Two waves on one string are described by the wave functions y 1 = 2.5 cos(4.5 x

Question

Two waves on one string are described by the wave functions

y1 = 2.5 cos(4.5x ? 1.0t)
y2 = 4.5 sin(4.5x ? 1.0t)

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.)

Two waves on one string are described by the wave functions y1 = 2.5 cos(4.5x ? 1.0t) y2 = 4.5 sin(4.5x ? 1.0t) 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 (b) x = 1.00, t = 0.500 cm (c) x = 0.500, t = 0 cm

Explanation / Answer

This is easy, just plug in the numbers and then add then up:
(a) x=1.00, t=1.00

y1(1, 1) = 2.5 * cos(4.5*1 - 1.0*1) = 2.5 * cos(3.5)
y2(1, 1) = 4.5 * sin(4.5*1 - 1.0*1) = 4.5 * sin(3.5)

Evaluate and then just add these two values, this is the answer, superposition just means that you arithmetically add the two waves.

y1 ~ ?2.34114171
y2 ~ -1.57852452

So just add (to do the superposition):

y1 + y2 ~ -3.91966623

(b) x=1.00, t=0.5

y1(1, 1) = 2.5 * cos(4.5*1 - 1.0*0.5) = 2.5 * cos(4.0)
y2(1, 1) = 4.5 * sin(4.5*1 - 1.0*0.5) = 4.5 * sin(4.0)

y1 ~ 2.49391012
y2 ~ 0.31390413

y1 + y2 ~ 2.80781425

(C) x=0.5, t=0

y1(1, 1) = 2.5 * cos(4.5*0.5 - 1.0*0) = 2.5 * cos(2.25)
y2(1, 1) = 4.5 * sin(4.5*0.5 - 1.0*0) = 4.5 * sin(2.25)

y1 ~ 2.49807259
y2 ~ 0.17666917

y1 + y2 ~ 2.67474176

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