The simplest two-dimensional continuous system is the flexible membrane which can be considered as the two-dimensional counterpart of the flexible string. As for strings, the assumption of flexibility implies that restoring forces in membranes arise from the in-plane tensile (or stretching) forces and that there is no resistance to bending and shear. In essence, it is the two-dimensional characteristics that distinguish this case from the case of the string.
In this regard, we note that now the tension at a point must be specified in terms of the pull across an elementary length of line drawn through the point:
this tension must equal the force tending to split the membrane along this line.
This pull will be proportional to the length ds of the line element and the proportionality factor T is a force per unit length (in units of N/m) which may or may not be perpendicular to the line element. In other words, in the general case, T is a vector which is a function of the position and orientation of the line element. The common simplifying assumption is to consider membranes for which this tensile stress is the same at every point on the membrane and for every orientation of the line element: in this circumstance T is a constant and represents also the outward pull across each unit length of the membrane’s boundary, i.e. its perimeter. With this in mind, we can picture a stretched membrane under the action of the in-plane stress T and consider a rectangular element of area dxdy (Figs 8.5(a), (b)).
Fig. 8.5 Membrane area element: (a) plane view; (b) side view.
If we call the coordinate used to measure the deflection of the membrane in the z direction it is easy to obtain from Fig 8.5(b) and from a similar figure in the y–z plane
(8.107a)
where s is the membrane mass per unit area and the angles θx and θy are given by ∂w/∂x and ∂w/∂y respectively. Note that small deflections have been assumed, so that etc. and the area of the deflected element can still be written as dxdy. Equation (8.107a) results in the equation of motion
(8.107b)
Obviously, we can arrive at eq (8.107b) by using Hamilton’s principle and by considering that the Lagrangian density in this case is given by
(8.108)
We will not do it here, but we can follow an analogous line of reasoning as in Chapter 3 to arrive at the Euler-Lagrange equation for the membrane which reads (e.g. Morse and Ingard [1]; Meirovitch [9])
(8.109)
where we write for simplicity etc.
Now, since the Laplacian operator ∇2 in rectangular coordinates is expressed by
we can write eq (8.107b) as
(8.110)
where has the dimension of a velocity. Equation (8.110) is the two-dimensional wave equation and has the advantage of being written in a form valid for all types of coordinates; the shape of the boundary (unless it is irregular) suggests which type of coordinates to use.
In order to obtain a solution of eq (8.110) we can once more adopt the method of separation of variables; we look for a solution in the form w=ug where u is a function of the space variables alone and g is a function of time alone. We substitute this solution into eq (8.110), divide both members by ug and arrive at
where the left side is a function of the space variables only and the right side is a function of time only and therefore both members must be equal to a constant, which we call –k2. The time equation becomes
(8.111) where and the space equation is the so-called Helmholtz equation
(8.112) which assumes different explicit forms depending on the type of coordinates we decide to use. Note that is the familiar wavenumber. Equation (8.111) is well known and its solution is represented by a sinusoidal function of time; hence, we turn to eq (8.112).
As an example we will now consider the calculation of the natural frequencies and eigenfunctions of a circular membrane of radius R clamped at its outer edge. The geometry of the problem suggests the use of polar coordinates (r, θ) so that the Laplacian is written
and eq (8.112) reads explicitly
(8.113)
If now we assume a solution of the form eq (8.113) leads to the two equations
(8.114)
where we call γ2 the separation constant. Now, by noting that the solution of the first equation is harmonic in θ, the continuity of membrane displacement requires that This periodicity condition can only be met if γ is an integer; hence the second of eq (8.114) becomes
(8.115) which is Bessel’s equation of order n having the solution
(8.116) where Jn and Yn are the Bessel functions of the first and second kind, respectively. The functions Yn approach infinity as so, for a complete (i.e. without a hole in the centre) membrane, we must eliminate them and write If the membrane is clamped at its outer edge r=R, we must satisfy the boundary condition f(R)=0, this means
(8.117) which is the frequency equation. The zeros of Bessel’s function can be found in table form in various texts (e.g. Abramowitz and Stegun [10]) and some of the zeros are as follows:
It should be noted that x=0 is also a root for all Bessel’s functions of order but this leads to trivial solutions for w and is therefore excluded.
The natural frequencies of our system can then be written as
(8.118) which means that for each value of n there exist a whole sequence of solutions labelled with the index m. For example, the frequencies are the solutions of the frequencies are the solutions of and so on; ω01 is the fundamental frequency and is given by
(8.119)
The overtones (which are not harmonics, i.e. are not integer multiples of the fundamental frequency) are, in increasing order
The mode shapes are obtained by the product of the two spatial solutions:
for the frequency ω0m (m=1, 2, 3,…) we have the eigenfunction
(8.120) where A0m is a constant and we note that u is a function of r alone. For each value of m, the corresponding mode has (m–1) nodal circles. A schematic representation for the first few modes can be given as in Fig 8.6.
For each one of the frequencies we have the
two eigenfunctions (Anm and are arbitrary constants)
(8.121)
which have the same shape and differ from one another only by an angular rotation of 90°. This is an example of degeneracy (two or more eigenfunctions belonging to the same eigenvalue) which occurs frequently in two- and three- dimensional systems.
Fig. 8.6 First few modes for n=0.
Fig. 8.7 First few modes for n>0.
A schematic representation of a few of the functions (8.121) is shown in Fig 8.7 where we note that the (n, m)th mode has n nodal diameters and (m–1) circular nodes.
A final word for the curious reader. From the foregoing discussions it may seem that—provided that the mathematics is manageable—we can tackle any boundary value problem by the method of separation of variables which, in its own right, is a widely adopted method for finding solutions of boundary value problems for partial differential equations. However, strictly speaking, this is not so. In fact, although it is not our case, it is worth knowing that the same equation may allow a separation of variables in one system of coordinates but not in another; for example, the Helmholtz equation separates into ordinary differential equations in eleven different orthogonal coordinate systems [11], which, in fact, is sufficient to solve a large number of problems of physical significance.