Theoretical and Mathematical Physics, 192(1): 974–981 (2017)
FAMILIES OF EXACT SOLUTIONS FOR LINEAR AND NONLINEAR
WAVE EQUATIONS WITH A VARIABLE SPEED OF SOUND AND
THEIR USE IN SOLVING INITIAL BOUNDARY VALUE PROBLEMS
E. V. Trifonov
We propose a procedure for multiplying solutions of linear and nonlinear one-dimensional wave equations,
where the speed of sound can be an arbitrary function of one variable. We obtain exact solutions. We
show that the functional series comprising these solutions can be used to solve initial boundary value
problems. For this, we introduce a special scalar product.
Keywords: exact solution, wave equation, B¨acklund transformation
We consider linear and nonlinear spatially one-dimensional wave equations describing acoustic waves.
For them, we derive B¨acklund autotransformations that we use to obtain families of exact solutions, which
we then use to approximate initial boundary value problems. We note that individual functions in the
families generally fail to satisfy the boundary conditions, but we can construct a functional series from
them such that these conditions are satisﬁed with suﬃcient accuracy.
Diﬀerent versions of B¨acklund transformations have been previously used in the physics of continuous
media. In the papers devoted to this topic, the nonlinear equations were linearized by the hodograph
transformation. For a linear system with variable coeﬃcients, conditions on the coeﬃcients were derived in
the case where there exist B¨acklund transformations leading to equations of the canonical form. A detailed
review of this approach can be found in . We note that the above conditions lead to constraints on the
form of the equation of state and correspondingly on the speed of sound.
The B¨acklund transformations we obtain here are distinguished by, ﬁrst, being autotransformations
and, second, holding for any functional dependence of the speed of sound.
2. Basic equations
Because the form of nonlinear acoustical wave equations is not unique, we focus on their derivation.
In addition, the ﬁrst-order systems obtained in the derivation are used later in seeking exact solutions.
The dynamics of the medium in a sound wave can be considered adiabatic and therefore barotropic.
Institute of Automation and Control Processes, Far Eastern Branch, RAS, Vladivostok, Russia; Far Eastern
Federal University, Vladivostok, Russia, e-mail: firstname.lastname@example.org.
Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 192, No. 1, pp. 41–50, July, 2017. Original
article submitted July 21, 2016; revised November 16, 2016.
2017 Pleiades Publishing, Ltd.