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Waves of Arbitrary Shape. (a) Explain why any wave described by a function of the form y(x, t)=f(x-v t) moves in the x -direction with speed v . (b) Show that y(x, t)=f(x-v t) satisfies the wave equation, no matter what the functional form of f . To do this, write y(x, t)=f(u), where u=x-v t . Then, to take partial derivatives of y(x, t), use the chain rule: (∂ y(x, t))/(∂ t)=(d f(u))/(d u)(∂ u)/(∂ t)=(d f(u))/(d u)(-v) (∂ y(x, t))/(∂ x)=(d f(u))/(d u)(∂ u)/(∂ x)=(d f(u))/(d u) (c) A wav
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