" The tympanic membrane - or eardrum as it is more commonly known - is a sensitive membrane separating the outer ear from the middle ear (shown in Figure 1). The membrane vibrates in response to incoming sound waves and transduces this motion to movement of the ossicles in the middle ear. Large pressure changes, such as from loud music, can damage or even rupture the tympanic membrane leading to loss of hearing. Typically, tympanic membrane damage will occur if the maximum displacement exceeds 1.5 mm.

The membrane dynamics can be modelled using the wave equation (see Equation 1). For simplicity this can be performed in 1D without significant loss of accuracy:

(∂^2/∂t^2)(u(x,t)) = c^2( ∂^2/∂x^2)(u(x,t))

where u(x,t) is the displacement of the membrane at location x, at time t. The length of the average eardrum is L = 9 mm and can be treated as fixed (i.e. zero displacement) at the end points. It has a density, ρ = 1.1g/cm, and wave speed of c = 1540 m/s. As part of your analysis, you are to model the tympanic membrane numerically. This will help you to consider the impact of pressure forcing and natural damping on the membrane response."

Section 1: Preparatory Work

For the initial part of your report you will perform a simple numerical analysis of the membrane and derive an analytical solution against which you can validate your numerical solution. In this section of your report you must:

- State clearly the Boundary Conditions you will apply based on the brief. Assuming an Initial Condition where the displacement of the eardrum resembles the 5th sinusoidal mode, state this Initial Condition mathematically. (5%)
- To prove to the company that the numerical algorithm you are planning to use will accurately capture the dynamics of the tympanic membrane (otherwise your advice could cause people serious harm!), you must present a validation case. A validation case is a benchmark, whereby a known solution is used to test the performance and accuracy of a numerical method. T
- Solve the wave equation analytically (showing all critical working and steps) using the param- eters provided in the brief. Remember to implement the Boundary and Initial Conditions, taking the amplitude of the mode to be A = 0.5 mm. Be sure to state clearly the eigenvalue, eigenfunction, and final solution. (15%)
- Choose a numerical scheme to solve Equation 1. State the chosen scheme clearly in mathe- matical form and identify what this scheme is called. Additionally, state the order of accuracy in space and time for your chosen stencils. You must provide a justification for using this scheme. (5%)
- From your previous work you know that there is a stability criteria that puts an upper limit on the time step you can use for this stencil. State clearly this criteria and use it to determine the maximum stable time step size for a grid spacing ∆x/L = 2^−3. (5%)