Question

At the x = 0 end of an semi-infinite rope, someone moves the end of the rope up and down sinusoidally as y[x = 0, t] = A Cos[ω t + π/4]: The speed of propagation down the string is given by c. (a) Write down a general formula for the resulting wave that propagates down the string. (b) What power is supplied by the person at the end of the rope? (c) At what frequencies ω must the person move the string so that at x = L the string moves with transverse harmonic motion given by y(x = L, t) = A Sin[ω t - π/4] ?

Answer 1

a) After t seconds, the wave will reach only up to the distance x = c*t After this length the string is stationary.

So, the resulting wave on the string is

b) The energy associated with one wavelength of the string is given by

Where A is the amplitude of the wave

is the angular frequency

and is the mass per unit length of the wire.

Where c is the wave propagation velocity.

c)

At x=L,

expanding the cosine and sine sums,

Since ,

(using x=ct)