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MA50281:  Mathematical Modelling for Industry

Problem Sheet 1

Deadline: 19/10/2022

1.  2D  Wave Equation:

Consider a flat 2D membrane with a density per unit area of ρ pulled taught with tension T per unit length.  Let u(x,y,t) be the displacement of the membrane in the vertical direction at position (x,y) and time t. Now consider a small element A of the membrane:

(a) Derive the force balance equation

\ ρ  dx = \T  dS.

A                            ∂A

(b) Deduce the equation of motion (wave equation)

∂2u

∂t2

where   c2  = T    and   ∆ =  ∂2   +  ∂2 

2. Perform a dimensionality check on equation (1), making sure that that the dimensions of the parameters T, ρ and c are clearly specified.

3. Python Questions:  Complete PS1 Python .ipynb.