Adding Two Wave Functions

In summary, the conversation is discussing how to calculate the amplitude of a sum wave given two waves with equal wavelength and amplitude but with a phase difference, using the equations y1(x) and y2(x) and the parameters A, λ, and ϕ.
  • #1
ColdFusion85
142
0
I am not looking for any answers, just some guidance.

Consider case (c), (case (c) involved two waves with equal wavelength and amplitude, but with some arbitrary phase difference), and write the two waves as

[tex]y1(x) = Acos((\frac{2*\pi*x}{\lambda}))[/tex]
[tex]y2(x) = Acos((\frac{2*\pi*x}{\lambda}) + \phi) [/tex]

where [tex]\lambda[/tex] and [tex]A[/tex] are the common wavelength and amplitude of the two waves and [tex]\phi[/tex] is their phase difference. Calculate the sum wave [tex]y(x) = y1(x) + y2(x)[/tex] and find an expression for the amplitude of the sum wave in terms of [tex]A[/tex] and [tex]\phi[/tex].

Find an expression for Amplitude in terms of the amplitude?? What exactly is this question asking?
 
Physics news on Phys.org
  • #2
The question is asking you to calculate the amplitude of the sum wave given by the expression y(x) = y1(x) + y2(x). To do this, you will need to use the two given equations for y1(x) and y2(x), as well as the parameters A, λ, and ϕ. After calculating the sum wave, you can then use the equation for the amplitude of a wave (A = √(P/ρ)) to find an expression for the amplitude in terms of A and ϕ.
 
  • #3


The question is asking for an expression that relates the amplitude of the resulting sum wave to the amplitude and phase difference of the two individual waves. This can be derived by using the trigonometric identity for the sum of two cosine functions. By substituting the given equations for y1(x) and y2(x) into the sum wave equation, we get:

y(x) = Acos((\frac{2*\pi*x}{\lambda})) + Acos((\frac{2*\pi*x}{\lambda}) + \phi)

Using the trigonometric identity, cos(a) + cos(b) = 2cos(\frac{a+b}{2})cos(\frac{a-b}{2}), we can simplify the expression for y(x) to:

y(x) = 2Acos((\frac{2*\pi*x}{\lambda} + \frac{\phi}{2}))cos(\frac{\phi}{2})

This shows that the amplitude of the resulting sum wave is 2Acos(\frac{\phi}{2}), which is dependent on the amplitude A and the phase difference \phi of the two individual waves. This expression can be further simplified to Amplitude = 2Acos(\frac{\phi}{2}) = 2Acos(\frac{\pi*\phi}{\pi}) = 2Acos(\frac{\pi*\phi}{2\pi}), which gives us a final expression for the amplitude of the sum wave in terms of A and \phi.
 

1. How do you add two wave functions?

To add two wave functions, you first need to understand the concept of superposition. This means that when two or more waves overlap, their amplitudes are added together at each point in space and time. To add two wave functions, simply add the amplitudes of each function at each point in space and time to obtain the new wave function.

2. Can two wave functions with different frequencies be added?

Yes, two wave functions with different frequencies can be added. The resulting wave function will have a combination of the frequencies of the two original wave functions. This is known as a beat phenomenon, where the amplitude of the resulting wave function will vary at a frequency equal to the difference between the two original frequencies.

3. What happens when two wave functions with the same frequency are added?

When two wave functions with the same frequency are added, the resulting wave function will have an amplitude that is the sum of the amplitudes of the two original wave functions. This is known as constructive interference, where the amplitudes of the two waves reinforce each other.

4. How do you calculate the phase difference between two wave functions?

The phase difference between two wave functions can be calculated by finding the difference in the phase angles of the two functions at a specific point in space and time. This can be done by comparing the equations of the two wave functions and finding the difference in the values of their phase constants.

5. Can the amplitude of the resulting wave function be greater than the amplitudes of the two original wave functions?

Yes, it is possible for the amplitude of the resulting wave function to be greater than the amplitudes of the two original wave functions. This is known as constructive interference and occurs when the two original wave functions are in phase, meaning their peaks and troughs line up and reinforce each other to create a larger amplitude in the resulting wave function.

Similar threads

  • Introductory Physics Homework Help
Replies
10
Views
808
  • Introductory Physics Homework Help
Replies
5
Views
1K
  • Introductory Physics Homework Help
Replies
7
Views
233
Replies
3
Views
1K
  • Introductory Physics Homework Help
Replies
4
Views
864
  • Introductory Physics Homework Help
Replies
17
Views
224
  • Introductory Physics Homework Help
Replies
8
Views
477
  • Introductory Physics Homework Help
Replies
7
Views
1K
  • Introductory Physics Homework Help
Replies
20
Views
2K
Replies
4
Views
719
Back
Top