Vibrating Flat Things 
Vibrating Flat ThingsMembranes, plates and shells have practical applications,
as well as furnishing good mathematical examples
There is a special counting word, jang, for flat (square) things
in Chinese. They say 'three flat thing paper' since there are no plurals
in the language, and numbers must be accompanied by a counting word, to
show that multiple things are meant. A flat thing has one dimension much
less than the other two. A membrane, from the Latin for a sheet
of parchment, is a flat thing with little stiffness, that can easily be
bent and folded. A plate is indeed flat, the word coming from the
Old French for flat. Sheets of glass come to mind. A plate resists bending,
and has complex internal stresses. A shell is a curved flat thing,
like an eggshell or bell. All such bodies can vibrate mechanically, the
membrane requiring a stress to give restoring force, which in the others
can be supplied by stiffness.
The membrane is a very simple case, because the internal stresses are simple. If we imagine a free body, a small part of the membrane imagined to be cut out and the forces that the rest of the membrane previously exerted on it replaced by forces around its boundary, these forces at any point are always in the tangent plane at that point. Since such a membrane cannot resist shearing forces, they are also normal to the edge of the element. A further assumption often made is that these tensions are equal in any direction. By considering a small triangular element, we then see that it is sufficient to specify the tensions in two perpendicular directions to be equal, for it to be the same in any direction. This is not necessary. A membrane can be stretched unequally in two directions, which will cause a lack of isotropy. We proceed with the case of an isotropically stretched membrane with a tension T dyne/cm and a density of σ gm/cm^{2}. The ratio T/σ has dimensions (cm/s)^{2}, which we recognize as the square of a wave phase velocity, which we confidently expect will be the speed of transverse waves in the membrane. This is easily verified by applying Newton's Law to an element of the membrane, just as in the case of the string, the onedimensional analogue of the present problem. In the Figure, the net force dF normal to an element of area dA = dydx is derived. Note that this can be expressed in terms of the principal radii of curvature of the surface. Normal displacements of the membrane then obey the simple wave equation. The surface tension of liquids makes them behave as if the surface were a membrane. Of course, it is not, since it would be a very curious membrane whose tension did not increase on extension. The surface of a liquid has air on one side, liquid on the other, so its motions are not those of a membrane alone. A soap film, with two surfaces, is much more like a material membrane, but is still a peculiar one. The force on an element dA = dydx of the membrane in terms of the principal radii of curvature can be directly interpreted as due to a difference in pressure dF/dA on the two sides of the surface. The static shape of the membrane, expressed by z(x,y), is the solution of Laplace's equation. This is true only for small displacements z, since we have used the approximation sin x = tan x in deriving the wave equation. A soap film whose boundary is given assumes a shape of minimum surface area, since this minimizes the free energy of the surface. A rubber sheet with the same boundary is an imperfect analogy, since the tension must be adjusted until it is isotropic, and the tension increases with strain. The surface is then one that minimizes strain energy, which is equivalent to minimum area only if all parts of the sheet are equally strained. Everything like this deserves a careful examination to determine what is happening. It is interesting to find the potential energy of the displaced membrane. This can be considered as the increase in area times the tension, since dyne/cm can be considered as erg/cm^{2}, or surface energy, as is often done in the case of surface tension of liquids. The original area dydx stretches to [1 + (dz/dx)^{2} + (dz/dy)^{2}]^{1/2}dydx = [1 + (dz/dx)^{2} + (dz/dy)^{2} + ...]dydx, so the potential energy V is the integral of (T/2)[(dz/dx)^{2} + (dz/dy)^{2}]dydx over the membrane, very closely. The kinetic energy T is likewise the integral of (σ/2)(dz/dt)^{2}dydx over the membrane. We can obtain vibration solutions by analysis for a few very simple geometries, and gain insight. We can also obtain solutions by numerical computation, and gain a lot of numbers. When we have a numerical solution, there are few ways of knowing if it is reasonable or absurd by cursory examination. Let's start, then, by assuming a time dependence e^{iωt}, as we always do for a linear equation, for then we can superimpose solutions to fit different problems. In this case, the result is a partial differential equation (because we have two independent variables x,y). Since we have rectangular coordinates, a rectangular membrane seems appropriate, since then the boundary conditions can be expressed simply. In fact, we have z(0,y) = z(a,y) = z(x,0) = z(x,b) = 0 if the membrane of size a x b is clamped all around its boundary, taking the origin at the lower left corner. Should the solution be of the form y(x,y) = X(x)Y(y), it will be easy to apply the boundary conditions, for then X(0) = X(a) = Y(0) = Y(b) = 0 will do the job. Substituting the product solution into the equation for the amplitude, we find ω^{2}/c^{2} = (d^{2}X/dx^{2})/X + (d^{2}Y/dy^{2})/Y. The first term on the right depends only on x, the second only on y. Since x and y can vary independently, they must each separately equal some constant. Let X"/X = c^{2}, and Y"/Y = d^{2}, where the primes stand for differentiation (c is not the phase velocity!). Now we have X" + c^{2}X = 0, Y" + d^{2}Y = 0, and c^{2} + d^{2} = k^{2}, where k = ω/c is the wavenumber. The ordinary differential equations are easily solved: X = A sin cx + B cos cx, and Y = C sin dy + D cos dy. The boundary conditions force B = D = 0 and sin ca = 0, sin db = 0. Therefore, c = mπ/a, d = nπ/b, where m, n = 1,2,3,4,.... This, then, means that (k/π)^{2} = (m/a)^{2} + (n/b)^{2} = (2f/c)^{2}, and so the frequencies of oscillation are found. The process we have just used is called separation of variables. It is not a powerful technique, since it can be used only in a few cases, but is extremely convenient in those few cases. The lowest frequency, the fundamental, is given by m = n = 1. Then f = f_{11} = (c/2)[(1/a)^{2} + (1/b)^{2}]^{1/2}. If, in addition, a = b, f = 2^{1/2}c/2a = (T/2aσ)^{1/2}. This is somewhat higher than the lowest frequency of a string of length a. The first overtone will have m = 2, n = 1, and a frequency f_{21} = (5/2) ^{1/2} f_{11}. The ratio of frequencies, 1.58, is not quite the 1.5 of a harmonic fifth. The overtones of a membrane, like those of many vibrating systems, are not harmonic. The string and pipe are exceptional in this respect. All modes except the fundamental have nodal lines along which the displacement is always zero. If sand is scattered on the membrane, it tends to collect along the nodal lines, rendering them visible. The sand does not appear to be attracted to the nodal lines, but it is the only place where they can rest undisturbed. Oddly, very light powders, such as lycopodium spores, teng to heap up at points of greatest motion, due to air currents driving them there. We find solutions to new problems by assuming the nodal lines to be boundaries at which the membrane is clamped, for example the modes of a triangular membrane. Another shape we may expect to be able to solve is a circular membrane, clamped at r = a. The usual method of procedure in engineering books is to 'express the Laplacian in polar coordinates' by a formal mathematical procedure. It is much easier and clearer to proceed directly, by deriving the wave equation in polar coordinates from first principles. We now take z = z(r,θ), and an element of area dA = rdrdθ. The Figure shows how to proceed, exactly as with rectangular coordinates. We simply find the change in the vertical component of the tension when we cross the element from side to side. Products of small quantities are neglected. Again, we assume that z = R(r)Θ(θ) and proceed by separation of variables. The θ equation is Θ" + n^{2}Θ = 0. The corresponding solutions are A cos (nθ/2π) + B sin (nθ/2π), with n integral, so the solution will be singlevalued as we go round and round the axis. Then the r equation is R" + R'/r + (k^{2}  n^{2}/r^{2})R = 0. This is Bessel's equation, and its solutions that are finite at r = 0 are called J_{n}(kr). Hence our solutions are z = C sin (nθ/2π)J_{n}(kr), and the boundary conditions demand that J_{n}(ka) = 0. Therefore, k can have only those values for which this equation is satisfied, and that determines the frequency of vibration. The roots of J_{0}(kr) are kr = 0.7655π, 1.7571π, 2.7546π, ..., roughly (m  1/4)π. The roots of J_{1}(kr) are kr = 1.2197π, 2.2330π, 3.2383π, roughly (m + 1/4)π. More values of roots are given in mathematical tables. The fundamental of the circular membrane has f = (0.7655/2a)(T/σ)^{1/2}. This frequency is 1.53 times greater than the fundamental of a string of length equal to the diameter of the membrane. Again, the first overtone is not harmonic with the fundamental, the frequency ratio being 1.59, four semitones. The patterns of vibration for the lowest modes are shown in the Figure. Note the nodal circles and lines, and the + and  that show relative phases on two sides of a node. The modes are numbered according to the number of radial and angular nodal lines. There are two possibilities for the 11 and 21 modes, corresponding to the choice of the sine or cosine for the angle variation. These are said to be degenerate modes, since they have exactly the same frequencies. Any lack of isotropy in the membrane will separate their frequencies, or lift the degeneracy. This is very much like what happens in quantum states, where frequency corresponds to energy. The free vibrations of a membrane were first successfully studied by Poisson, 1829. The theory was complete in Clebsch's Theory of Elasticity, 1862. Membranes are more affected by the surrounding air than are strings. A kettledrum produces unequal reactions on the two sides. Such a drum is usually struck halfway along a radius, not at the centre, so vibrations with a nodal diameter are favoured. Rayleigh remarks that the four lowest modes with a nodal diameter are harmonically consonant, which may be an explanation for this. It is usually difficult to achieve uniform tension in an actual membrane. Forced vibrations of a membrane resonate with the natural modes as the frequency of forcing slowly changes, and nodal lines also move, coinciding with those for the natural modes at resonance. There are no nodal lines below the fundamental frequency, but they begin to appear as soon as this boundary is crossed. In a vibrating plate, the restoring force is provided by the elasticity of the plate instead of by an applied tension. It is easier to get isotropy, but the analysis is much more difficult. The potential energy of a strained plate is given by V = A(1/R_{1}^{2} + 1/R_{2}^{2} + 2B/R_{1}R_{2}), where A and B are elastic constants and R_{2} and R_{2} are the principal radii of curvature. For a homogeneous plate of isotropic material, A = Yh^{3}/3(1  μ^{2}) and B = μ, where h is the thickness of the plate, Y Young's modulus, and μ the shear modulus. The Figure shows how to express the strain energy in terms of the derivatives of the displacement. Note how the principal curvatures are expressed in terms of the derivatives of z, especially that the sum of the principal curvatures is the Laplacian of z. The matter is obviously quite complicated, so we will not proceed with finding the equation of motion here. It is all worked out in Rayleigh. Lamb presents only the highlights, as we shall do here. A rectangular plate cannot vibrate like a bar, with nodal lines parallel to a pair of opposite edges, since the other edges will curl up with an opposite curvature. The ratio of the principal curvatures will be Poisson's ratio. It is also difficult to set up the boundary conditions at a free edge, since the strain distribution is peculiar. The frequency of a normal mode is given by an expression like ω^{2} = Yh^{2}m^{4}/3ρ, where h is the thickness, ρ the volume density, Y Young's modulus, and m is an inverse length given by a transcendental equation. The frequency then varies directly as the thickness, and inversely as the square of the lateral dimensions, for similar plates. For a disc, the nodal circle in the lowest circularly symmetrical mode has a radius of 0.678a, where a is the radius of the plate, and (ma)^{2} = 8.8897, assuming Poisson's ratio to be 1/4. Kirchhoff showed that the fundamental mode has two nodal diameters and no nodal circle, and a frequency of 0.517(h/a^{2})(Y/ρ)^{1/2} if Poisson's ratio is 1/3. The nodal lines are shown in the Figure. A circular plate clamped at the edges is a problem with applications. Lamb gives a way to find an approximation for the natural frequency based on Rayleigh's method. The shape of the mode is assumed, and then the kinetic and potential energies are found in terms of the normal coordinate q and its time derivative dq/dt. The ratio of the coefficient of q^{2}/2 in V to the coefficient of (dq/dt)^{2}/2 in T is then an upper limit to the square of the angular frequency. If there is an adjustable parameter, the ratio is minimized with respect to this parameter. Here, we simply have to make a good guess, since there is no adjustable parameter. Nevertheless, the error in the frequency is only a bit over one percent. The process is shown in the Figure. The principal radii of curvature are used to get the potential energy. 1/R_{1} is the second derivative of z with respect to r. 1/R_{2} is the distance along the normal of the curve to the axis of symmetry. These expressions are used in the earlier expression for V in terms of them, which is integrated over the disc. T is, of course, easily obtained from dz/dt. Such a disc may be used to transmit or receive underwater vibrations. A 7" disc 1/8" thick had a resonant frequency of 1013 Hz in air, but only 550 Hz in water, due to the inertia of the water on one side. The coupling to the water is also strong, leading to rapid dissipation of energy by radiation. When plates are curved into shells, additional complication arises because extensional and flexural strains are no longer distinct. When the thickness is imagined to approach zero, some frequencies tend to finite limits, and correspond to extensional vibrations. Others tend to zero, and correspond to flexural vibrations. The flexural modes are of lower frequency, and are the main ones of acoustic interest. The nodal lines in an axially symmetric shell are parallels and meridians, but at a flexural node there is maximum tangential motion. Two methods of excitation can be recognized. One is typified by running a wet finger around the rim of a fine glass to make it sing, the other by striking the shell transversely, as in ringing a bell. The ordinary bell is a very complicated case, which was never studied until Rayleigh considered it in the 1890s. The shape of the bell apparently helps the excitation of certain modes, and resists that of others. In particular, the fundamental, or hum tone, is suppressed. The pitch of a bell in many cases was found to be judged as one octave below the fifth partial tone. Bells made in different locations to different designs may behave differently. ReferencesH. Lamb, The Dynamical Theory of Sound, 2nd ed. (London: Edward Arnold, 1925), Chapter V.Lord Rayleigh, The Theory of Sound, 2nd. ed. (London: Macmillan, 1926). Vol. 1, Chapter IX. F. E. Relton, Applied Bessel Functions (London: Blackie & Son, 1946). This happens to be the reference I have on hand. There are many others that give the properties of Bessel functions useful in applications. There is no reason for the student to be frightened of Bessel functions. Composed by J. B. Calvert

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