Two waves on one string are described by the wave functions y1 = 2.0 cos(4.5x -

Question

Two waves on one string are described by the wave functions
y1 = 2.0 cos(4.5x - 1.9t)
y2 = 5.0 sin(4.5x - 3.0t)
where x and y are in centimeters and t is in seconds. Find the superposition of the waves y1 + y2 at the following points. (Remember that the arguments of the trigonometric functions are in radians.)
(a) x = 1.00, t = 1.00
cm

(b) x = 1.00, t = 0.500
cm

(c) x = 0.500, t = 0
cm

Explanation / Answer

Two waves in one string are described by the wave functions: y(1)=3.0cos(4.0x-1.6t) and y(2)=4.0sin(5.0x-2.0t) where x and y are in centimeters and t is in seconds. Find the superposition of the waves y(1)+y(2) at the points: x=1.00 and t=1.00 x=1.00 and t=.500 x=.500 and t=0 ANS y(1)=3.0cos(4.0x-1.6t) and y(2)=4.0sin(5.0x-2.0t) y1 = 3 cos 4[x-0.4t] = 3 cos (4p) = 3 sin (4p+pi/2] y2 = 4 sin 5[x-0.4t] = 4 sin (5p) This is superposition of 2 sinusoidal waves in string with different amplitudes (3 , 4), different group velocities {v in k(x -vt)} and having a constant phase difference of (pi/2)> At superposition these 2 wavelets or disturbances in the string interfere constructively or destructively depending upon the path difference. the resultant displacement y = y1+y2 {which can be solved like interference method) y = 3 cos (4p) + 4 sin (5p) angles in radian -------------------- x=1.00 and t=1.00 p = x - 0.4 t = 0.6 y = 3 cos (2.4) + 4 sin (3) y = 3 *(- 0.7374) + 4 *0.1411 y = - 2.2122 + 0.5644 = - 1.6478 (minima) ========== you try -------------------- x=0.5 and t= 0 p = x - 0.4 t = 0.5 y = 3 cos (2.0) + 4 sin (2.5) y = 3 *(- 0.4161) + 4 *0.5984 y = - 1.2483 + 2.3935 = 1.1452 (maxima).. hope it helps

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