The perturbation (small-parameter) method is used to obtain the first three approximations for the problem of a harmonic longitudinal plane wave propagating in a quadratic nonlinear hyperelastic material described by the classical Murnaghan model. The subsequent approximations are discussed. The contribution of each approximation to the overall wave pattern is analyzed. It is shown that the third approximation corrects the prediction of the evolution of the initial wave profile. Which of the harmonics dominates depends on the distance traveled by the wave: the second harmonic is generated first, then it transforms into the fourth harmonic, and finally, as the distance increases, the eighth harmonic shows Keywords: harmonic longitudinal plane wave, quadratic nonlinear wave equation, perturbation method, generation, fourth harmonic, eighth harmonic, subsequent harmonicsIntroduction. This paper deals with the well-known physical phenomenon of second-or third-harmonic generation by a harmonic wave propagating in a nonlinear medium of arbitrary nature (water, gas, thermomechanical, optical, or acoustic medium, etc.). The essence of the phenomenon is often and very effectively demonstrated in a second-harmonic generation experiment where an optical wave propagating in an ammonium dihydrogen phosphate crystal changes from ruby light (red) to ultraviolet [26]. More abstractly, a harmonic wave enters a quadratic or cubic nonlinear medium, interacts with itself while propagating, and leaves the medium as a second or third harmonic, respectively. Thus, experiments confirm that a harmonic wave does generate a second harmonic in a quadratic nonlinear medium [1-3, 8, 26]. This phenomenon is theoretically described as a property of the solution of the corresponding nonlinear wave equation. The generation of the second harmonic by a longitudinal harmonic plane wave in a quadratic nonlinear hyperelastic material described by the classical Murnaghan model was described in [5, 17] by a method traditional for nonlinear optics (Van der Pol's method of slowly varying amplitudes). Let us outline this approach.The following nonlinear wave equation is known [2, 3, 5, 10, 17] to describe the motion of a quadratic nonlinear longitudinal plane wave: