Sonar echo analysis

Pulse mode sonar operation is analyzed under the assumption that the scattering object Γ lies in the far fields of both the transmitter and the receiver. It is shown that, in this approximation, the sonar signal is a plane wave s ( x · θ 0 – t ) near Γ, where θ 0 is a unit vector directed from the transmitter toward Γ, and similarly the echo is a plane wave e ( x · θ – t ) near the receiver, where θ is a unit vector directed from Γ toward the receiver. Moreover, if Γ is stationary with respect to the sonar system then it is shown thatwhere š(ω) is the Fourier transform of S (τ) and T +(ωθ,ωθ 0 ) is the scattering amplitude in the direction θ due to the scattering by Γ of a time‐harmonic plane wave with frequence ω and propagation direction θ 0 . A generalization of this relation is derived for moving scatterers.