$$L^2$$-boundedness for 1-D wave equations with time variable coefficients

Abstract We consider the $$L^{2}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:math> -boundedness of the solution of the Cauchy problem for a wave equation with time-dependent wave speeds. We treat it in the one-dimensional Euclidean space $$\textbf{R}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>R</mml:mi> </mml:math> . We adopt a simple multiplier method by using a special property of the one dimensional space.