Question

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 1.3 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.

Answer 1

a) Let a₁ = amplitude of one wave

a₂ = amplitude of second wave

R = amplitude of resultant wave

= phase difference between the two waves

Amplitude of the resultant wave is given by

R = ( a₁² + a₂²_{​​​​​​} + 2a₁​​​​​​a₂cos )^1/2

As the amplitude of the two waves are equal

Let, a₁ = a₂ = a

R = ( a² + a² + 2a²​​​​​​cos )

1.3a = ( a² + a² + 2a²​​​​​​cos )^1/2

1.69a² = 2a² + 2a²​​​​​​cos

-0.31a² = 2a²​​​​​​cos

cos = -0.31 / 2

cos = -0.155

= 98.91°

b) As = 98.91°

= 98.91 × / 180

= 1.726 rad

c) Let = wavelength of the wave

2 rad has path difference

1.726 rad has path difference ( /2 )×1.726

Path difference = ( / 2 ) × 1.726

Path difference = 0.2745×

Path difference = 0.275×

Path difference = (275/1000)

Path difference = (11/40)