journal of acoustics

THE MATHEMATICAL THEORY OF VIBRATING

BY R. C. Cowv, ^N) J. K. STF, W^RT Wst Virginia University

By a very slight extension of Wheatstone's equation t regarding a vibrating membrane with free 'edges, it is possible to demonstrate that many of the nodal lines of a Chladni plate are given approximately by the formula.

m,rx n,ry n,rx m,ry A cos -- cos B cos-- co = 0. (1)

Although we have plotted all the values of this equation between m-1, n-1 and m= 12, n-12, only a few typical curves are given in this paper. It is our purpose to show the relations between these different curves and to demonstrate that the very beautiful and complicated sand figures formed upon a square plate are either made up of many very much simpler figures or follow a regular gradation in complexity as m and n are increased. Throughout this paper A is taken equal to B.

z! 4.4 Fro. 1.

In all the diagrams of Figure (1), m-n=0, so that the nodal lines are straight lines regularly spaced and parallel to the sides of the square. The sum of m and n will give the number of nodal lines.

In Figure (2), m-n= 1. Under this condition there is a single straight line diagonal from the upper left hand corner to the lower right hand corner of the square, with certain wavy lines parallel to this diagonal. The total number of lines on the plate is always m+n if the diagonal is included. Thus for m= 11, n= 12, there will be eleven wavy lines on

Rayleigh--Sound, Vol. 1, Art. 227. Phil. Mag., Vol. 12, no. 76, p. 320.

591

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592 JOURNAL OF THE ACOUSTICAL SOCIETY [Am, each side of the diagonal or twenty-three lines in all. It will also be no- ticed that if one number is odd, the other must be even. Hence, any two consecutive values for m and n, will always give a pattern similar to those of Figure 2.

1.2

Fro. 2.

In Figure 3, both m and n are odd numbers and m-n-2. In this case, there are two straight lines through the center parallel to the edges. The diagonal from the lower left hand corner to the upper right cuts across m +n lines provided both lines at the center are counted.

The nodes of Figure 4, are plotted for even values of m and n. Under these conditions, any plate may be divided into four equal parts each of which represents the vibration of m/2 and n/2. Thus the plate 2, 4 is


1932] R. C. COLWELL AND J. K. STEWART 593 derived from the vibration 1, 2; 4, 6 from 2 Figure 2 with which they should be compared.

, 3 and 6, 8 from 3, 4 in

Fro. 4.

The nodes of Figure 5 show what happens when m is held constant and n is increased by regular increments. Since m and n are both odd, the characteristic straight line nodes divide the plates into four equal parts. As n is increased by its increment to (nq-2), (n+4), etc., a new nodal line appears in each of the four squares, but the general appear- ance of any figure is remarkably like the preceding one.

L3

I.F 19 1.11 Fro. 5.

If m is held odd and constant and n increased regularly as before, but kept even, another succession of somewhat similar figures is ob- tained as shown in Figure 6.


594 JOURNAL OF THE ACOUSTICAL SOCIETY [Avmz,

1.

1.8

1.4

1,10 Fro. 6.

1,0

1.12

The nodal lines of Figure 7, show how the plate divides when m and n are multiples of lower numbers. Thus plate 4, 8 is made up of sixteen regular squares all vibrating in the mode 1, 2; 3, 9 is made up of nine squares vibrating in the mode 1, 3; 2, 10 is formed from the plate 1, 5 and so on.

)oo( ) oo

4.8 2,10 3. Fro. 7.

Figure 8 is another illustration of m even, n odd when m is held equal to 2 and n is increased by equal increments. The figures become more complicated for higher values of n; but the resemblance to the lower modes is very striking.

In addition to obtaining these curves theoretically, we have also been able to produce many of them with a valve oscillator which has been


1932] R. C. COLWELL AND J. K. STEWART 595

2,7 Fro. 8.

described in the paper referred to above. It is possible to get many of these curves with a violin bow, provided the plate is supported at the proper nodal points.

The mathematical equations for all the curves shown in this paper are found by substituting the two numbers given under each figure into equation (1). The equations are then reduced by means of trigonometric identities and the resulting algebraic equations are solved by Horner's method. The reduced equations for some of the Chladni curves are given below, where u and v represent cos 'xfa and cos 'yfa respectively. The numerals to the right of each equation correspond with the numbers in the different figures. (u + v){2uv - 1} - 0 (1, 2) (uv) { u4(64v 2 - 48) h- uS(64v 4 - 160v 2 h- 80) - 48v 4 h- 80v 2 - 30} = 0 (3, 5) (u -t- v){u4(128v 4 - 128v 2 q- 16) - v a - 24v

- u2(128v 4 - 136v s h- 20) h- u(24v a - 20v) q- 16v 4 - 2v 2 h- 5} = 0 (4, 5) (u2+ v2){u(128v - 128v2 + 16)- u2(128v - 120v + 12)

-t- 16v - 12v2 -t - 1} = 0 (4, 6) 2(uv){32u - 56u + 28u 2 + 32v - 56v+ 28v 2 - 7} = 0 (1, 7) (u -t- v){u7(128v) - u(128v 2) h- u*(128v - 156v) - u(128v - 256v 2)

h-u(128v * - 256v-1 - 160v) -u(128v - 256v-1 - 160v ) h-u(128v 7 - 256v *q- 160v a - 32v) q- 1} = 0. (1, 8) The curves above m=8, n-8 were calculated by a simple method

which will be discussed in another paper. It depends upon the symmetri- cal distribution of the points in those Chladni figures for which A = B.


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