Consider the following generic equation of motion for an arbitrary wave represented by y(x, t): 2² y any 2 = α² (9) Ət²

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Consider the following generic equation of motion for an arbitrary wave represented by y(x, t): 2² y any 2 = α² (9) Ət² Əxn Where is the nth derivative of y with respect to x. (The conventional wave equation we : 2.) მოყ dxn studied in class corresponds to n = (a) Let's assume we have a traveling single frequency wave of the form: y = e²(kx-wt) (10) Find the frequency, w, of this wave as a function of k for an arbitrary n. (b) Consider the specific case of n = 3. This is not a valid equation for a uniform density plane wave: explain why. What happens to this wave? (Hint: √i = .) 1+i √2 (c) Find the phase and group velocity of waves for the case n = 6, which does support waves. Assume a is positive, and express your answer in terms of a and k. (d) = For the conventional wave equation (n = 2), we found that an arbitrarily shaped wave which traveled at the correct speed was a valid solution. (In other words, y f(x - vt) was a solution for an arbitrary f, provided v = ta.) Can we do something similar for n = = 6, perhaps with a different velocity? Explain why or why not.