7.10 Wave Interference

When two waves come together the result is the sum of both waves at every point. A special case of superposition is created when two waves of the same frequency (coherent waves) meet, called interference. When the crests of two coherent waves meet, they add together and we get a crest which is larger than either of the initial ones, which is called constructive interference. When a crest of one wave meets a trough of another, they cancel out and we get a much smaller amplitude, or even nothing if the initial amplitudes match. This is destructive interference. Let’s look at this mathematically. Consider two waves of the same amplitude and frequency travelling in the same direction in 1D. Lets look at the amplitude at a point $x_{i}$ which is a distance $x_{1}$ from the source of the first wave and a distance $x_{2}$ from the source of the second. The resulting amplitude is

y(x_{i},t)=A\cos(kx_{1}-\omega t+\phi_{0,1})+A\cos(kx_{2}-\omega t+\phi_{0,2}).

(7.153)

Now, we will make use of the addition formula

A\cos(\theta_{1})+A\cos(\theta_{2})=2A\cos\left(\frac{\theta_{1}+\theta_{2}}{2% }\right)\cos\left(\frac{\theta_{1}-\theta_{2}}{2}\right).

(7.154)

Then the amplitude at $x_{i}$ becomes

	$\displaystyle y(x_{i},t)$	$\displaystyle=2A\cos\left(\frac{k(x_{1}+x_{2})-2\omega t+\phi_{0,1}+\phi_{0,2}% }{2}\right)\cos\left(\frac{k(x_{1}-x_{2})+\phi_{0,1}-\phi_{0,2}}{2}\right)$		(7.155)
		$\displaystyle=2A\cos\left(\frac{\Delta\phi}{2}\right)\cos(k\bar{x}-\omega t+% \bar{\phi}_{0}),$		(7.156)

where we have defined the total phase difference $\Delta\phi=k(x_{2}-x_{1})+\phi_{0,2}-\phi_{0,1}$ , the average distance from $x_{i}$ to each source $\bar{x}=\frac{x_{1}+x_{2}}{2}$ , and the average initial phase $\bar{\phi}_{0}=\frac{\phi_{0,1}+\phi_{0,2}}{2}$ . The maximum amplitude of the superposition is $2A\cos\left(\frac{\Delta\phi}{2}\right)$ , so we see that this is maximised — i.e. we get destructive interference — for $\Delta\phi=2m\pi$ for integer $m$ . Likewise the maximum amplitude goes to zero and we get destructive interference when $\Delta\phi=(2m+1)\pi$ .

To extend this to more than one dimension, we must keep in mind that wavefronts are circular in 2D and spherical in 3D. If we are very far away from the source, the curvature of the waves becomes negligible and we get plane waves. This is known as the far-field limit.

7.10.1 Array of Point Sources

Consider an colinear array of $n$ wave sources all separated by a distance $d$ . Let’s put ourselves in the far-field limit, so that all the incoming wavefronts from the array will be parallel. The path difference (the extra distance one wave has travelled relative to another) is given by $h=d\sin\theta$ . Then the phase difference is just the path difference multiplied by the wavelength $\Delta\phi=kh=\frac{2\pi}{\lambda}d\sin\theta$ . If this phase difference is a multiple of $2\pi$ , we get constructive interference. This is given by the condition

d\sin\theta_{m}=m\lambda,

(7.157)

where $\theta_{m}$ labels the angles where we get maxima in amplitude. The value of $m=0,1,2,...$ is called the order of the maxima. Inbetween the maxima, we get secondary maxima with amplitude given by

s(\theta)=\varepsilon\frac{\sin\left(n\frac{\Delta\phi}{2}\right)}{\sin\left(% \frac{\Delta\phi}{2}\right)}=\varepsilon\frac{\sin\left(n\frac{kd\sin\theta}{2% }\right)}{\sin\left(\frac{kd\sin\theta}{2}\right)},

(7.158)

where $n$ is the number of sources and $\varepsilon$ is the amplitude of a single source. The intensity will be proportional to the square of the amplitude.

I(\theta)=I_{1}\frac{\sin^{2}\left(n\frac{kd\sin\theta}{2}\right)}{\sin^{2}% \left(\frac{kd\sin\theta}{2}\right)},

(7.159)

where $I_{1}$ is the intensity from a single source.

7.10.2 Bragg Scattering

Constructive interference is quite often used to examine structures which are too small to observe with visible light, as was historically the case with Bragg scattering. Lawrence and William Henry Bragg used constructive interference of X-rays to measure the distance between atoms in crystals in the early 1910s. Consider a crystal structure with layers of atoms separated by a distance $d$ . When X-rays enter the crystal, they scatter off the layers. Consider two parallel incident rays which reflect off the top two layers. The path difference between the two rays will be $2d\sin\theta$ , so therefore we get constructive interference when the path difference is an integer multiple of the wavelength:

2d\sin\theta=n\lambda.

(7.160)

7.10.3 Diffraction

Interference also gives rise to the phenomenon known as diffraction, which is where waves are observed to curve around apertures and obstructions in their path. Each part of the wavefront in the gap or around the barrier becomes a secondary source of spherical waves, and these waves interfere to give a curved interference pattern. This method of analysis is known as the Huygens-Fresnel principle.

Consider a plane wave moving through a gap of height $d$ . Let’s look at a thin slice of the gap of length $\operatorname{\mathrm{d}}\!y$ at a distance $y$ from the middle as a point source. The amplitude due to this small slice at a far away point $P$ is $\operatorname{\mathrm{d}}\!s=\varepsilon_{R}\operatorname{\mathrm{d}}\!y\cos(% kr-\omega t)$ , where $r$ is the distance from the thin slice to $P$ and $R$ is the distance from the middle of the gap to $P$ . If $R\gg d$ , then $r\approx R$ so $\varepsilon_{R}$ only depends on $R$ . It can be shown that $r\approx R-y\sin\theta$ , which we will need to use because small differences in the distance will make a big difference for the phase of the wave at $P$ . Integrating over the whole gap, we get

	$\displaystyle s(\theta)=\int\operatorname{\mathrm{d}}\!s$	$\displaystyle=\varepsilon_{R}\int_{-\frac{d}{2}}^{\frac{d}{2}}\cos(k(R-y\sin% \theta)-\omega t)\operatorname{\mathrm{d}}\!y$		(7.161)
		$\displaystyle=2\varepsilon_{R}d\frac{sin\left(\frac{kd\sin\theta}{2}\right)}{% kd\sin\theta}\cos(kR-\omega t).$		(7.162)

The intensity is proportional to the time average of the square of the amplitude.

I(\theta)=4I_{0}\frac{\sin^{2}\left(\frac{kd\sin\theta}{2}\right)}{k^{2}d^{2}% \sin^{2}\theta},

(7.163)

where $I_{0}$ is the intensity at $\theta=0$ . Note that in this situation minima occur at integer multiples of the wavelength, not maxima.

7.10.4 Beating

If we superpose two sounds with a very similar frequency, we will hear a periodic variation in sound intensity. Consider two longitudinal sound waves:

	$\displaystyle s_{1}(x,t)$	$\displaystyle=A\cos(k_{1}x-\omega_{1}t)$		(7.164)
	$\displaystyle s_{2}(x,t)$	$\displaystyle=A\cos(k_{2}x-\omega_{2}t),$		(7.165)

where $k_{1}\approx k_{2}$ and $\omega_{1}\approx\omega_{2}$ . Then using the addition formula, the superposition is

	$\displaystyle s_{1}+s_{2}$	$\displaystyle=2A\cos\left(\frac{k_{1}+k_{2}}{2}x-\frac{\omega_{1}+\omega_{2}}{% 2}t\right)\cos\left(\frac{k_{1}-k_{2}}{2}x-\frac{\omega_{1}-\omega_{2}}{2}t\right)$		(7.166)
		$\displaystyle=2A\cos(\bar{k}x-\bar{\omega}t)\cos(k_{\text{mod}}x-\omega_{\text% {mod}}t),$		(7.167)

where we have defined the average wavenumber and frequency $\bar{k}$ and $\bar{\omega}$ , and a modulation wavenumber and modulation frequency $k_{\text{mod}}$ and $\omega_{\text{mod}}$ . This superposition is a product of a wave with the average frequency of the two original ones and a low frequency modulation (since the two original frequencies are close, the modulation frequency is very low). The intensity rises and falls twice per period, so the beat frequency is twice the modulation frequency, $f_{\text{beat}}=2f_{\text{mod}}=f_{1}-f_{2}$ .

7.10.5 Modulation

Now consider the case where we allow the phase velocities of the waves to be different ( $v_{1}=\frac{\omega_{1}}{k_{1}}\neq v_{2}=\frac{\omega_{2}}{k_{2}}$ ). Then we can write

	$\displaystyle s_{1}(x,t)$	$\displaystyle=A\cos((k_{0}+\Delta k)x-(\omega_{0}+\Delta\omega)t)$		(7.168)
	$\displaystyle s_{2}(x,t)$	$\displaystyle=A\cos((k_{0}-\Delta k)x-(\omega_{0}-\Delta\omega)t),$		(7.169)

and we get

	$\displaystyle s_{1}+s_{2}$	$\displaystyle=2A\cos\left(\frac{2k_{0}}{2}x-\frac{2\omega_{0}}{2}t\right)\cos% \left(\frac{2\Delta k}{2}x-\frac{2\Delta\omega}{2}t\right)$		(7.170)
		$\displaystyle=2A\cos(k_{0}x-\omega_{0}t)\cos(\Delta kx-\Delta\omega t).$		(7.171)

This is the product of a higher frequency wave with a lower frequency envelope. The higher frequency wave has phase speed $v_{\text{crest}}=\frac{\omega_{0}}{k_{0}}$ and the envelope has speed $v_{\text{env}}=\frac{\Delta\omega}{\Delta k}$ . If $\omega\propto k$ , then these speeds are always the same. However, if this is not the case then the wave crests travel at a different speed to the envelope. If the range of wavenumbers in a superposition is small, then the group velocity is the speed of the envelope (the largest amplitude) and we have $v_{\text{gr}}=\left.\frac{\operatorname{\mathrm{d}}\!\omega}{\operatorname{% \mathrm{d}}\!k}\right|_{k_{0}}$ .