Massachusetts Institute of Technology

In the opening of the paper The Dynamics of Thin Sheets of Fluid II: Waves on Fluid Sheets, Taylor notes the analogy between small disturbances on the surface of deep water and waves on thin fluid sheets. Just as surface waves can be analyzed into their Fourier components to simplify the analysis, this same type of analysis can be applied to a sheet of uniform thickness. In the case of the thin sheet, eight parameters are required to completely specify the system rather than the four that are necessary in the case of surface waves. In both cases, the relationship between the velocity potential and the surface displacement can be selected such that the pressure condition is satisfied when the wave propagates at a certain speed. The two types of waves that are observed are classified as symmetrical (figure 1b) and antisymmetrical (figure 1a). The governing equations for both types of waves will be derived in what follows. The Mach angle will also be discussed, along with observations of these waves.

To first order in y, the velocity of the sheet in the x-direction can be expressed as . Using the fact that the fluid is incompressible and integrating with respect to y, one can write (to leading order) . Neglecting viscosity and neglecting nonlinear terms as small due to the small size of y, the Navier Stokes equations reveal the form of the pressure:

Using the fact that surface tension determines the pressure along with conservation of mass, the equations of motion are:

For a small perturbation h to the half-thickness, h, one sees a small induced velocity expressed by:

If one assumes h is of the form e^{ikx+w t}, then w ² is of the form: , which is the case for dispersive waves.

In this case, v_y>>v_x, so that we can write , and , where . Combining these equations gives. The pressure on either side of the sheet is given by the expressions p_top = - g h’’ and p_bottom = g h’’, so that the general expression is of the form . Combining this with the relation between pressure and the velocity potential, we see that . The acceleration of the displacement from equilibrium can then be expressed as , which is the equation for sound. This expression, then, can be used to check the effect of small obstacles in sound.

Any h of the form h(x – ut) satisfies the wave equation above. Plugging this into the wave equation, one arrives at the expression , where M is known as the Mach number. Rearranging the terms, we arrive at . This is essentially the wave equation, with any h of the form satisfying the equation. If a disturbance passes through the thin sheet, the effect is localized in a wedge (or a cone in the three-dimensional case) of angle a . As can be seen from the diagram in figure 1, , so that .

In a radial sheet, there is a constant flux given by the expression, so that . Taylor asserts at this point that u is constant, while t varies inversely with r. This assumption is verified by the results, and can be motivated by considering Bernoulli’s principle inside the sheet so that the fluid obeys the equation. If k is small, we see that u² is proportional to p, which is constant. This implies that Taylor’s assumption is a reasonable one. Combining the expression for flux with the value of sina , we see that

,

where is the scale governing transition from supersonic to subsonic behavior. It should be familiar to the reader from the paper on water bells.

Though Taylor uses a rather complicated geometric argument to arrive at the cardoid form of r/R, this effect can also be seen by a more simple argument. We first note that the waves obey the wave equation

.

If we define x as the ratio r/R, this expression becomes

.

Consider h of the form . If we consider localized disturbances (such that l _dist<<r) and assume geometric optics, such that (F ¢ )² >> F ¢ ¢ , this equation becomes

.

Þ

Þ Þ .

The characteristics of this equation yield the expression 2x – 1 = cos(q - q _o), which is the expression for cardoids at which Taylor arrived.

If one assumes that the fluid that flows into the free edges mixes so rapidly with the fluid already present there that the velocity within the edge is uniform at any given point and is given by the value qu, continuity requires that

where y is the angle of the velocity of a drop relative to the direction of the stream, s is the arclength, and m is the mass per unit length of the fluid. The rate of increase in momentum relative to the edge is given by , so that one can express the conservation of momentum parallel to the edge as

.

The conservation of the component of momentum perpendicular to the edge is given by

.

Defining the half-thickness of the sheet at r = R (where hR = h_or) as h_o, and integrating siny dt = r dq to arrive at mq = 2r h_oRq , the expression for conservation of momentum parallel to the edge becomes

,

while the fact that yields

for the conservation of momentum perpendicular to the edge. Combining these two expressions with the fact that and conditions at the obstacle (at r = r₁) are sufficient for determining the shape of the edge.

A flat impact disc with eight equally spaced radial nicks was used by Taylor to produce an easily recognizable wave pattern that could be compared to the predicted cardoids. The lighting was arranged so that a camera placed near the axis of symmetry would be able to see the waves, thus allowing the patterns to be compared with those predicted by theory. To compare the waves, the negative of a picture of the wave was projected onto the predicted curves and the magnification was adjusted so that the cardoids lay as nearly as possible to the waves.

Since both types of waves are present at any given time, it was necessary to find a method for observation of one type of wave in the absence of the other. The theory predicts that symmetrical and antisymmetrical waves will propagate at different speeds, so that antisymmetrical waves should be found at a larger angle to the flow than symmetrical waves. The amplitude of antisymmetrical waves may be many times greater than the thickness of the sheet (as they are analogous to waves propagated on a stretched string of diameter much smaller than the wave amplitude), so that rays of light are reflected through a much greater range of angles than transmitted light which would hardly experience any deflection by waves that leave the thickness of the sheet unchanged. Since symmetrical waves cause change the thickness of the sheet by as much as half the thickness, transmitted waves will be deflected through an angle which is greatest where the rate at which the thickness changes is greatest. These facts implied that symmetrical waves could be observed in photographs where antisymmetrical waves were invisible.

Assuming that the initial value of the fluid mass per unit length is zero at the position of the obstacle, the angle between the two parts of the sheet is defined by , as seen above. This theory was tested by photographing the edges that resulted when a wire obstacle was placed in the path of the expanding sheet at a number of different radii. The angle between the two edges was measured at the obstacle. Data showed a strong correspondence to theory, as can be seen in Taylor’s paper in a plot of as a function of r₁.

The theory discussed by Taylor in his paper describes the phenomena of waves on thin sheet fairly accurately, both for shapes of waves and for measurements of the angle a , created in the wake of a disturbance. The antisymmetrical waves were shown to be non-dispersive and well described by the cardoids predicted by theory, while the angle a agreed well with theoretical predictions for the diameter of obstacle used in a certain range of r₁. Despite the fact that this paper was published 41 years ago, research continues into phenomena related to thin sheets, as is demonstrated by a 1995 Physical Review Letter on the subject of the viscous bursting of suspended films. Taylor’s contributions in this paper represent a strong framework upon which a considerable amount of knowledge could be built both through his own research and the research of those who followed him.