An experiment with waves.

Figure 18.2: Double slit experiment with waves

The experiment that was being done using bullets is now repeated replacing the device that sprays bullets with a source that produces waves. The wave amplitude $A$ can be increases continuously and the intensity $I \propto A^2$ also increases continuously. This is to be contrasted with the bullet count which can increase only by an integer number.

Further, using $I_1$ to denote the intensity of the waves at screen B when only slits $1$ is open (Figure 18.2) and $I_2$ when only slit $2$ is open, as shown in the figure the intensity $I_{12}$ when both slits are open is not the sum of $I_1$ and $I_2$ ie. $I_{12} \not= I_1 + I_2$. At some points the intensity $I_{12}$ is more than $I_1 + I_2$ whereas it is less than this at some other points on screen B. The intensity at any point $x$ on screen B is given by

$$I_{12} = I_1 +I_2 + 2 \sqrt{I_1 I_2} \cos \left[ \delta (x) \right] \,.$$ (18.1)

The value of the wave at $x$ is calculated by superposing $\tilde A_1$ and $\tilde A_2$ from slits and $2$ respectively, and we use this to calculate the intensity

$$I_{12} =\left\vert\tilde A_1 + \tilde A_2 \right\vert^2 \,.$$ (18.2)

These two contributions have a phase difference which varies with $x$

$$\delta(x) = \phi_2 (x) - \phi_1 (x) \,.$$ (18.3)

The two waves add constructively when$\tilde A_1$ and $\tilde A_1$ are in phase $\delta(x)=0, 2 \pi,..$a nd $I_{12}$ is more than $I_1 + I_2$, they add destructively when they are out of phase $\delta(x)=\pi,3 \pi,...$ and $I_{12}$ is less than $I_1 + I_2$. This is the familiar phenomenon of interference discussed in detail earlier.