INTRODUCTION

The interaction of perturbations with the front of a nonlinear wave in a dissipative medium, described by equations of the Burgers type, are considered in [1–4].

In [1], the interaction of weak deterministic and random perturbations with smooth areas of the sawtooth wave profile was calculated. For nonplane waves in an inhomogeneous medium, in the nonlinear geometric acoustics approximation, the model of the generalized Burgers equation is valid. This model yields an evolution equation for regular perturbations and an equation for transformation of the correlation function of noise perturbations. Analytical solutions are found for an arbitrary shape of the initial signal or correlation function. Examples demonstrate the stability of the saw shape with a small perturbation of its rectilinear area.

In [2], perturbations localized at the shock front were considered. A simple mathematical model of a front with a variable slope in a dissipative medium is proposed. Quantitative characteristics of the interaction between the evolving shock front and weak perturbations of various types are obtained. A detailed review of wave interactions is given in [3].

Article [4] presents the results of detailed studies as a continuation of [1, 2]: the interaction of noise and regular signals with a front whose steepness increases or decreases due to nonlinear distortion of the profile of an intense pumping wave. At the leading edge, the signal amplitude increases and its time scale decreases. At the trailing edge, there is a decrease in amplitude, an increase in scale, and a slowdown in the rate of evolution. It is shown that, due to competition between amplification at the shock front and high-frequency damping, the distance dependence of the noise intensity is nonmonotonic; at large distances, the intensity tends to zero, and the correlation time, to a finite value.

However, the interactions of perturbations with a nonlinear wave profile as a whole (containing a shock front and smooth areas) have not been fully studied. It is most interesting to consider these interactions with the perturbations deemed not weak. The presence of an exactly solvable Burgers equation makes this possible.

1 INTERACTION OF A SHORT PULSE WITH A STATIONARY WAVE FRONT

Let us analyze the processes using model of the Burgers equation describing the propagation of nonlinear nondispersive waves in a dissipative medium:

$$\frac{{\partial V}}{{\partial z}} - V\frac{{\partial V}}{{\partial \theta }} = \Gamma \frac{{{{\partial }^{2}}V}}{{\partial {{\theta }^{2}}}}.$$
(1)

The equation is written in standard dimensionless notation. If we bear in mind the propagation of acoustic pressure waves \(p\) in a medium of finite amplitude \({{p}_{0}}\), the physical sense of the normalized variables and constants will be

$$\begin{gathered} V = \frac{p}{{{{p}_{0}}}},\,\,\,\,{{\theta }} = {{\omega \tau }},\,\,\,\,z = \frac{x}{{{{l}_{{{\text{SH}}}}}}}, \\ {{l}_{{{\text{SH}}}}} = \frac{{{{c}^{3}}{{\rho }}}}{{{{\varepsilon \omega }}{{p}_{0}}}},\,\,\,\,{{l}_{{{\text{DIS}}}}} = \frac{{2{{c}^{3}}{{\rho }}}}{{b{{{{\omega }}}^{2}}}},\,\,\,\,\Gamma = \frac{{{{l}_{{{\text{SH}}}}}}}{{{{l}_{{{\text{DIS}}}}}}}. \\ \end{gathered} $$
(2)

Here, \({{\omega }}\) is the characteristic wave frequency of the wave or the reciprocal duration of the pulse signal, and \({{\tau }} = t - {x \mathord{\left/ {\vphantom {x c}} \right. \kern-0em} c}\) is time in a coordinate system moving together with the wave in the direction of the \(x\) axis at sound speed \(c\). The parameters of the medium are as follows: density \({{\rho }}\) and nonlinearity and dissipation coefficients \({{\varepsilon }}\) and \(b\). \({{l}_{{{\text{SH}}}}},\,\,\,{{l}_{{{\text{DIS}}}}}\) denotes the characteristic lengths of nonlinearity (formation of a discontinuity in the initial smooth wave profile) and dissipation (linear absorption of the wave by the medium). The ratio \(\Gamma \) of these lengths is known as the reciprocal acoustic Reynolds–Goldberg number [5].

The stationary solution to Eq. (1) for a wave traveling at the sound speed and not changing its shape during propagation has the form

$$V = {\text{tanh}}\left( {\frac{\theta }{{2\Gamma }}} \right).$$
(3)

Let us write the general solution to Eq. (1), which describes the initial wave profile with an arbitrary shape:

$$\begin{gathered} V = 2\Gamma \frac{\partial }{{\partial {{\theta }}}}\ln U,\,\,\, \\ \,U\left( {z,\,{{\theta }}} \right) = \int\limits_{ - \infty }^\infty {{{U}_{0}}\left( {{{\theta }}{\kern 1pt}^ {{'}}} \right)} \,G\left( {z,\,{{\theta }} - {{\theta }}{\kern 1pt}^ {{'}}} \right)\,d{{\theta }}{\kern 1pt}^ {{'}}. \\ \end{gathered} $$
(4)

Here, the Green’s function is

$$G\left( {z,\,{{\theta }}} \right) = \frac{1}{{\sqrt {4{{\pi }}\,\Gamma z} }}\exp \left( { - \frac{{{{{{\theta }}}^{2}}}}{{4\,\Gamma z}}} \right).$$
(5)

Conditions at the boundary of the medium for the auxiliary function \(U\) and sought function \(V\) are denoted as

$${{U}_{0}} = \exp \left[ {\frac{1}{{2\Gamma }}\int\limits_{}^{\theta {\kern 1pt} '} {{{V}_{0}}\left( {\theta {\kern 1pt}^ {"}} \right)\,d\theta {\kern 1pt}^ {"}} } \right],\,\,\,\,{{V}_{0}}\left( \theta \right) = V\left( {z = 0,\,\theta } \right).$$
(6)

Let us “impose” a short pulse in the form of the delta function on stationary wave (3). Conditions (6) at the boundary take the form

$$\begin{gathered} {{V}_{0}} = \operatorname{th} \left( {\frac{{{\theta }}}{{2\Gamma }}} \right) + {{\alpha \delta }}\left( {{{\theta }} - {{{{\theta }}}_{1}}} \right), \\ {{U}_{0}} = \cosh \left( {\frac{{{\theta }}}{{2\Gamma }}} \right)\exp \left[ {\frac{{{\alpha }}}{{2\Gamma }}H\left( {{{\theta }} - {{{{\theta }}}_{1}}} \right)} \right]. \\ \end{gathered} $$
(7)

Here \(H\) is the Heaviside unit function, which is zero for negative values of the argument and one for positive values. The center of pulse is shifted to the position \({{\theta }} = {{{{\theta }}}_{1}}\).

Note that in the absence of a stationary “step,” an isolated \({{\delta }}\)-function evolves in accordance with the self-similar solution [6]

$$\begin{gathered} V = 4\Gamma \beta \frac{{\exp \left( { - {{\xi }^{2}}} \right)}}{{\sqrt {4\pi \,\Gamma z} }}{{\left[ {1 + \beta \,{\text{erf}}\left( \xi \right)} \right]}^{{ - 1}}}, \\ \xi = \frac{\theta }{{\sqrt {4\Gamma z} }},\,\,\,\,\,\beta = {\text{tanh}}\left( {\frac{\alpha }{{4\Gamma }}} \right). \\ \end{gathered} $$
(8)

Here, \({\text{erf}}(\xi ) = \frac{2}{{\sqrt \pi }}\int_0^\infty {\exp \left( { - \xi {\kern 1pt} {{'}^{2}}} \right)d\xi } \) is the error function. Already at small distances, a δ-pulse with positive polarity acquires an asymmetric bell-like shape. Its leading edge is steeper than its trailing edge. The asymmetry is more pronounced at \({{\beta }} = \tanh \left( {{{{\alpha }} \mathord{\left/ {\vphantom {{{\alpha }} {4\Gamma }}} \right. \kern-0em} {4\Gamma }}} \right)\) close to one. As the distance traveled increases, the pulse duration increases and its peak value decreases.

The interaction of a short pulse and a stationary wave with boundary condition (7) is described by the expression

$$\begin{gathered} \,U = \cosh \left( {\frac{{{\theta }}}{{2\Gamma }}} \right) + \frac{{{\beta }}}{2}\left[ {\exp \left( {\frac{{{\theta }}}{{2\Gamma }}} \right)\operatorname{erf} \left( {\frac{{{{\theta }} - {{{{\theta }}}_{1}} + z}}{{\sqrt {4\Gamma z} }}} \right)} \right. \\ \left. { + \,\,\exp \left( { - \frac{{{\theta }}}{{2\Gamma }}} \right){\text{erf}}\left( {\frac{{{{\theta }} - {{{{\theta }}}_{1}} - z}}{{\sqrt {4\Gamma z} }}} \right)} \right]. \\ \end{gathered} $$
(9)

The process is shown in Figs. 1a, 1b. For Fig. 1a, the following parameters are chosen: \({{\beta }}\) = 0.98, Г = 0.02, θ1 = 1. The distance is taken as z = 0.01, 0.1, 0.5, 1, 2 for curves 1–5, respectively. Curve 1 is close in shape to the initial pulse disturbance at some distance from the front. This perturbation travels at supersonic speed, faster than a stationary wave, and at some distance “catches up” with the front. In this case, the pulse broadens, decreases in magnitude, and merges with the front. Curves 3 and 4 show that in the merging process, a shock front is formed with a pressure jump elevated in comparison to the stationary value. This shifts the front forward compared to the initial position. After complete absorption of the pulse, the wave again takes the shape of the stationary solution (curve 5), while the front stops and does not move with further propagation. For Fig. 1b, the following parameters were chosen: \({{\beta }}\) = 0.8, Г = 0.2, θ1 = 1. The distance is taken as z = 0.01, 0.1, 0.5, 1, 2 for curves 1–5. Here, the details of the merging process are seen more clearly. Final curve 5 is also a stationary solution with the front shifted forward as a result of interaction.

Fig. 1.
figure 1

Interaction of short pulse and stationary front: (a) \({{\beta }}\) = 0.98, Г = 0.02, (b) \({{\beta }}\) = 0.8, Г = 0.2. Parameter θ1 = 1. Curves 1–5 correspond to distances z = 0.01, 0.1, 0.5, 1, 2.

Full size image

Let us additionally note the following peculiarities in the behavior of the solution. With an increase in parameter Г, which corresponds to broadening of the front of the stationary solution, the pulse amplitude increases at small distances. However, this is accompanied by faster process pulse dissipation, and its complete absorption occurs at nearly the same distance for different Г values. A decrease in parameter β leads to a decrease in the pulse amplitude and, as a consequence, a decreased shift of the shock front in the final profile.

2 INTERACTION OF A SHORT PULSE WITH AN EVOLVING FRONT

During the formation of a shock front, the pulse undergoes a more complex evolution. As is known, the simplest exact solution to the Burgers equation describes a straight area of a profile with a variable slope. As the front forms, the steepness increases; the solution in this case looks as follows:

$$V\left( {z,\,{{\theta }}} \right) = \frac{{{\theta }}}{{{{z}_{0}} - z}}.$$
(10)

The initial slope of straight line (10) is determined by the constant \({{z}_{0}}\). During propagation, the slope increases, and with \(z \to {{z}_{0}}\), the straight line tends to a “vertical” position. Of course, dissipation should limit the steepness of the front to a value of \(2\Gamma \) (see (3)); this happens at distances smaller than \({{z}_{0}}\).

To calculate the interaction of a pulse with such a front, instead of (7), we set a boundary condition of the following form:

$$\begin{gathered} {{V}_{0}} = \frac{\theta }{{{{z}_{0}}}} + \alpha \delta \left( \theta \right), \\ {{U}_{0}} = \exp \left( {\frac{{{{\theta }^{2}}}}{{4\Gamma }}} \right)\exp \left[ {\frac{\alpha }{{2\Gamma {{z}_{0}}}}H\left( \theta \right)} \right]. \\ \end{gathered} $$
(11)

Solving the equation for U and returning to V, we obtain

$$\begin{gathered} V = \frac{\theta }{{{{z}_{0}} - z}} + 4\Gamma \beta \frac{{\exp \left( { - \frac{{{{\theta }^{2}}}}{{4\Gamma f\left( z \right)}}} \right)}}{{\sqrt {4\pi \Gamma f\left( z \right)} }} \\ \times \,\,{{\left[ {1 + \beta {\text{erf}}\left( {\frac{\theta }{{\sqrt {4\Gamma f\left( z \right)} }}} \right)} \right]}^{{ - 1}}}\,. \\ \end{gathered} $$
(12)

Here

$$f\left( z \right) = z\left( {1 - \frac{z}{{{{z}_{0}}}}} \right).$$
(13)

Let us compare the second term in formula (12) with formula (8), which describes the evolution of a separate \({{\delta }}\)-pulse that does not interact with the front. We see that the difference is that in (12), instead of the distance \(z\), there is a reduced distance (13), which determines both the time duration of the pulse and its peak value. The transformation of pulse profile (12) without taking into account the linear front is shown in Fig. 2 with the following parameters: \({{\beta }}\) = 0.98, Г = 0.02, z0 = 1. Curves 1–9 correspond to distances z = 0.01, 0.02, 0.05, 0.15, 0.5, 0.85, 0.95, 0.98, 0.99. The duration for \(z = 0\) was zero (the peak value was infinite), then the pulse broadened and the peak value decreased due to dissipative effects (curves 1–4). The pulse reaches the maximum duration \({{{{\theta }}}_{ * }} = \sqrt {\Gamma {{z}_{0}}} \) (minimum amplitude) at a distance \(z = {{{{z}_{0}}} \mathord{\left/ {\vphantom {{{{z}_{0}}} 2}} \right. \kern-0em} 2}\) (curve 5). In the domain \({{{{z}_{0}}} \mathord{\left/ {\vphantom {{{{z}_{0}}} 2}} \right. \kern-0em} 2} < z < {{z}_{0}}\), the inverse process occurs (curves 6–9 replicate corresponding curves 1–4, but for different distances). The pulse begins to narrow due to interaction with the evolving front and at \(z \to {{z}_{0}}\) turns back into the delta-function. This trend was noted in [1–4]. Obviously, on the smoothing front \(V\left( {z,\,{{\theta }}} \right) = {{ - {{\theta }}} \mathord{\left/ {\vphantom {{ - {{\theta }}} z}} \right. \kern-0em} z}\), the steepness of which decreases, a monotonic process of an increase in the pulse duration and decrease in its amplitude will occur, since the interaction with the front and dissipation in this case act in the same direction.

Fig. 2.
figure 2

Interaction of short pulse and evolving front; transformation of pulse profile (12) without taking into account linear front. Curves 1–9 correspond to distances z = 0.01, 0.02, 0.05, 0.15, 0.5, 0.85, 0.95, 0.98, 0.99. Parameters \({{\beta }}\) = 0.98, Г = 0.02, z0 = 1.

Full size image

In concluding this section, let us consider the well-known property of solutions to the Burgers equation, which is based on its invariance with respect to the projective transformation [7]

$${{V}_{2}} = \frac{\theta }{{{{z}_{0}} - z}} + \frac{1}{{1 - {z \mathord{\left/ {\vphantom {z {{{z}_{0}}}}} \right. \kern-0em} {{{z}_{0}}}}}}{{V}_{1}}\left( {\frac{z}{{1 - {z \mathord{\left/ {\vphantom {z {{{z}_{0}}}}} \right. \kern-0em} {{{z}_{0}}}}}},\frac{\theta }{{1 - {z \mathord{\left/ {\vphantom {z {{{z}_{0}}}}} \right. \kern-0em} {{{z}_{0}}}}}}} \right).$$
(14)

Its meaning is as follows: if the function \(V = {{V}_{1}}(z,\,\theta )\) is an exact solution to the Burgers equation, then the function \(V = {{V}_{2}}(z,\,\theta )\), defined by formula (14) will also be an exact solution. Using (14) in application to solution (8), we immediately obtain solution (12).

3 INTERACTION OF A PULSE WITH A SINGULARITY OF THE WAVE PROFILE

As is known, the Burgers equation has exact solutions with singularities. Such solutions, at first glance, have no physical meaning. The simplest is the function \(V = {{2\Gamma } \mathord{\left/ {\vphantom {{2\Gamma } \theta }} \right. \kern-0em} \theta }\). Since there is no distance dependence here, it would seem that this solution should describe a stationary wave. However, as shown in [8], a solution with a singularity cannot be stationary, since the singularity is destroyed (see Fig. 3).

Fig. 3.
figure 3

Destruction of stationary solution with singularity (for \({{\gamma }} = 0\)). Curves 1–5 correspond to distances z = 0.01, 0.1, 0.5, 1, 2. Parameter Г = 0.2.

Full size image

Conditions at the boundary of type (7), (11) will be taken for this problem:

$$\begin{gathered} {{V}_{0}} = \frac{{2\Gamma }}{\theta } + \alpha \delta \left( {\theta - {{\theta }_{1}}} \right), \\ {{U}_{0}} = \left| \theta \right|\exp \left[ {\frac{\alpha }{{2\Gamma }}H\left( {\theta - {{\theta }_{1}}} \right)} \right]. \\ \end{gathered} $$
(15)

The solution to the Burgers equation for (15) has the form

$$\begin{gathered} U = \exp \left( { - {{\xi }^{2}}} \right) + \gamma \exp \left( { - {{{\left( {\xi - {{\xi }_{1}}} \right)}}^{2}}} \right) \\ + \,\,\sqrt \pi \xi \left[ {\gamma + {\text{erf}}\left( \xi \right) + \gamma {\text{erf}}\left( {\xi - {{\xi }_{1}}} \right)} \right]\,, \\ V = \sqrt {\frac{{\pi \,\Gamma }}{z}} \frac{1}{U}\left\{ {\gamma \left[ {1 + {\text{erf}}\left( {\xi - {{\xi }_{1}}} \right)} \right]} \right. \\ \left. { + \,\,{\text{erf}}\left( \xi \right) + \frac{2}{{\sqrt \pi }}\gamma {{\xi }_{1}}\exp \left( { - {{{\left( {\xi - {{\xi }_{1}}} \right)}}^{2}}} \right)} \right\}. \\ \end{gathered} $$
(16)

For brevity, it is denoted here as

$$\gamma = \frac{1}{2}\left[ {\exp \left( {\frac{\alpha }{{2\Gamma }}} \right) - 1} \right],\,\,\,\xi = \frac{\theta }{{\sqrt {4\Gamma z} }},\,\,\,{{\xi }_{1}} = \frac{{{{\theta }_{1}}}}{{\sqrt {4\Gamma z} }}.$$
(17)

Solution (16) for the destruction of an individual singularity in the absence of perturbation \({{\gamma }} = 0\) is simplified:

$$\begin{gathered} V = \sqrt {\frac{{\pi \Gamma }}{z}} \frac{{{\text{erf}}\left( \xi \right)}}{{\exp \left( { - {{\xi }^{2}}} \right) + \sqrt \pi \,\xi \,{\text{erf}}\left( \xi \right)}}, \\ {{\left. V \right|}_{{z \to 0}}} = {{\left. V \right|}_{{\xi \to \infty }}} = \sqrt {\frac{\Gamma }{z}} \frac{1}{\xi } = \frac{{2\Gamma }}{\theta }. \\ \end{gathered} $$
(18)

It is shown in Fig. 3 for the parameter Г = 0.2 and distances z = 0.01, 0.1, 0.5, 1, 2 (curves 1–5). Note that this solution is odd.

Figure 4 shows the “incidence” of a short pulse on a singularity and its absorption in the vicinity of a singular point, which proceeds simultaneously with the destruction of the singularity due to simultaneous nonlinear and dissipative effects. Figures 4a and 4b plot pulses with a relatively small amplitude. In this case, the pulse is quickly absorbed. The profile formed at large distances is similar to the case of absence of an incident pulse, but it becomes asymmetric: the peak positive values exceed the peak negative ones. Figure 4c shows the case of incidence of a short pulse with a large amplitude. In this case, the formation of a shock wave is clearly visible, which propagates at supersonic speed and displaces the shock front.

Fig. 4.
figure 4

Interaction of pulse with singularity for (a) \({{\gamma }} = 0.4\), (b) \({{\gamma }} = 2\), and (c) γ = 109. Parameters Г = 0.2, θ1 = 1.

Full size image
Fig. 4.
figure 5

(Contd.)

Full size image

In all the cases considered, the short pulse vanishes in the vicinity of the initial singularity of the profile and does not prevent the its destruction, although it significantly modifies this process.