Two identical traveling waves, moving in the same direction, are out of phase by /5.0 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.)
To find the amplitude of the resultant wave, we need to consider the interference between the two waves.
When two identical waves with the same amplitude combine, they can either interfere constructively or destructively, depending on their phase difference.
In this case, the waves are out of phase by 5/0 rad. Since the phase difference is not a multiple of π, the waves will interfere destructively.
When waves interfere destructively, the resulting amplitude is the difference between the amplitudes of the two waves.
Let's denote the common amplitude of the two combining waves as ym. Since the waves are identical, we can consider one wave to have an amplitude of ym, and the other wave to have an amplitude of -ym (because they interfere out of phase).
Thus, the resultant amplitude (AR) can be calculated as:
AR = ym - (-ym) = ym + ym = 2ym
So, the amplitude of the resultant wave is 2 times the common amplitude of the two combining waves.
Therefore, the ratio of the total amplitude (2ym) to the common amplitude (ym) is:
Ratio = (2ym) / ym = 2
So, the amplitude of the resultant wave is twice the common amplitude.
When two waves combine, the resultant wave's amplitude can be calculated using the principle of superposition.
In this case, two identical traveling waves are moving in the same direction, but they are out of phase by π/5 rad (approximately 0.628 rad). Let's call the common amplitude of the two waves ym.
The resultant wave can be calculated using the following formula:
Amplitude of Resultant Wave = 2 * ym * cos(Δφ/2)
Where:
Δφ = Phase Difference between the two waves
Given that Δφ = π/5 rad, we can substitute this value into the above equation:
Amplitude of Resultant Wave = 2 * ym * cos(π/5 / 2)
Now, let's simplify the expression:
Amplitude of Resultant Wave = 2 * ym * cos(π/10)
The cosine of π/10 can be evaluated to be approximately 0.9659.
So, the Amplitude of Resultant Wave = 2 * ym * 0.9659
Finally, the ratio of the total amplitude to the common amplitude is:
Total Amplitude / Common Amplitude = 2 * 0.9659 = 1.9318 (approximately)
Therefore, the amplitude of the resultant wave is approximately 1.9318 times the common amplitude (ym) of the two combining waves.