Two waves on one string are described by the wave functions
y1 | = | 2.5 cos(3.5x − 1.3t) |
y2 | = | 3.5 sin(4.5x − 2.5t), |
where x and y are in centimeters and t is in seconds. Find the values 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
(c) x = 0.500, t = 0
Given Two waves are
y1 = 2.5 cos(3.5x-1.3t)
y2 = 3.5 sin(4.5x-2.5t)
y1+y2 = 2.5 cos(3.5x-1.3t)+3.5 sin(4.5x-2.5t)
we know that from trignometric functions
Cos (A-B) = Cos A CosB + Sin A Sin B
Sin (A-B) = Sin A Cos B + Cos A Sin B
expanding these waves we get
y1+y2 = 2.5 (cos(3.5x)cos(1.3t)+ sin (3.5x) sin(1.3t))+3.5( sin(4.5x)cos(2.5t) - cos(4.5x)sin(2.5t))
substituting the values of x and t
a)
x = 1.00, t= 1.00
y1+y2 = 2.5 (cos(3.5*1)cos(1.3*1)+ sin (3.5*1) sin(1.3*1))+3.5( sin(4.5*1)cos(2.5*1) - cos(4.5*1)sin(2.5*1)) cm
y1+y2 = 1.7113 cm
b)
x = 1.00 , t = 0.500
y1+y2 = 2.5 (cos(3.5*1.00)cos(1.3*0.500)+ sin (3.5*1.00) sin(1.3*0.500))+3.5( sin(4.5*1.00)cos(2.5*0.500) - cos(4.5*1.00)sin(2.5*0.500))
y1+y2 = -2.7732 cm
c)
x = 0.500, t= 0
y1+y2 = 2.5 (cos(3.5*0.500)cos(1.3*0)+ sin (3.5*0.500) sin(1.3*0))+3.5( sin(4.5*0.500)cos(2.5*0) - cos(4.5*0.500)sin(2.5*0))
y1+y2 = 2.2776 cm
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