# Two identical traveling waves, moving in the same direction, are out of phase by 5.0 rad. What is...

## Question:

Two identical traveling waves, moving in the same direction, are out of phase by 5.0 rad. What is the amplitude of the resultant wave in terms of the common amplitude ym of the two combining waves? (Give the answer as the ratio of the total amplitude to the common amplitude.)

Waves have both amplitude and phase. So mathematically they may be represented by phasors. A phasor is a rotating arrow. Its length represents the amplitude of the wave, its frequency of rotation matches the angular frequency of the wave and the angle it makes with the x-axis is the phase. Phasor addition may be done just like vector addition using the parallelogram law. If two phasors have amplitudes or lengths {eq}\displaystyle {a} {/eq} and {eq}\displaystyle {b} {/eq} and if their relative phase angle is {eq}\displaystyle {\theta} {/eq} then the resultant phasor has an amplitude {eq}\displaystyle {\sqrt{a^2+b^2+2ab \cos \theta}} {/eq}.

Here it is given that two identical waves travelling in the same direction superpose with a phase difference of {eq}\displaystyle {\theta=5} {/eq} radians. It is given that the amplitude of each wave is {eq}\displaystyle {y_m} {/eq}. Hereafter we replace {eq}\displaystyle {y_m} {/eq} with {eq}\displaystyle {a} {/eq}. That is, let the amplitude of each wave be {eq}\displaystyle {a} {/eq}.Then using the parallelogram law for phasor addition the amplitude of the resultant wave is,

{eq}\displaystyle { \sqrt{a^2+b^2+2ab \cos \theta}=\sqrt{a^2+a^2+2aa \cos5}=\sqrt{2a^2(1+0.284)}=1.6a} {/eq}.

Hence the ratio of the amplitude of the resultant to the common amplitude {eq}\displaystyle {a} {/eq} is {eq}\displaystyle {\color{blue}{1.6}} {/eq}.