Transverse and Longitudinal waves in a wire

Question

The question is :

The speed of longitudinal wave is ten times the speed of transverse waves in a tight brass wire. If the Young's Modulus of the wire is Y, then strain in the wire is?

I have read about transverse waves in a wire. I also know that the velocity is equal to $\sqrt{T/μ}$ where $T$ is the tension in the wire and $μ$ is the mass per unit length. My doubt is about longitudinal waves. Are longitudinal waves even possible in a stretched wire? And if they are possible, how can I derive the expression for its velocity?

Note that while this is a homework question, my doubt is about a concept and not the solution of the problem.

Answer 1

Longitudinal waves are possible. To see the effect in action, take a slinky and stretch it out on the ground. Holding one end of the stretched Slinky, quickly move your hand a short distance toward the other end of the slinky, and immediately move your hand right back to the original position. You will create a longitudinal wave pulse that travels away from your hand. That is a nice visible model of how a longitudnal wave in a wire happens.

To derive the speed of longitudnal wave, consider a small segment of wire that has been displaced longitudinally. The longitudinal forces on the segment come from stress in the wire, which is proportional to strain, which is the derivative of the displacement. Set the sum of forces on the segment equal to the longitudnal acceleration of the segment and you will get differential equation that will reveal the spead of the longitudinal wave.