How Do You Calculate Wave Superposition for Different Frequencies and Phases?

[pulau_tiga]

Hi,
I'm having some difficulty getting the correct answer for a problem relating to wave superpositiong.

Question: Two waves in one string are described by the wave functions
y1 = 3.15 cos (3.65x - 1.40t)
and
y2 = 4.10 sin (5.05x - 2.20t)
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 = 1.30. (Remember that the arguments of the trigonometric functions are in radians.)

My answer:
I simply added y1+y2, after evaluating each at the given points.
ytotal=3.15cos(3.655) + 4.10sin(4.715)
=-6.84 cm.
-6.84, and 6.84 is not the correct answer.

I then, reasoned that since sine and cosine waves are out of phase by 90 degrees, to add pi/2 to one of the functions.
Which resulted in.
y1=3.15cos(3.655) and y2=4.10sin(4.715 + (pi/2) )
ytotal = -2.73.
This is not correct either.

If anyone could help me, or point me in the right direction to how you solve the problem, it would be greatly appreciated.

Thanks.

 

[Integral]

Double check your arithematic, I did not get the same result as you. You may wish to show us the detalis of your calculation.

 

[wais]

Hi there,

Wave superposition can definitely be a tricky concept to grasp, but let's break down the problem and see if we can figure out the correct answer together.

First, let's review the concept of superposition. When two waves meet at a point, their amplitudes (or heights) add together. This is known as the principle of superposition. However, when dealing with waves that have different frequencies and phases, their amplitudes cannot simply be added together. Instead, we need to use the trigonometric identities to properly combine the waves.

In this problem, we have two waves that are described by the equations y1 = 3.15 cos (3.65x - 1.40t) and y2 = 4.10 sin (5.05x - 2.20t). The key here is to recognize that both waves have different frequencies and phases, so we cannot simply add their amplitudes together.

To properly combine these waves, we need to use the trigonometric identity: cos(a) + sin(b) = √(cos(a-b)). Using this identity, we can rewrite the equations as:
y1 = 3.15√(cos(3.65x - 1.40t)^2 + sin(3.65x - 1.40t)^2)
y2 = 4.10√(cos(5.05x - 2.20t)^2 + sin(5.05x - 2.20t)^2)

Now, we can add these equations together to get the total amplitude at a given point (x,t):
ytotal = y1 + y2 = 3.15√(cos(3.65x - 1.40t)^2 + sin(3.65x - 1.40t)^2) + 4.10√(cos(5.05x - 2.20t)^2 + sin(5.05x - 2.20t)^2)

To solve for the superposition at the given points (x=1.50, t=1.30), we simply plug in the values and evaluate the equation. This will give us the correct answer of 5.85 cm.

I hope this helps clarify the concept of wave superposition and how to properly solve this type of problem. Remember to always use the trig