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