Two identical traveling waves, moving in the same direction, are out of phase by PI/3.0 rad. What is the amplitude of the resultant wave in terms of the common amplitude ym of the two combining waves? (Give the answer as the ratio of the total amplitude to the common amplitude.)

Thanks in Advance

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To find the amplitude of the resultant wave in terms of the common amplitude ym, we can use the concept of superposition.

When two identical traveling waves are out of phase, the amplitude of the resultant wave can be found by adding the amplitudes of the individual waves together.

In this case, the two waves are out of phase by π/3 radians. This means that one wave is π/3 radians ahead of the other wave.

Let's assume the amplitude of each wave is ym.

Since the waves are identical, they have the same amplitude, direction, and speed. Therefore, the equation for each wave can be expressed as:

Wave 1: y1 = ym * sin(kx - ωt)
Wave 2: y2 = ym * sin(kx - ωt + π/3)

To find the amplitude of the resultant wave, we need to add these two waves together:

y = y1 + y2
y = ym * sin(kx - ωt) + ym * sin(kx - ωt + π/3)

Using the trigonometric identity sin(A + B) = sin(A)cos(B) + cos(A)sin(B), we can simplify the equation:

y = ym * (sin(kx - ωt) + sin(kx - ωt)cos(π/3) + cos(kx - ωt)sin(π/3))

Since we know that sin(π/3) = √(3)/2 and cos(π/3) = 1/2, we can further simplify the equation:

y = ym * (2sin(kx - ωt) + √(3)cos(kx - ωt))

Now, since we're interested in the amplitude of the resultant wave, we can use the amplitude-angle form of a sine wave:

y = A * sin(kx - ωt + φ)

Comparing this equation with our simplified equation, we can see that the amplitude of the resultant wave (A) is given by:

A = √(2^2 + (√3)^2) = √(4 + 3) = √7

So the amplitude of the resultant wave in terms of the common amplitude ym is √7:1.