====== Superposition ======
Interference is simply a consequence of the principle of //superposition//. This means that at any point in space, the total amplitude of the disturbance due to two separate traveling waves is the sum, or superposition, of the amplitudes of the two separate waves.
Mathematically, we can write two sinusoidal traveling waves of equal amplitude, $y_1$ and $y_2$, at point $x = 0$ as
| $y_1 = A\cos (\omega t + \varphi_1)$\\ $y_2 = A\cos (\omega t + \varphi_2)$, | (1) |
where $\varphi_i$ may be a function of position, but not of time. The superposition of these waves yields the total wave $y_T$,
| $y_T = y_1 + y_2 = 2A\cos\left(\dfrac{\varphi_2-\varphi_1}{2}\right)\cos \left(\omega t + \dfrac{\varphi_2 + \varphi_1}{2}\right)$. | (2) |
This gives a total intensity of
| $I_t = \left< y_T^2\right> = 2A^2\cos^2\left(\dfrac{\varphi_2-\varphi_1}{2}\right) = 2A^2\cos^2\left(\dfrac{\Delta\varphi}{2}\right)$. | (3) |
Note that the total intensity depends critically on the phase difference $\Delta\varphi$ between the two waves. If $\Delta\varphi$ is zero or an integral multiple of $2\pi$, then the waves are in phase and $I_T$ is twice as large as the sum $I_1+I_2$ (constructive interference). If $\Delta\varphi =(2n+1)\pi$, where $n = 0, 1, 2, \dots$, then the waves are 180 degrees out of phase and $I_T=0$ (destructive interference).
A sinusoidal traveling wave, propagating in the $x$-direction and produced at $x_0$ with phase $\varphi_0$, has the form
| $Y = A \cos\left(\omega t - k(x-x_0)+\varphi_0 \right) = A \cos \left(2\pi\left(ft - \dfrac{x-x_0}{\lambda}\right) + \varphi_0\right)$, | (4) |
where
| $\omega = 2\pi f$\\ $k = \dfrac{2\pi}{\lambda}$\\ $v = f\lambda = \dfrac{\omega}{k}$ | (5) |
so that the phase at any point $x$ is $\varphi = \varphi_0 - k(x-x_0)$ If two such waves interfere, $\Delta\varphi$ at the point of interference will depend on the distance traveled by each wave.
For example, suppose two such waves with identical $\omega$ and $\varphi_0$ are produced on the $x$-axis at $x_1$ and $x_2$. Then, at a distant point $x$, the phase difference will be
| $\Delta \varphi = k(x_2-x_1) = 2\pi\left(\dfrac{x_2-x_1}{\lambda}\right)$ | (6) |
and the intensity of the total wave at $x$ will be
| $I_T = 2A^2\cos^2\left(\pi\left(\dfrac{x_2-x_1}{\lambda}\right)\right)$. | (7) |
This analysis applies equally to transverse waves (microwaves and light) and to longitudinal waves (sound). Note that the two waves must have identical frequency and a fixed phase difference at their point of production.
====== Interference of waves from two parallel sources ======
One of the simpler ways to observe interference is to consider the strength of the sound wave produced by two parallel audio speakers operated at the same intensity and frequency. An experimental arrangement is shown in Fig. 1.
Since the two identical speakers are driven by the same audio oscillator, the frequency of the waves produced by each will be identical. The phase relationship between the two waves will be fixed as long as the position between the two speakers is held fixed. The microphone will then detect the sum of the two audio signals produced by the two speakers. We will consider the following three cases shown in Fig. 1:
* //Case 1,// the speakers side by side;
* //Case 2,// the two speakers separated by one-half wavelength; and
* //Case 3,// the two speakers separated by one wavelength.
In each case, the waves from each of the speakers is drawn, and the resultant wave is shown below them.
| {{ phylabs:lab_courses:phys-120_130-wiki-home:spring-experiments:image2019-7-10_11-55-4.png?direct&400 |}}\\ //Case 1//: Waves in phase |
| {{ phylabs:lab_courses:phys-120_130-wiki-home:spring-experiments:image2019-7-10_11-55-10.png?direct&400 |}}\\ //Case 2//: Waves out of phase by 180 degrees |
| {{ phylabs:lab_courses:phys-120_130-wiki-home:spring-experiments:image2019-7-10_11-55-14.png?direct&400 |}}\\ //Case 3//: Waves out of phase by 360 degrees |
| **Figure 1**: Addition of waves from two identical sources |
It is apparent from Fig. 1 and from Eq. (7), that the signal at the microphone is a maximum (the waves reinforce) when the path lengths traversed by the two component waves differ by some multiple of the wavelength (including zero), i.e. when
| $x_2-x_1 = n\lambda\; (\mathrm{for\;} n = \mathrm{\;0,1,2,\dots})$. | (8) |
A minimum of the signal occurs when the path lengths differ by an odd multiple of the half-wavelength, corresponding to a phase difference between the two waves of 180 degrees.
===== Beats =====
We have considered interference from two sources, i.e., two loudspeakers driven at the same frequency by the same audio oscillator. Figure 1 shows how the waves from the two sources add (interfere) as the relative phase of the two sources changes. It is also possible to drive the two loudspeakers at slightly different frequencies. In this case, the phase //difference// would change continuously. The resulting sum of the two waves would cycle from //Case 1// (maximum amplitude) to //Case 2// (minimum amplitude) to //Case 3// (maximum amplitude), etc. This slow variation in the amplitude of the sum of the two frequencies is called //beating//. The frequency of the beat is given by the difference between the two frequencies which drive the loudspeakers. The production of beats is shown in Fig. 4.
| {{ phylabs:lab_courses:phys-120_130-wiki-home:spring-experiments:image2019-7-10_12-1-21.png?direct&400 |}} |
| **Figure 4**: Beats formed by the addition of waves with two different frequencies |
You will use two audio oscillators, producing sine waves, and a pair of headphones. The left headphone is to be driven by one oscillator and the right by the other.
====== Interference with a Michelson interferometer ======
A very well-known device for observing interference between light waves is the Michelson Interferometer (shown in Fig. 2).
| {{ phylabs:lab_courses:phys-120_130-wiki-home:spring-experiments:image2019-7-10_11-57-27.png?direct&400 |}} |
| **Figure 2**: Michelson interferometer (rays offset for clarity) |
Its operation is as follows: A wave coming from the source impinges on the half-reflecting mirror. Half is reflected to mirror M-1, while the remaining half continues to mirror M-2. These two mirrors in turn reflect the wave back to the half-reflecting mirror. (The rays are shown displaced so you can trace them). The wave reflected from mirror M-2 is now reflected by the half-reflecting mirror, while that from M-1 passes through the half-reflecting mirror, also to the detector. Thus, the original beam was split into two equal parts, traveled different paths, and then recombined. The condition for interference is clearly once again the same, i.e., one gets constructive interference whenever the path differences are integer multiples of the wavelength. The path difference is changed by displacing a mirror along the light path. (Note that when a mirror is displaced the path length is changed by twice the displacement of the mirror.)
The optical Michelson interferometer used in this experiment allows one to detect changes in path length on the order of 0.0001 cm, and is thus a very sensitive instrument.