# Derivation about the wave interference

• christang_1023
In summary, the conversation discusses the integration of a single wave and the average energy change that can be observed. The calculation involves triangular identity and approximation, leading to the result that only the average energy change can be observed. The conversation then moves on to a more complex case involving two waves, and the integration leads to a result that does not indicate the sign of bright and dark patterns. The question is raised about whether this is due to an incorrect approximation or integration, and whether there are any circumstances where interference can be detected.

#### christang_1023

Starting from the simple case, there is a single wave ##e=a\cos(2\pi ft+\frac{2\pi}{\lambda}x+\phi_0)##, and integrate in such a way, where ##T_{eye}## stands for the response time of human eyes' response time towards energy change:
$$I=\int_{0}^{T_{eye}}e^2dt$$
The calculation includes triangular identity and approximation(i.e. there is a part ##\frac{\sin(a)-sin(b)}{4\pi f}\approx 0##, due to ##f>>2##). The result of the integration is ##I\approx \frac{1}{2}a^2T_{eye}##, showing that we can only observe the average energy change of a wave.

Then, I consider a more complex case in which there are two waves, ##e_1=a_1\cos(2\pi ft+\frac{2\pi}{\lambda}x+\phi_0),e_2=a_2\cos(2\pi ft)##.
$$I=\int_{0}^{T_{eye}}(e_1+e_2)^2dt$$,
After expanding it, ##I=\frac{1}{2}(a_1^2+a_2^2)T_{eye}+\int_{0}^{T_{eye}}2(e_1e_2)dt##. Concerning the integration, I use the same approximation mentioned above, such that ##\int_{0}^{T_{eye}}2(e_1e_2)dt\approx 0##.

Finally, I get ##I=\frac{1}{2}(a_1^2+a_2^2)T_{eye}##, which doesn't indicate the sign of bright and dark pattern.

Is there anything wrong with my approximation or integration?

You seem to have a wave traveling in the x direction and a wave that isn't travelling. Is this a circumstance you would expect to lead to visible interference? Or, indeed, one you would expect to exist at all?

Under what circumstances would you expect detectable interference?

Ibix said:
You seem to have a wave traveling in the x direction and a wave that isn't travelling.
Yes. this is a strange model.
The value of the second 'oscillation' is independent of position so can it be a wave?

## 1. What is the basic concept of wave interference?

The basic concept of wave interference is the interaction between two or more waves that results in a new wave pattern. This occurs when waves of the same or different frequencies meet in the same space. The resulting wave can have a higher or lower amplitude depending on the type of interference.

## 2. What is the difference between constructive and destructive interference?

Constructive interference occurs when two waves with the same frequency meet and their amplitudes add up, resulting in a higher amplitude wave. Destructive interference, on the other hand, occurs when two waves with the same frequency meet and their amplitudes cancel each other out, resulting in a lower amplitude or even a flat wave.

## 3. How is the phase difference between two waves related to their interference pattern?

The phase difference between two waves is the difference in their starting points or the difference in their peaks and troughs. It determines whether the two waves will interfere constructively or destructively. When the phase difference is a multiple of 2π, the two waves will interfere constructively, and when it is an odd multiple of π, they will interfere destructively.

## 4. Can interference occur with waves of different frequencies?

Yes, interference can occur with waves of different frequencies. In this case, the resulting wave will have a complex pattern that is a combination of the individual frequencies and their respective interference patterns. This phenomenon is known as beats.

## 5. How is the concept of wave interference applied in real-life situations?

Wave interference has many practical applications, such as in noise-cancelling headphones, where destructive interference is used to cancel out unwanted sound waves. It is also used in engineering and architecture to reduce the impact of earthquakes on buildings by creating destructive interference between seismic waves. Interference also plays a crucial role in the fields of optics, acoustics, and communication systems.