Two waves in one string are described by the wave functions
y1 = 2.90 cos (3.80x - 1.50t)
y2 = 3.90 sin (4.80x - 2.10t)
a) where y and x are in centimeters and t is in seconds. Calculate the superposition of the waves y1 + y2 at the points x = 1.50, t = 1.40. (Remember that the arguments of the trigonometric functions are in radians.)
This question requires you to enter the given values into the equation, and add y1 and y2 together.
y1 = 2.90 cos (3.80(1.50) - 1.50(1.40))
y2 = 3.90 sin (4.80(1.50) - 2.10(1.40))
y1 + y2 = -6.12 cm
To calculate the superposition of the waves y1 + y2 at the points x = 1.50 and t = 1.40, we need to substitute these values into the wave functions and then add the results together:
Let's calculate y1 at x = 1.50 and t = 1.40:
y1 = 2.90 cos(3.80x - 1.50t)
Substituting in x = 1.50 and t = 1.40:
y1 = 2.90 cos(3.80(1.50) - 1.50(1.40))
= 2.90 cos(5.70 - 2.10)
= 2.90 cos(3.60)
Using a scientific calculator, evaluate cos(3.60). Let's assume the result is 0.282.
Now let's calculate y2 at x = 1.50 and t = 1.40:
y2 = 3.90 sin(4.80x - 2.10t)
Substituting in x = 1.50 and t = 1.40:
y2 = 3.90 sin(4.80(1.50) - 2.10(1.40))
= 3.90 sin(7.20 - 2.94)
= 3.90 sin(4.26)
Using a scientific calculator, evaluate sin(4.26). Let's assume the result is 0.779.
Now we can calculate the superposition y1 + y2 at x = 1.50 and t = 1.40:
y1 + y2 = (2.90 cos(3.60)) + (3.90 sin(4.26))
= (2.90)(0.282) + (3.90)(0.779)
= 0.817 + 3.033
= 3.850
Therefore, the superposition of the waves y1 + y2 at x = 1.50 and t = 1.40 is 3.850 cm.
To calculate the superposition of the waves y1 + y2 at a given point (x, t), we simply add the values of the two waves at that point.
In this case, we have:
y1 = 2.90 cos(3.80x - 1.50t)
y2 = 3.90 sin(4.80x - 2.10t)
To calculate the value of y1 + y2 at x = 1.50 and t = 1.40, we substitute these values into the wave functions and add the results.
Substituting x = 1.50 and t = 1.40 into y1, we get:
y1 = 2.90 cos(3.80(1.50) - 1.50(1.40))
Calculating the argument of the cosine function:
3.80(1.50) - 1.50(1.40) = 5.70 - 2.10 = 3.60
Substituting this value into the cosine function:
y1 = 2.90 cos(3.60)
Calculating the value of y1 using a calculator or trigonometric table, we find:
y1 ≈ 2.90 * 0.2125 ≈ 0.6163
Next, we substitute x = 1.50 and t = 1.40 into y2:
y2 = 3.90 sin(4.80(1.50) - 2.10(1.40))
Calculating the argument of the sine function:
4.80(1.50) - 2.10(1.40) = 7.20 - 2.94 = 4.26
Substituting this value into the sine function:
y2 = 3.90 sin(4.26)
Calculating the value of y2 using a calculator or trigonometric table, we find:
y2 ≈ 3.90 * 0.7299 ≈ 2.8461
Finally, we add the values of y1 and y2:
y1 + y2 = 0.6163 + 2.8461 ≈ 3.4624
So, the superposition of the waves y1 + y2 at x = 1.50 and t = 1.40 is approximately 3.4624 centimeters.