Consider a string fixed to a wall which has a linear mass density u1 in the first half of the string and a distinct linear mass density u2 in the latter half.

A wave propagating through the string decomposes into a reflected and transmitted wave after the incident wave reaches the interface of the two linear mass densities, say x=0. With x=0, the three waves are modeled by y_i=A*cos(-w*t), y_r=B*cos(w*t) and y_t=C*cos(-w*t). Why is 'w', the angular frequency, the same for all three waves?

Question

Consider a string fixed to a wall which has a linear mass density u1 in the first half of the string and a distinct linear mass density u2 in the latter half. 

A wave propagating through the string decomposes into a reflected and transmitted wave after the incident wave reaches the interface of the two linear mass densities, say x=0. With x=0, the three waves are modeled by y_i=A*cos(-w*t), y_r=B*cos(w*t) and y_t=C*cos(-w*t). Why is 'w', the angular frequency, the same for all three waves?

fifciol · Answer

The string is stretched so it has the same tension everywhere on the spring. There is no net force on every particle in the horizontal direction, so the particles move only up and down. The junction of those halfs of the string is fixed, that means if the wave from first half reaches second half the particles move at exactly that frequency, you cannot change it. What do changes is the velocity and therefore the amplitude. 
so $$\omega =v_1k_1=v_2k_2$$