[SOLVED] Regarding the Superposition of Two Plane Waves

  • #1
Slimy0233
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My professor was teaching me about the superposition of two waves and after this derivation, he marked ##2Acos(\frac{dk}{2}x -\frac{d\omega}{2}t)## as the oscillation part and ##sin (Kx-\omega t)## as the oscillation part, I don't understand why? Any answers regarding this would be considered helpful.

My main question would be, why did he choose sin part as the oscillation and why not the cos part and more importantly, why not both? I mean, my bad intuition tells me, that I should include both.
 
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  • #2
The answer is in the assumption that ##\mathrm{d}k \ll k## and ##\mathrm{d} \omega \ll \omega##. First, think about ##y## as function of ##x## at a fixed time. The cos-factor has a spatial period of ##4 \pi/\mathrm{d} k## and the sin-factor one of ##2 \pi/k\ll 4 \pi/\mathrm{d} k##. So the first cos factor is much slower varying than the sin factor as a function of ##x##. So you can interpret this as something oscillating in space with a wave length ##\lambda=2 \pi/k## and a position dependent amplitude, where the dependence of this amplitude on ##x## is much slower, i.e., the corresponding wave-length of these variations is ##\lambda'=4 \pi/\Delta k \gg \lambda##.

The analogous arguments hold also for the variations of the factors with time at a fixed position in space.
 
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  • #3
vanhees71 said:
The answer is in the assumption that ##\mathrm{d}k \ll k## and ##\mathrm{d} \omega \ll \omega##. First, think about ##y## as function of ##x## at a fixed time. The cos-factor has a spatial period of ##4 \pi/\mathrm{d} k## and the sin-factor one of ##2 \pi/k\ll 4 \pi/\mathrm{d} k##. So the first cos factor is much slower varying than the sin factor as a function of ##x##. So you can interpret this as something oscillating in space with a wave length ##\lambda=2 \pi/k## and a position dependent amplitude, where the dependence of this amplitude on ##x## is much slower, i.e., the corresponding wave-length of these variations is ##\lambda'=4 \pi/\Delta k \gg \lambda##.

The analogous arguments hold also for the variations of the factors with time at a fixed position in space.
beautiful analogy! thank you!
 
  • #4
[SOLVED]
 
  • #5
Slimy0233 said:
My main question would be, why did he choose sin part as the oscillation and why not the cos part and more importantly, why not both? I mean, my bad intuition tells me, that I should include both.
I am not very comfortable with the derivation. While I understand what your prof is trying to do, his/her methods seem unnecessarilly capricious. In particular exact derivations can be found for adding waves. Let's add two waves $$f(x,t)=cos(k_1x-\omega_1t)+cos(k_2x-\omega_2t) $$ then using trig identities $$ =2cos\left( \frac {(k_2-k_1)x-(\omega_2-\omega_1)t} 2 \right)cos\left( \frac {(k_2+k_1)x-(\omega_2+\omega_1)t} 2 \right)$$ $$=2f_1(x,t)f_2(x.t)$$
Typically the differences are smaller than the sums and f1 "modulates" f2. For your ear at x=0 this will give sound "beats" at the small difference frequency in the usual way.
 
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  • #6
hutchphd said:
I am not very comfortable with the derivation. While I understand what your prof is trying to do, his/her methods seem unnecessarilly capricious. In particular exact derivations can be found for adding waves. Let's add two waves $$f(x,t)=cos(k_1x-\omega_1t)+cos(k_2x-\omega_2t) $$ then using trig identities $$ =2cos\left( \frac {(k_2-k_1)x-(\omega_2-\omega_1)t} 2 \right)cos\left( \frac {(k_2+k_1)x-(\omega_2+\omega_1)t} 2 \right)$$ $$=2f_1(x,t)f_2(x.t)$$
Typically the differences are smaller than the sums and f1 "modulates" f2. For your ear at x=0 this will give sound "beats" at the small difference frequency in the usual way.
thank you for this sir!
 
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FAQ: [SOLVED] Regarding the Superposition of Two Plane Waves

What is the principle of superposition in the context of plane waves?

The principle of superposition states that when two or more plane waves overlap in space, the resultant wave at any point is the sum of the displacements of the individual waves at that point. This principle applies to all linear wave equations.

How do you mathematically represent the superposition of two plane waves?

Mathematically, the superposition of two plane waves can be represented as \( \Psi(x,t) = \Psi_1(x,t) + \Psi_2(x,t) \), where \( \Psi_1 \) and \( \Psi_2 \) are the individual wave functions of the two plane waves. Each wave function can be expressed in the form \( \Psi(x,t) = A \cos(kx - \omega t + \phi) \), where \( A \) is the amplitude, \( k \) is the wave number, \( \omega \) is the angular frequency, and \( \phi \) is the phase.

What are the conditions for constructive and destructive interference for two plane waves?

Constructive interference occurs when the two plane waves are in phase, meaning their phase difference \( \Delta \phi \) is an integer multiple of \( 2\pi \) (\( \Delta \phi = 2n\pi \), where \( n \) is an integer). Destructive interference occurs when the two waves are out of phase by an odd multiple of \( \pi \) (\( \Delta \phi = (2n+1)\pi \)). In constructive interference, the amplitudes add up, while in destructive interference, they cancel each other out.

What is the resulting wave pattern when two plane waves of the same frequency and amplitude interfere?

When two plane waves of the same frequency and amplitude interfere, the resulting wave pattern depends on their phase difference. If they are in phase, the result is a wave with the same frequency but with an amplitude that is the sum of the individual amplitudes (constructive interference). If they are out of phase by \( \pi \), the result is a wave with the same frequency but with an amplitude that is the difference of the individual amplitudes (destructive interference). For any other phase difference, the resulting wave will have an amplitude that varies between these two extremes.

How does the superposition of two plane waves relate to the formation of beats?

Beats are formed when two plane waves of slightly different frequencies interfere. The resulting wave is a modulated wave with an amplitude that varies at a frequency equal to the difference between the two original frequencies. This variation in amplitude, or "beating," is perceived as a periodic increase and decrease in sound intensity when the

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