Question

Assumptions Modeling: Membrane, Two-Dimensional Wave Equation

Physical Assumptions 1. The mass of the membrane per unit area is constant (“homogeneous membrane”). The membrane is perfectly flexible and offers no resistance to bending.

2. The membrane is stretched and then fixed along its entire boundary in the xy-plane. The tension per unit length T caused by stretching the membrane is the same at all points and in all directions and does not change during the motion.

3. The deflection of the membrane during the motion is small compared to the size of the membrane, and all angles of inclination are small.

Question:

Which part of Assumption 2 cannot be
satisfied exactly? Why did we also assume that the
angles of inclination are small?

Answer 1

Assumptions 1,2 and 3 cannot be realized exactly. They hold accurately for small transverse vibrations of a thin elastic membrane, so that we shall obtain a good model.

(1). The tension per unit length T caused by stretching the membrane is the same at all points cannot be satisfied exactly during motion. Thus we need to assume Tension T does not change appreciably during motion.

(2). we did also assume that the angles of inclination are small in view of the fact that deflection of the membrane during the motion is small compared to the size of the membrane.