Question

Two sinusoidal waves are identical except for their phase. When these two waves travel along the same string, for which phase difference will the amplitude of the resultant wave be a maximum?

Answer 1

Answer:

zero or 2π is maximum

Explanation:

Sine waves can be written

      x₁ = A sin (kx -wt + φ₁)

     x₂ = A sin (kx- wt + φ₂)

When the wave travels in the same direction

      Xt = x₁ + x₂

      Xt = A [sin (kx-wt + φ₁) + sin (kx-wt + φ₂)]

We are going to develop trigonometric functions, let's call

     a = kx + wt

     Xt = A [sin (a + φ₁) + sin (a + φ₂)

We develop breasts of double angles

     sin (a + φ₁) = sin a cos φ₁ + sin φ₁ cos a

    sin (a + φ₂) = sin a cos φ₂ + sin φ₂ cos a

Let's make the sum

     sin (a + φ₁) + sin (a + φ₂) = sin a (cos φ₁ + cos φ₂) + cos a (sin φ₁ + sinφ₂)

to have a maximum of the sine function, the cosine of fi must be maximum

     cos φ₁ + cos φ₂ = 1 +1 = 2

the possible values ​​of each phase are

     φ1 = 0, π, 2π  

     φ2 = 0, π, 2π,  

so that the phase difference of being zero or 2π is maximum