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, we can use the principle of superposition. When two waves are out of phase, we can add their amplitudes to find the total amplitude of the resultant wave.

In this case, the two waves are identical, so they have the same amplitude, which we'll call ym. The phase difference between them is given as π/3 radians.

When two waves are out of phase by π/3 radians, we can use the cosine function to calculate the amplitude of the resultant wave:

Amplitude of Resultant Wave = 2 * ym * cos(π/6)

Since cos(π/6) = √3/2, we can substitute this value in:

Amplitude of Resultant Wave = 2 * ym * (√3/2)

Simplifying further:

Amplitude of Resultant Wave = √3 * ym

Therefore, the amplitude of the resultant wave in terms of the common amplitude ym is √3 times the common amplitude.

To find the amplitude of the resultant wave in terms of the common amplitude, we can use the principle of superposition. The superposition principle states that when two waves meet, the displacement at any point is the sum of the displacements of the individual waves.

In this case, we have two identical traveling waves that are out of phase by π/3 rad. Let's call these waves Wave A and Wave B.

Wave A can be represented by the equation:

A = ym * sin(kx - ωt)

Wave B, which is out of phase by π/3 rad from Wave A, can be represented by the equation:

B = ym * sin(kx - ωt + π/3)

Now, let's find the resultant wave by adding Wave A and Wave B:

Resultant wave = A + B
= ym * sin(kx - ωt) + ym * sin(kx - ωt + π/3)

To simplify this expression, we can use the trigonometric identity sin(A + B) = sin(A)cos(B) + cos(A)sin(B):

Resultant wave = ym * [sin(kx - ωt)cos(π/3) + cos(kx - ωt)sin(π/3)]
= ym * [sin(kx - ωt) * 1/2 + cos(kx - ωt) * √3/2]

We can further simplify this expression using the trigonometric identity sin(π/3) = √3/2 and cos(π/3) = 1/2:

Resultant wave = ym * [1/2 * sin(kx - ωt) + √3/2 * cos(kx - ωt)]

Finally, we can rewrite this expression in the form of the common amplitude, ym:

Resultant wave = ym * [1/2 * sin(kx - ωt)/ym + √3/2 * cos(kx - ωt)/ym]

Therefore, the amplitude of the resultant wave in terms of the common amplitude ym of the two combining waves is:
1/2 * ym + √3/2 * ym = (1/2 + √3/2) * ym

So, the ratio of the total amplitude to the common amplitude is:
(1/2 + √3/2) : 1

In simplified form, the amplitude of the resultant wave is:
(1 + √3) : 2