An asymptotic description of the acoustic signature of a crack breaking the surface of an otherwise homogeneous, isotropic elastic material for a line focus, scanning acoustic microscope is constructed. The Debye approximation is used to calculate an incident focused beam whose profile falls off continuously at its edges. The wavefields scattered from the surface are constructed as Fourier integrals that are approximated asymptotically. Included in the asymptotic approximations are the leaky Rayleigh waves, which play a crucial part in the acoustic signature. Explicit expressions for the incident and scattered wavefields are given. The acoustic signature is calculated by using an electromechanical reciprocity identity to relate the wavefields in the coupling fluid to the voltage at the terminals of the microscope's transducer. Several ways of evaluating this identity for an unbroken surface are explored and are shown to be asymptotically equivalent. The acoustic signature of a surface-breaking crack is then calculated by assigning to the crack reflection and transmission coefficients for the leaky Rayleigh wave and then using geometrical elastodynamics to construct the scattered wavefields. Explicit expressions for the acoustic signature of the cracked surface are given. Moreover, an explicit expression for the reflection coefficient of a Rayleigh wave reflected from a surface-breaking crack is given.