# Interference - two waves of different frequencies - beat velocity

• I
• LmdL
In summary, the conversation discusses the "beats" phenomena and interference of two plane waves in a medium with a refractive index n(ω). The interference pattern along the z-axis is described by the sum of the initial fields contributions and a modulation speed is calculated. It is noted that this speed does not necessarily have to be limited by the speed of light, as it is not a physical movement but rather a feature in a pattern. The concept is further illustrated with an example of a rotating laser on a sphere. The conversation then discusses the behavior of a particle placed in the overlapping area of the waves, confined to the modulation wave maxima, and whether it would move if the modulation speed exceeds the speed of light. It is concluded that
LmdL
Hello,
I'm considering the "beats" phenomena. I have two plane waves in some medium with a refractive index n(ω), one propagates in a z direction and second in a direction making an angle θ with z axis. Waves have frequencies ω1, ω2 (not necessarily equal) and k-vectors k1, k2 (not necessarily equal):

Now, I'm interested in interference (in the area where both waves overlap) along the z axis direction. I can write the initial fields contributions along the z axis as

So their sum is

First cosine is the envelope modulation and second is the carrier wave. Now, my question is regarding the modulation speed. As far as I understand, the above describes the beats phenomena. Or, alternatively, an interference pattern with a period of 2π/Δk that moves in time. If I'm right, the modulation speed reads

And here is the question: this speed can exceed c (speed of light), once I change the θ. For some θ I can even get an infinite vm (when denominator vanishes). On the other hand, modulation speed should be limited by the speed of light. What I'm missing here?
The possibilities that I thought of:
1. This modulation speed is a sinusoidal and cannot transfer any information, thus isn't limited by c.
2. The expression for vm is correct, but for values of θ that lead to vm>c it should be constrained vm=c .
3. The expression for vm is an expression for phase velocity and I should use d(Δω)/d(Δk) instead (group velocity). However, in this case I also get velocities that exceed c (checked numerically).

Any advice or thoughts are appreciated.
Thank you.

The modulation speed need not be limited by c because it is not the speed of a particle or wave. It is the speed of a feature in a pattern, which is an abstract, non-physical concept, and hence not limited by the speed of light.
As another example of how this can happen, consider a powerful, perfect laser mounted perpendicularly on an axle in the middle of a sphere with radius one light second and pointing directly at the sphere's inner surface. The laser rotates on the axle at a rate of one full rotation per 3 seconds, so that the dot of light it makes on the sphere's inner surface traverses a full equatorial circle of the sphere in that time - distance covered = 2 pi light seconds - more than twice the speed of light. Yet the laser itself is only moving quite slowly. This can happen because the only thing moving on the sphere's inner surface is a feature in a pattern - not a physical thing limited by c.

LmdL
Thank you for your reply. I got your point. I have an additional question: let's suppose I place a particle on the z axis (in the waves overlapping area) and restrict it to move along z axis, for example by adding two transparent parallel to z axis plates, one slightly above it and one slightly below. Particle is then confined to one of the modulation wave maxima (similar to optical tweezers). What will happen now? I suppose that if vm is below c, the particle will move with that speed along the z axis, being in the area of the modulation maximum it was confined to from the beginning. What if vm>c? Particle will "slip" to the next maximum to the left, and then again, and again, and so forth? Is this correct?

Why would the particle move at all? What force are you expecting to act on the particle to accelerate it out of a stationary state?

Never mind. I need to review my understading of optical tweezers. Thank you for your help!

## 1. What is interference between two waves of different frequencies?

Interference is a phenomenon that occurs when two or more waves meet and interact with each other. In the case of two waves of different frequencies, they will interfere with each other, resulting in a new wave with a frequency that is equal to the difference between the two original frequencies.

## 2. How does interference between two waves of different frequencies affect their velocity?

The velocity of a wave is determined by its wavelength and frequency. When two waves of different frequencies interfere, their wavelengths also change, resulting in a new velocity for the resulting wave. This velocity can be calculated using the formula v = λf, where v is the velocity, λ is the wavelength, and f is the frequency.

## 3. What is the beat frequency in interference between two waves of different frequencies?

The beat frequency is the difference between the frequencies of the two interfering waves. It is the frequency at which the amplitude of the resulting wave oscillates between maximum and minimum values. This beat frequency can be calculated by subtracting the lower frequency from the higher frequency.

## 4. How does the phase difference between two waves of different frequencies affect their interference?

The phase difference between two waves of different frequencies determines the type of interference that occurs. If the waves are in phase (their crests and troughs align), they will interfere constructively, resulting in a wave with a higher amplitude. If the waves are out of phase (their crests and troughs do not align), they will interfere destructively, resulting in a wave with a lower amplitude.

## 5. Can interference between two waves of different frequencies produce silence?

Yes, interference between two waves of different frequencies can produce silence. This occurs when the two waves have the same amplitude but are 180 degrees out of phase. In this case, the waves will interfere destructively, resulting in complete cancellation and silence.

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