Question

Two waves on one string are described by the wave functions

y₁ = 2.5 cos(4.5x − 1.3t)
y₂ = 4.5 sin(5.5x − 1.5t)

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

Answer 1

(a) x= 1 , t = 1
y₁ = 2.5 cos(4.5x − 1.3t)
On putting the above values
y₁ = 2.5Cos3.2 = -2.4957
y₂ = 4.5 sin(5.5x − 1.5t)
y₂ = 4.5Sin(4) = -3.406
Now
y₁ + y₂ = -2.4957 +(-3.406) = -5.9013
(b) x = 1 , t = 0.5
y₁ = 2.5 cos(4.5x − 1.3t)
On putting the above values
y₁ = 2.5Cos3.85 = -1.8985
y₂ = 4.5 sin(5.5x − 1.5t)
y₂ = 4.5Sin(4.75) = -4.497
Now
y₁ + y₂ = -1.8985 +(-4.497) = -6.3955
(c) x = 0.5 , t = 0
y₁ = 2.5 cos(4.5x − 1.3t)
On putting the above values
y₁ = 2.5Cos2.25 = -1.5704
y₂ = 4.5 sin(5.5x − 1.5t)
y₂ = 4.5Sin(2.75) = 1.7175
Now
y₁ + y₂ = 1.7175 +(-1.5704) = 0.1471