Speed of longitudinal waves on a stretched horizontal spring

Abstract In introductory calculus-based physics, students learn that traveling waves are solutions of the wave equation. By applying Newton’s second law to a disturbance in an elastic medium, a wave equation can be derived and the speed of the associated traveling waves identified. Many textbooks use this procedure to deduce the speed of transverse waves on a string. Instructors often demonstrate such waves in class by having a student hold the far end of a rope while they quickly shake their end up and down once to send a transverse pulse along the string. Another common demonstration does the same on a stretched slinky. Using a spring instead of a string permits one to alternatively launch a longitudinal pulse by moving one’s hand quickly forward and backward once (rather than up and down). Here it is shown how to efficiently derive the wave equation for this case also from Newton’s second law, and the resulting longitudinal wave speed is compared to the transverse formula. Student learning is reinforced by deducing the one-dimensional wave equation twice from the familiar starting point of Newton’s second law, once for transverse and then again for longitudinal waves.