Longitudinal Waves on a Thin Elastic Rod

(302) |

In the limit , this becomes

It is assumed that the strain is small: that is, .

Here, is a constant, with the dimensions of pressure, which is known as the

Consider the motion of a thin section of the rod lying between and . If is the mass density of the rod then the section's mass is . The stress acting on the left boundary of the section is . Since stress is force per unit area, the force acting on the left boundary is . This force is directed in the minus -direction, assuming that the strain is positive (i.e., the force acts to lengthen the section). Likewise, the force acting on the right boundary of the section is , and is directed in the positive -direction, assuming that the strain is positive (i.e., the force again acts to lengthen the section). Finally, the mean longitudinal (i.e., -directed) acceleration of the section is . Hence, the section's longitudinal equation of motion becomes

(305) |

In the limit , this expression reduces to

(306) |

or

where

(308) |

is a constant having the dimensions of velocity, which turns out to be the propagation speed of longitudinal waves along the rod (see Section 7.1), and use has been made of Equation (303). Equation (307) has the same mathematical form as Equation (254), which governs the motion of transverse waves on a uniform string. This implies that longitudinal and transverse waves in continuous dynamical systems (i.e., systems with an infinite number of degrees of freedom) can be described using the same mathematical equation.

In order to solve Equation (307), we need to specify boundary conditions at the two ends of the rod. Suppose that the left end of the rod is fixed: that is, it is clamped in place such that it cannot move. This implies that . Suppose, on the other hand, that the left end of the rod is free: that is, it is not attached to anything. This implies that , because there is nothing that the end can exert a force (or a stress) on, and vice versa. It follows from Equations (303) and (304) that . Likewise, if the right end of the rod is fixed then , and if the right end is free then .

Suppose, for the sake of argument, that the left end of the rod is free, and the right end is fixed. It follows that , and . Let us search for normal modes of the form

where , , , and are constants. The preceding expression automatically satisfies the boundary condition . The other boundary condition is satisfied provided

(310) |

which yields

(311) |

where is an integer. As usual, the imposition of the boundary conditions leads to a quantization of the possible mode wavenumbers. Substitution of Equation (309) into the equation of motion, (307), yields the normal mode dispersion relation

(312) |

The preceding dispersion relation is consistent with the previously derived dispersion relation (301), given that and . Here, is the interplane spacing, the mass of a section of the rod containing a single plane of atoms, and the effective force constant between neighboring atomic planes.

It follows, from the previous analysis, that the th longitudinal normal mode of an elastic rod, of length , whose left end is free, and whose right end is fixed, is associated with the characteristic displacement pattern

where

Here, and are constants that are determined by the initial conditions. It can be demonstrated that only those normal modes whose mode numbers are positive integers yield unique displacement patterns. Equation (313) describes a standing wave whose nodes (i.e., points where for all ) are evenly spaced a distance apart. The boundary condition ensures that the right end of the rod is always coincident with a node. On the other hand, the boundary condition ensures that the left hand of the rod is always coincident with a point of maximum amplitude oscillation [i.e., a point where ]. Such a point is known as an

Because Equation (307) is linear, its most general solution is a linear combination of all of the normal modes: that is,

(315) |

The constants and are determined from the initial displacement,

and the initial velocity,

It can be demonstrated that [cf., Equation (277)]

Thus, multiplying Equation (316) by , and integrating in from 0 to , we obtain

(319) |

where use has been made of Equations (318) and (278). Likewise, Equation (317) gives

(320) |

Finally, and .

Suppose, for the sake of example, that the rod is initially at rest, and that its left end is hit with a hammer at in such a manner that a section of the rod lying between and (where ) acquires an instantaneous velocity . It follows that . Furthermore, if , and otherwise. It can be demonstrated that these initial conditions yield , ,

(321) |

and

(322) |

where . Figure 33 shows the time evolution of the normalized rod displacement, , calculated from the preceding equations using the first 100 normal modes (i.e., ), and choosing . The top-left, top-right, middle-left, middle-right, bottom-left, and bottom-right panels correspond to , , , , , , , and , respectively. It can be seen that the hammer blow generates a displacement wave that initially develops at the free end of the rod ( ), which is the end that is struck, propagates along the rod at the velocity , and reflects off the fixed end ( ) at time with no phase shift. (The wave front is traveling from the left to the right in all panels except the final one, where it is traveling from right to left.)