Question

A wave propagates along a string and is reflected at the free end of the string. If we set the free end of the string as x=0, the wave can be described by y=0.2sin(1.5*pi*x-pi*t); here y is in unit of meters, and t is in unit of seconds.

(a) What is the resultant wave equation when the reflected wave combines with the incoming wave?

(b) What would be the resultant wave equation if the end of string (x=0) is not free but fixed?

Answer 1

a) if the wave is reflected at the free end, then the reflected wave will be in phase with the incoming wave.

let y1 be the incoming wave equation and

let y2 be the wave equation of the reflected wave.

The resultant wave equation be Y

=> y = y1 + y2 = 0.2sin(1.5*pi*x-pi*t) + 0.2sin(1.5*pi*x-pi*t)

=> y = 2*y1 = 0.4sin(1.5*pi*x-pi*t)

B) if the wave is reflected at the fixed end then then reflected wave will 180 degrees out of phase with the incoming wave.

so y2 = 0.2*sin(1.5*pi*x - x*t + pi)

y2 = - 0.2 * sin(1.5*pi*x - x*t)

y = y1 + y2 = 0.2*sin(1.5*pi*x - x*t + pi) - 0.2*sin(1.5*pi*x - x*t + pi)

y = 0.

so the resultant wave is zero.