The equation of a wave travelling on a string is y = sin[20πx + 10πt], where x and t are distance and time in SI units. The minimum distance between two points having the same oscillating speed is:
❓ Question The equation of a wave travelling on a string is y = sin ( 20 π x + 10 π t ) , where x x and t t are distance and time in SI units. The minimum distance between two distinct points on the string having the same oscillating speed (transverse velocity) is: (find it). 🖼️ Question Image ✍️ Short Solution We interpret “oscillating speed” as the instantaneous transverse velocity of a point of the string, i.e. ∂ y ∂ t \dfrac{\partial y}{\partial t} . Given y ( x , t ) = sin ( 20 π x + 10 π t ) . Compute transverse velocity: v ( x , t ) = ∂ y ∂ t = 10 π cos ( 20 π x + 10 π t ) . Two points x 1 x_1 and x 2 x_2 have the same transverse velocity at the same instant t t t when cos ( 20 π x 1 + 10 π t ) = cos ( 20 π x 2 + 10 π t ) . Using the cosine identity cos A = cos B \cos A=\cos B , we have two families of solutions: (i) 20 π x 1 + 10 π t = 20 π x 2 + 10 π t + 2 π n 20\pi x_1 + 10\pi t = 20\pi x_2 + 10\pi t + 2\pi n ⇒ 20 π (...