The first continuous system whose normal modes were known, the first use of Bessel functions, and a demonstration of Rayleigh's Principle
This is not a problem with great practical application, but it is of great historical interest and it is interesting to solve. By a chain we mean an extended, uniform, flexible, massive body, of which a chain is an excellent example. The links are to be much shorter than the length of the chain, so that their length plays no role. The chain is suspended at its upper end. Since the chain is to be ideally flexible, it does not matter if the chain is hung on a peg or clamped, just so that the end point remains fixed. Let the he chain have a linear density of ρ gm/cm, and a total length L cm. It hangs under the influence of gravity of strength g cm/sec2. When disturbed, it swings from side to side. In the mode of lowest frequency, it assumes a curve of increasing curvature as one goes away from the point of support. If forced to swing faster, modes can be found that cross the line that the chain assumes when at rest. These crossings are nodes of the oscillation. We will consider small oscillations of the chain.
Let x be the vertical coordinate, measured upwards from the equilibrium position of the end of the chain, and let y(x,t) be the displacement of the point x of the chain to the right of its equilibrium position. The displacements are so small that we do not need to consider the difference between distances measured along the chain and distances measured along x (they are of second order). The tension in the chain is ρgx, and the difference in the horizontal components of the tension at the ends of a small interval dx of chain is the accelerating force. This is just like the vibrating string, except that the tension is variable, not constant. The differential equation of motion that results is shown in the Figure. If we assume a time dependence ejωt, we have an ordinary differential equation for the amplitude y. By the substitution x = gz2/4, this equation can be put into standard form, as shown.
The solution of this equation that is finite at z=0 is J0(ωz), the Bessel function of zero order and the first kind, whose series expansion is shown in the Figure to the right, and whose form is suggested in the Figure on the left below. It is an oscillating function of slowly decreasing amplitude. It is the most common 'special function' found in mathematical analysis, very useful for problems in cylindrical coordinates. Special functions are those that cannot be expressed in terms of the elementary functions, and which are often defined by infinite series or definite integrals. To each root of the equation J0(ωz) = 0 corresponds a normal mode of the swinging chain, and the value of the root determines the frequency of oscillation. The roots are close to (n - 1/4)π, n = 1, 2, 3, .... An arbitrary motion of the chain can be expressed as a linear combination of the normal modes, since the governing equation is linear. Results such as this have been of inestimable value in mathematical physics, notably in vibrations and in quantum mechanics. Normal modes resemble quantum states a great deal, since the mathematical formalism is very much the same. Daniel Bernoulli (1700-1782) determined the normal modes of the hanging chain in 1732, and it was further discussed by Euler in 1781. F. W. Bessel (1784-1846) investigated the functions that bear his name.
The period of oscillation in the lowest mode is T = 5.225(L/g)1/2. Recall that the period of a simple pendulum of length L is T = 6.283(L/g)1/2, so the chain oscillates slightly quicker than a pendulum of the same length. The period of a rigid bar of the same length is 5.130(L/g)1/2, so the bar oscillates slightly quicker than a chain of the same length. This reflects a general principle, that the application of any constraint to a system increases the frequency of its lowest mode of vibration.
We were able to solve this problem because we could solve the differential equation that gave us the shape of the mode. In some practical cases, this either cannot be done, or is very difficult. However, it is still possible to get a good estimate of the frequency of the lowest mode by a method due to Rayleigh. We shall carry this out for a system with one degree of freedom, such as the hanging chain oscillating in its fundamental mode. If q is the generalized coordinate describing the motion of the system, the kinetic energy T can be put in the form T = a(dq/dt)2/2, and the potential energy in the form V = cq2/2, where the coefficients a and c depend on the form of the normal mode. For a mass m suspended by a spring of constant k, T = mv2/2, and V = kx2/2, so a = m and c = k. The ratio of c to a, c/a = k/m, which we recognize as the square of the angular frequency of oscillation.
Rayleigh's method is to assume some plausible form for the normal mode that contains an adjustable constant, and then to determine the coefficients c and a. The ratio c/a is then minimized with respect to the adjustable constant. This, in effect, gives the best approximation to the normal mode that can be obtained from the assumed form, and the minimum of the ratio c/a is an upper limit to the square of the angular frequency of oscillation. Let us carry this out for the hanging chain.
Let us assume y(x,t) = q[x/L + β(x/L)2], where x is now the distance down from the support, and q is the generalized coordinate. y(L,t) = q[1 + β] is the displacement of the end of the chain. The kinetic energy is easy to find; we simply integrate ρdx(dy/dt)2/2 from 0 to L, obtaining T = ρL(dq/dt)2(10 + 15β + 6β2)/60. The potential energy is more difficult. When the chain is displaced to the side, each element is raised a small amount, and the sum of the changes of gravitational potential along the chain is what we want. This must turn out proportional to q2. To carry this out, let us start by replacing x by s for integration along the length of the chain, and use x for vertical distance. We first find the relation of x to s for any point on the chain, and then integrate to find the potential energy from the amount each element is lifted.
Since the oscillations are small, we can expand in powers of dy/dx, and keep only the lowest terms. The procedure is shown in the Figure. The result is V = gρq2(3 + 4β +2β2)/12. We now have ω2L/g = (15 + 20β + 10β2)/(10 + 15β +6β2). The minimum of a ratio u = (A + 2Hx + Bx2)/(a + 2hx + bx2) is given by the quadratic equation (ab-h2)u2 - (aB + bA -2hH)u + (AB - H2) = 0. In the present case, the minimum value is 1.4460, so the period T = 5.225(L/g)1/2, which agrees exactly with the precise result we obtained earlier. The corresponding value of β is 0.6382 + j0.0490. The small imaginary part is curious, and I do not know if it has any meaning. It is not a result of round-off error, but is really there. The ratio c/a is, however, strictly real.
H. Lamb, The Dynamical Theory of Sound, 2nd ed. (London: Edward Arnold, 1925), pp. 84-88. I have expanded a little on his treatment.
F. E. Relton, Applied Bessel Functions (London: Blackie and Son, 1946), pp. 91-92.
Composed by J. B. Calvert
Created 26 June 2000