Mathematical Physics Vol 1

Chapter 7. Partial differential equations

340

2 ◦ The membrane is deformed so that its boundary remains fixed to the xy –plane. At such deformation, the stress in the membrane T is the same at all points and directions and it does not change over time. 3 ◦ The deflection of the membrane u ( x , y , t ) (displacement of the points of the membrane in the z – axis direction) is small relative to membrane dimensions. 4 ◦ The slopes of the membrane at each point are small. Imagine that we have cut a particle of the membrane ∆ P (see Fig. 7.2). This part remains in the same state as before cutting, if the influence of the removed part is replaced by appropriate forces.

Figure 7.2: Membrane.

The projection of these forces on the xz and yz planes, respectively, expressed in terms of the stress T , can be written as S xz = T · ∆ y , that is S yz = T · ∆ x . Projection of the equation of motion ( ∆ m a = ∑ i S i ) on the z direction is

∂ 2 u ∂ t 2

∆ m

= S xz ( sin β x − sin α x )+ S yz ( sin β y − sin α y )

(7.12)

where ∂ 2 u ∂ t 2 is the projection of acceleration on the z – axis, and ∆ m = ρ ∆ x ∆ y . As in this case, similarly to the case of the vibrating wire (according to assumption 4 ◦ ), we have sin β x ≈ tg β x = ∂ u ( x + ∆ x , y ) ∂ x , sin α x ≈ tg α x = ∂ u ( x , y ) ∂ x that is sin β y ≈ tg β y = ∂ u ( x , y + ∆ y ) ∂ y , sin α y ≈ tg α y = ∂ u ( x , y ) ∂ y ,