Propagation of Transient Sound Signals into a Viscous Fluid

Abstract

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 usi-•g 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.

On the relation between the wavefront speed and the group velocity concept

“…In 1957 Hanin presented a solution to the above problem although he had great difficulties since the model did not lead to a finite value for the phase velocity at high frequencies and consequently, a wavefront could not be found and the problem was ill-posed. [11][12][13][14][15][16][17][18] The Maxwell model for dissipative fluids, however, yields a well-posed problem and for future use we here give the expression for the phase velocity for the case of a single relaxation process 12…”

“…[10][11][12][13][14][15][16] In terms of analytic function theory this corresponds to application of Jordan's lemma which can be related to the Tauberian theorem also known as the initial value theorem. 11,[15][16][17] What all this means is that in order to get a picture of the solution for short times, that is, at the wavefront, one has to look at the behavior of the integrand for high frequencies. 11 However, this means, from Eq.…”

“…10,12 To discuss the wavefront property and the associated velocity, we here quote some results found in the literature which emanates from solving wave propagation problems by means of evaluation of Fourier integrals. [10][11][12][13][14][15][16][17] The output pulse p(x,t) can formally be written as …”

“…First, the method of Landau is not applicable since it predicts undistorted propagation which is impossible for dispersive media. [12][13][14][15][16][17] The approximate method of steepest descent cannot be used in that it only yields information far behind the wave front. [16][17] The energy velocity interpretation by Reynolds requires sinusoidal waves and time as well as space averages which is of no help for this pulse problem.…”

“…14 He had great difficulties since the model did not lead to a finite value for the phase velocity at high frequencies and consequently, a wave front could not be found which means that the problem was ill-posed. [10][11][12][13][14][15][16][17] The causality constraint which states that the cause must precede its effects was thus violated. 11,18 The causality principle is the basis for the Kramers-Kronig dispersion relations which contain the value for the phase velocity at a limiting high frequency.…”