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?

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

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