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Longitudinal Waves in Rod

Let us consider the motion of a disturbance along a long elastic solid rod. As the rod is struck lengthwise, i.e. longitudinally, at one of its ends (at x = 0), the vibrations travel along the length of the rod; each section of the rod vibrates longitudinally as the disturbance passes through.

Consider the equation of motion of a thin slice AB of the rod, which in the undisturbed state, lies between position x and x + Δ x. The mass of this differential element is ρ A Δ x where A is area of cross-section of the rod and ρ is density.

As the disturbance reaches the part AB, the material in slice AB is put under stress; consequently it is both shifted and stretched. Suppose the longitudinal displacement of particles at x is y, and that of particles at x + Δ x is y + Δ y. Hence, the average strain of the slice is

Corresponding, we define the average stress on the slice Ab using the definition of Young’s modulus Y as,

gives the stress, or elastic force per unit area, at position x. The stress at position (x + Δ x) therefore is

Retaining only linear term in Δ x, we find

The above expression gives the net stress (force per unit area) on the slice AB. The equation of motion, therefore, is

The equation of motion of the particles in elastic rod shows a wave motion with velocity which depends only on the properties of the material, viz. its Young’s modulus and density.