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QUESTION 4, please
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QUESTION 4, please
QUESTION 4, please
3. (Adapted from Serway & Jewett) We stated that a displacement function y(x,t) is a wave if it is a solution to the linear wave equation: ²y 1 10²y = ax² v² at ² a) (2 pts) Show that the sinusoidal wave function y(x,t) = Asin (kx ± wt) is a solution to the linear wave equation, provided that v = w/k. b) (2 pts) Show that the wave function y(x,t) = x² + v²t² is a solution to the wave equation. c) (2 pts) Show that the function in part (b) can be written as f(x + vt) +g(x − vt) and determine the functions f and g. d) (4 pts) Repeat parts (b) and (c) for y(x,t) = sin(x)cos (vt) (Hint: sin (a + B) = sin acos ß ± cos asin ß) 4. (Extra Credit) Suppose functions f(x,t) and g(x,t) are solutions to the linear wave equation (see Problem 3). a) (2 pts) Show that m(x,t) = f(x,t) ± g(x,t) is also a solution to the linear wave equation. b) (2 pts) Show that m(x,t) = f(x,t)g(x,t) is a solution to the linear wave equation if afag dxdx lafag v² at ət c) (1 pt) Verify part b with the function of y(x,t) in part 3d.
3. (Adapted from Serway & Jewett) We stated that a displacement function y(x,t) is a wave if it is a solution to the linear wave equation: ²y 1 10²y = ax² v² at ² a) (2 pts) Show that the sinusoidal wave function y(x,t) = Asin (kx ± wt) is a solution to the linear wave equation, provided that v = w/k. b) (2 pts) Show that the wave function y(x,t) = x² + v²t² is a solution to the wave equation. c) (2 pts) Show that the function in part (b) can be written as f(x + vt) +g(x − vt) and determine the functions f and g. d) (4 pts) Repeat parts (b) and (c) for y(x,t) = sin(x)cos (vt) (Hint: sin (a + B) = sin acos ß ± cos asin ß) 4. (Extra Credit) Suppose functions f(x,t) and g(x,t) are solutions to the linear wave equation (see Problem 3). a) (2 pts) Show that m(x,t) = f(x,t) ± g(x,t) is also a solution to the linear wave equation. b) (2 pts) Show that m(x,t) = f(x,t)g(x,t) is a solution to the linear wave equation if afag dxdx lafag v² at ət c) (1 pt) Verify part b with the function of y(x,t) in part 3d.