What phase difference between two otherwise identical traveling waves, moving in the same direction along a stretched string, will result in the combined wave having an amplitude 1.5 times that of the common amplitude of the two combining waves? Express your answer in (a) degrees, (b) radians, and (c) as a fraction of the wavelength.

a) 82.8°

b)1.44 rad

c)0.23λ

Explanation:

Wave function form for each wave will be

-> wave 1:

(x,t)=sin( kx -  ωt)

-> wave 2:

(x,t)=sin( kx -  ωt +φ)

The summation of above two functions is the resultant wave.

Y(x,t)=sin( kx -  ωt)+sin( kx -  ωt +φ)

Using the trigonometric addition formula for sin i.e

sin(A)+ sin(B) = 2 sin(A+B/2) cos(A-B/2)

Y(x,t)=2cos(φ/2) sin(kx - ωt +φ)

When comparing to the general wave form, the amplitude is 2cos(φ/2)m

Also, cos(φ/2)= 1.5

φ/2=(1.5/s)

(a) φ= 82.8°

(b)φ= 82.8π/180 =>1.44 rad

(c) one complete wave is 2π radians

Therefore, for two waves,  φ/2π= 1.44/2π => 0.23 fraction of a complete wave from each other i.e they are separated by 0.23λ

as we know our atmosphere or air have different levels. every level has other gasses, and every gas has another permeability. when the light comes from lower permeability to higher permeability, it changes its direction. so that means that when the light comes from the moon to the earth, it changes it's direction many more times. it basically means that the shape of the light part of the moon is changed on the earth.

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