Propagation of Transient Sound Signals into a Viscous Fluid

An arbitrary excitation of the plane x = 0 sends sound signals into the half-space x ⩾ 0 occupied by the viscous fluid. The governing third-order partial differential equation is solved exactly using the Laplace transform on time and the sine transform on space. New expressions for the most general solution are derived. The specific inputs considered in detail are the Dirac delta function, the Heaviside unit function, a decaying exponential, and a sinusoidal excitation. The final expressions are given in the form of real integrals and of exact power series. Short-time approximations are also given for a general input and for the four aforementioned specific inputs. Previously obtained approximations for the case of small attenuation coefficients are corrected and extended for the decaying exponential and the sinusoidal input. The results found indicate that viscosity tends to reduce the sharpness of the propagating disturbance and thus to smooth out any initially imposed discontinuity. The presence of the disturbance is felt immediately everywhere in the medium, and, in fact, the short-time approximations satisfy the parabolic heat equation.