Question

(P3) [11 points] Two waves linear waves of equal amplitude and frequency have a phase difference ?, as they travel in the same medium. Their displacements can be represented by the equations: y1 = A sin(kx ? ?t) y2 = A sin(kx ? ?t + ?) (a) Use the trigonometric identity sin ?1 + sin ?2 = 2 sin 1/2 (?1 + ?2) cos 1/2 (?1 ? ?2) to show that the resultant wave is given by yR = 2A cos ?/2 sin(kx ? ?t + ?/2) (b) What is the amplitude of the resultant wave? Is the wave purely sinusoidal? (c) How does the amplitude vary with ?? (How can you get constructive or destructive interference?) (d) How does the mathematical result for (a) change if y2 has a + sign instead of a – sign between kx and ?t? Briefly describe what happens from a physical point of view.

          (P3) [11 points] Two waves linear waves of equal amplitude and frequency have a phase
difference ?, as they travel in the same medium. Their displacements can be represented by the
equations:
y1 = A sin(kx ? ?t)
y2 = A sin(kx ? ?t + ?)
(a) Use the trigonometric identity sin ?1 + sin ?2 = 2 sin 1/2 (?1 + ?2) cos 1/2 (?1 ? ?2)
to show that the resultant wave is given by
yR = 2A cos ?/2 sin(kx ? ?t + ?/2)
(b) What is the amplitude of the resultant wave? Is the wave purely sinusoidal?
(c) How does the amplitude vary with ?? (How can you get constructive or destructive
interference?)
(d) How does the mathematical result for (a) change if y2 has a + sign instead of a – sign between
kx and ?t? Briefly describe what happens from a physical point of view.

(P3) [11 points] Two waves linear waves of equal amplitude and frequency have a phase
difference ?, as they travel in the same medium. Their displacements can be represented by the
equations:
y1 = A sin(kx ? ?t)
y2 = A sin(kx ? ?t + ?)
(a) Use the trigonometric identity sin ?1 + sin ?2 = 2 sin 1/2 (?1 + ?2) cos 1/2 (?1 ? ?2)
to show that the resultant wave is given by
yR = 2A cos ?/2 sin(kx ? ?t + ?/2)
(b) What is the amplitude of the resultant wave? Is the wave purely sinusoidal?
(c) How does the amplitude vary with ?? (How can you get constructive or destructive
interference?)
(d) How does the mathematical result for (a) change if y2 has a + sign instead of a – sign between
kx and ?t? Briefly describe what happens from a physical point of view.

Added by Alejandro R.

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