A wall made of steel developed two corroded layers: the external layer of thickness d2 is heavily damaged and the internal layer of thickness d1 is relatively less damaged as can be seen in Fig. 2. The total thickness of the wall, L, is known but the thicknesses of the damaged layers are unknown. In order to determine these thicknesses without disrupting the wall, ultrasonic measurements are conducted by placing a transmitter of acoustic pulses at the point A and the a receiver at the point B as shown in Fig. 2. The system times measure the durations tp and ts needed for the p (pressure) – wave and s (sear) – wave to traverse the wall. The aim of this question is to construct a linear model (a set of linear equations) top determine the unknown thicknesses. You can proceed according to the following steps: 1. Model formulation. Suppose the thicknesses, d1 and d2 of the damaged layers are known. Write the equations to determine the times t p and t s needed for p and s waves to cross the wall. Assume that the total wall thickness, L, is known as well as the p – and s wave velocities: c0 p , c1 p , c2 p , c0 s , c1 s , c2 s of each layer and the undamaged material, see Fig. 2. 2. Test example. Suppose you know the thicknesses of the layers: d1 =2 mm and d2=1 mm. Determine the times t p and t s needed for p and s waves to traverse the wall. Assume L=10 mm, c0 p = 5 km /sec c1 p = 4.45721093750000 km /sec c2 p = 4.68033845625000 km /sec c0 s = 3.125 km /sec c1 s =1.61766128656250 km /sec c2 s = 2.05351933062500 km /sec 5 3. Model calibration. Suppose you know the durations t p and t s . Determine the thicknesses from the model built in step 1. Check if they coincide with the values for the thicknesses assumed in step 2. 4. Model analysis. Suppose that the measurements of the times are conducted with small errors: the value measured for t p is 0.1% lower than the one obtained in step 2, while t s is 0.2% higher than the one obtained in step 2. Determine the layer thicknesses with these slightly incorrect values for the times and find the error in the thickness determination. Use two direct methods and two iterative methods of your choice. For the iterative methods, use a tolerance (relative error) of 10-6 . How many iterations are required to achieve the specified tolerance. Compare your results to the thicknesses obtained in step 2. Comment on the agreement or discrepancy that you obtain when you use direct and iterative methods. Check the condition number and comment. 5. Improvement of the method. Suppose you conduct 10 measurements with random errors, independently and uniformly distributed within the interval (-0.2%, 0.2%) and then average the result. Has the situation improved? How many measurements you need to achieve a 10% accuracy?