A standing acoustic wave with shocks in a cubically nonlinear medium

• It is well known that transversal elastic waves in homogeneous solids satisfy a wave equation with a cubic nonlinearity. This equation with resonator boundary conditions can be transformed into a functional equation, which can be reduced to a second order partial differential equation with a cubic nonlinearity. From this equation, by specializing to steady state and integrating one step, we obtain a first order ordinary differential equation with three terms in addition to the derivative: a cubic and a linear term in the dependent variable and a known term (sinus). The coefficient of the derivative is proportional to the dissipation and assumed to be small. Among several cases the most complicated case, the coefficient of the linear term lying between zero and (0.5) (2/3) = 0.63, is treated in this paper. In each period the solution has two shocks. At one side of each shock it is necessary to introduce an intermediate boundary layer between the outer region and the inner region next to the shock. The intermediate solution is matched both outwards and inwards. The actual first order ordinary differential equation is also solved numerically both in the outer region and in the neighborhood of the shocks.

Ämnesord

TEKNIK OCH TEKNOLOGIER  -- Maskinteknik -- Strömningsmekanik och akustik (hsv//swe)

ENGINEERING AND TECHNOLOGY  -- Mechanical Engineering -- Fluid Mechanics and Acoustics (hsv//eng)

TEKNIK OCH TEKNOLOGIER  -- Maskinteknik -- Teknisk mekanik (hsv//swe)

ENGINEERING AND TECHNOLOGY  -- Mechanical Engineering -- Applied Mechanics (hsv//eng)

Nyckelord

cubic nonlinear media

nonlinear acoustic resonator

shocks

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