PHYS212 : Lectures

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Subsections

# 3.1 The Hanging Cable

## 3.1.1 The Equation of Motion

• The continuous limit of the hanging chain considered in Section 1.4.2 is a hanging cable or rope of linear mass density .

• In the limit of small displacements from equilibrium, the tension in the cable acts mainly in the vertical direction and is given by
 (3.1)

where is the vertical position relative to the free end of the cable.

• Any net transverse force on an infinitesimal mass element of the cable is due to a difference in the components of the tensions in the cable just above and just below the cable element, due to a bending of the cable element through an angle . Newton's second law yields
 (3.2)

which can be approximated via by
 (3.3)

• Differentiating Eq. 3.3 with respect to yields
 (3.4)

Recognizing as the linear mass density of the cable, we obtain the equation of motion of the cable
 (3.5)

## 3.1.2 The Solutions : Bessel Functions of Order Zero

• Like Eq. 1.55, the equation of motion for waves on a string, Eq. 3.5 is a second order linear partial differential equation, and we will try solving it using the method of separation of variables.

• You will show in homework that the product solution
 (3.6)

leads rapidly to the separation of Eq. 3.5 into
 (3.7)

which gives the usual time dependence, and
 (3.8)

• You will also show that with the enlightened'' substitution
 (3.9)

and with
 (3.10)

that Eq. 3.8 becomes
 (3.11)

• Eq. 3.11 is Bessel's differential equation of order zero. It has two independent solutions , the zeroth order Bessel function, and , the zeroth order Neumann functions.

• goes to at , so it does not contribute to the solutions for the hanging cable.

Figure 3.1: The zeroth order Bessel function and its first six roots.
 Root u 1 2.404825558 2 5.520078110 3 8.653727913 4 11.79153444 5 14.93091771 6 18.07106397

• The Bessel and Neumann functions cannot be simply written in terms of other elementary functions, but they have been extensively studied and their properties are readily available in the literature.3.1 Many scientific and mathematics software packages include them as mathematical functions like sine and cosine.

• To obtain standing wave solutions for the hanging cable, we apply the boundary condition , that the end of the cable at is fixed. This implies that the argument of the Bessel function must be one of the infinite set of roots of .

• Just as for the stretched string, we have a countably infinite set of standing wave modes. For the th standing wave mode,
 (3.12)

where is the th root of .

• Eq.'s 3.10 and 3.12 combine to give
 (3.13)

• The general solution, combining the time dependent and position dependent parts is
 (3.14)

Shapes of the first four standing wave modes are shown in Fig. 3.2.

• The resemblance between Fig.'s 3.2 and Fig. 1.7 showing the first three normal modes of a 20-link hanging chain is expected. However, the two systems are not identical. Fig. 1.8 shows the progression of the discrete system toward the cable solution with increasing link density for the second mode. The solid curve in that figure is the cable solution of Eq. 3.14. Note that the 100 link chain is already quite close to the continuous limit.

• Demo : a driven, hanging, beaded chain 1.0 m long. The predicted frequencies (Eq. 3.13) and periods for the first three modes are

 Copyright © 2003-2009, Lewis A. Riley Updated Wed Jan 18 09:51:28 2006