Answer to Question #162242 in Optics for Gill Allain

Question

Assume 2 identical sinusoidal waves are moving through the same medium in the same direction. Under what condition will the amplitude of the resultant wave be greater than either of the 2 original waves?

A. In all cases

B. Only if the waves have no difference in phase

C. Only if the phase difference is less than 90 degrees

D. Only if the phase difference is less than 120 degrees

Answer 1

Let the amplitude of two sinosoidal waves be, $A_1=A_2=a$

Case 1: When There is no phase differencei.e. $\phi=0^{\circ}$

Then amplitude of resulting wave

$A=\sqrt{a^2+a^2+2(a)(a)(cos0)}$

$=\sqrt{4a^2}=2a$

Amplitude of resulting wave is higher.

Case 2: When There is no phase differencei.e. $\phi=90^{\circ}$

Then amplitude of resulting wave

$A=\sqrt{a^2+a^2+2(a)(a)(cos90)}$

$=\sqrt{2a^2}=\sqrt2a$

Amplitude of resulting wave is higher.

Case 3: When There is no phase differencei.e. $\phi=120^{\circ}$

Then amplitude of resulting wave

$A=\sqrt{a^2+a^2+2(a)(a)(cos120)}$

$=\sqrt{2a^2-a^2}=\sqrt{a^2}=a$

Amplitude of resulting wave is equal to amplitude of either of them.

Hence In the case of phase difference $0^{\circ},90^{\circ}$ The amplitude of resulting wave is higher.