We study the acoustic scattering which occurs when a plane wave is incident on an infinite hollow elastic cylinder covered with an ideally bonded layer of viscoelastic and sound‐absorbing material. The structure is immersed in water and it contains air in the cavity. The normal‐mode solution is found in all four media and programmed for numerical evaluation in the exterior fluid. Series expressions for the differential and the sonar scattering cross sections are constructed and added by computer. Two size cylinders are investigated in a parametric study, and oscillations in the cross‐sectional values due to resonances in the shell and coating materials are found and plotted for the basically low‐frequency rangek1c<20. The effect of a variable viscosity in the coating is analyzed, and as one would expect, progressively higher viscosity values damp out the rapid cross‐sectional oscillations and reduce the amplitudes to smaller values, down to an optimum point, beyond which the reverse effect begins to take place. The way the oscillating cross‐section peaks are shifted in frequency or damped out in amplitude by changing shell or coating sizes or materials, can be quantitatively used at one’s convenience for various design purposes.