Question

Four waves are to be sent along the same string, in the same direction: y1(x, t) = (5.40 mm) sin(2πx - 410πt + 0.22π) y2(x, t) = (5.40 mm) sin(2πx - 410πt + 0.86π) y3(x, t) = (5.40 mm) sin(2πx - 410πt + 1.22π) y4(x, t) = (5.40 mm) sin(2πx - 410πt + 1.86π). What is the amplitude of the resultant wave?

Answer 1

All the four waves have same amplitude. Only difference is in their phase.

So, we can draw the phasor diagram and can add all the four as a vector. The resultant vector will give the phase angle for the resultant vector.

Resultant amplitude can be calculated as

                              

if the amplitudes are same

                               

where A₂ & A₁ are the amplitudes of the two waves and is phase difference between the two waves.

Now adding y₁ & y₃, we get

                             

adding y₂ and y₄, we get

                             

As both the resultants are zero. so,the final resultant amplitude will also be zero.