Two sinusoidal waves in a string are defined by the wave functions where x , y 1

Question

Two sinusoidal waves in a string are defined by the wave functions

where x, y1, and y2 are in centimeters and t is in seconds.

(a) What is the phase difference between these two waves at the point x = 5.00 cm at t = 2.00 s? (Your answer should be between 0° and 360°.)
___________°

(b) What is the positive x value closest to the origin for which the two phases differ by ± at t = 2.00 s? (At that location, the two waves add to zero.)
___________cm

Explanation / Answer

(a) at x = 5 cm and t = 2s

y1 = 1.50 sin(16x5 - 34x2) = 1.50 sin(12 rad)

y2 = 1.50 sin(30x5 - 44x2) = 1.50 sin(62 rad)

phase differene = 62 - 12 = 50 rad

in deg = 50 x 180 / pi = 2864.79 deg

in between 0 to 360 deg

= 2864.79 mod 360 = 344.8 deg .........Ans

(B) y1 + y2 = 1.50 sin(16x - 68) + 1.50 cos(30x - 88)

= 3 sin(16x - 68 + 30x - 88 / 2) cos(16x - 68 - 30x + 88 / 2)

= 3 sin(23x -78) cos(-7x + 10) = 0

23x - 78 = n pi

n can be any integer.

x = 0.113 cm

Or -7x + 10 = (2n + 1) pi / 2

7x = 10 - (2n + 1)pi/2

x = 0.307 cm

Ans: 0.113 cm

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