.

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