Check 2 waave displacement equations please

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Discussion Overview

The discussion revolves around the superposition of two wave displacement equations, specifically examining their frequencies, wavenumbers, and the process of combining them mathematically. The scope includes theoretical aspects of wave mechanics and mathematical reasoning related to harmonic oscillations.

Discussion Character

  • Exploratory
  • Mathematical reasoning

Main Points Raised

  • One participant questions whether the two given equations represent waves of the same frequency and wavenumber, noting the lack of a spatial variable in the equations.
  • Another participant clarifies that the equations represent harmonic oscillations with a phase difference of π/4 and identifies the angular frequency as 2π, leading to a frequency of 1 Hz.
  • A participant inquires about the process of superposing two harmonic oscillators with different phases, specifically whether to convert sine to cosine using a phase shift.
  • Responses indicate that the functions should be added directly according to the superposition principle, and that a trigonometric identity is not necessary for combining them, although one participant cautions against using the identity due to differing amplitudes.
  • It is suggested that the resulting expression can be represented as a single sine or cosine function, with a method proposed for determining the resultant amplitude and phase.
  • Phasor algebra is mentioned as an alternative method for simplifying the process, though familiarity with it is assumed.

Areas of Agreement / Disagreement

Participants generally agree on the process of superposition but express differing views on the necessity and validity of using trigonometric identities in the context of differing amplitudes. The discussion remains unresolved regarding the best approach to combine the wave equations.

Contextual Notes

There are limitations regarding the assumptions about the nature of the equations as waves, as well as the dependence on definitions of frequency and wavenumber. The discussion also highlights unresolved mathematical steps in determining the resultant amplitude and phase.

Lengalicious
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Got to calculate the superposition of a couple o' waves, wanted to double check if they are both the same frequency and wavenumber?x1(t) = 3 sin(2∏t + ∏/4 ) and x2(t) = 3 cos(2∏t).

I take it 2∏ is the frequency? And since Kx is not inside the function how can I know whether the wave numbers the same? :S
 
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These are not waves. The equations have to spatial variable. They may represent two harmonic oscillations with a phase difference of pi/4.
2∏ is the angular frequency, ω=2∏f. The frequency is 1 Hz.
 
Ok thanks, when 2 harmonic oscillators superpose do you just take the sum of the two even though they have different phases?
EDIT: Since one is in terms of sin and the other in cos, do i make sin(2pit+pi/4) into cos(2pit+pi/4+pi/2) or is that not valid, because i need to use the trig sum-product identity
 
Last edited:
Yes, you add the functions. You don't need a trig sum-product identity. Use the phase identity sin(x) = cos(x-pi/2)
 
Lengalicious said:
Ok thanks, when 2 harmonic oscillators superpose do you just take the sum of the two even though they have different phases?
EDIT: Since one is in terms of sin and the other in cos, do i make sin(2pit+pi/4) into cos(2pit+pi/4+pi/2) or is that not valid, because i need to use the trig sum-product identity
Depends what you mean by the "take the sum".
At any time, the total displacement is the sum of the two displacements (superposition principle):
x(t)=x1(t)+x2(t)
If you wish you can try to simplify or otherwise put the expression into a different form that shows something interesting.
You cannot use the trig identity as the two have difefrent amplitudes.
For the given case, you can write the result as a single cosine or sine,
x(t)=Asin(2∏t+ψ).
To find the values of A and ψ you can expand the sin and and identify the terms with the original components. You'll find two equations in the variables A and ψ.
Phasor algebra is a more straightforward method, if you are familiar with it.
 

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