Exam2soln - 1. Consider the following partial differential...

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1. Consider the following partial differential equation (called the “diffusion equation”): 2 2 (,) f xt D tx ∂∂ = a. Assuming t is time and x is distance, what are the units of D ? (5 pts) b. Show that this one partial differential equation of two variables can be written as two ordinary differential equations of one variable each: (10 pts) 2 2 () dux ux dx λ =− and dv t Dvt dt c. What are the units of ? (5 pts) d. Solve these ordinary differential equations for the MOST GENERAL solutions u(x) and v(t) . (10 pts) e. If 1 ( , 0) sin( ) n n n A x L π = == , then what are the coefficients A n ? [HINT: 0 2 sin( )sin( ) L nm nm xx d x LL L ππ δ = ] (10 pts)
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2. If a point source emits “monochromatic” light (of only one wavelength) isotropically incident on a Fabry-Perot etalon of fixed thickness, draw the pattern on the screen. (5 pts) 3. If a string held fixed at x=0 and x=L is driven with mechanical force at angular frequency ω=3 s π /L (s is wave speed) and location x 0 =L/3 , sketch a bar graph representing the magnitude of coefficients for the response of each of the normal modes. (15 pts)
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Exam2soln - 1. Consider the following partial differential...

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