Question

Which of the following should be equal to the path difference as the condition for constructive interference?
${\text{A}}{\text{.}}$ Odd integral multiple of wavelength
${\text{B}}{\text{.}}$ Integral multiple of wavelength
${\text{C}}{\text{.}}$ Odd integral multiple of half wavelength
${\text{D}}{\text{.}}$ Integral multiple of half wavelength

Answer 1

Hint: Here, we will proceed by defining the term wave. Then, we will discuss constructive and destructive interference between two waves. We will also mention the condition for the interference to be constructive or destructive.

Complete Step-by-Step solution:
Wave is defined as the regular and organized propagation of perturbations from place to place. Most familiar are surface waves traveling on water but all exhibit wavelike properties in sound , light, and the motion of subatomic particles. In the simplest waves the disturbance regularly oscillates with a fixed frequency and wavelength (see periodic motion). Mechanical waves, such as sound, need a moving medium, whereas electromagnetic waves (see electromagnetic radiation) do not need a medium and can be propagated by a vacuum. The propagation of a wave by means depends on the properties of the medium.

Constructive interference is defined as the interference of two or more waves of equal frequency and phase, resulting in mutual strengthening and producing a single amplitude equal to the sum of the individual waves' amplitudes. When two waves meet in such a way that their crests line up together, then constructional interference is called. The resulting wave is greater in amplitude. The phase difference between the waves must be zero for Constructive Interference.
Two waves of equal frequency and opposite phase interference, resulting in their cancelation where one's negative displacement often coincides with the other's positive displacement is called destructive interference. In disruptive interference, one wave's crest crosses another's trough, and the effect is a lower maximum amplitude.
Condition for destructive interference is a phase difference of 180 degrees between the waves being superimposed.
If a crest of one wave meets a trough of another wave, then the displacement magnitude is equal to the difference in the individual magnitudes; this is referred to as destructive interference.
For constructive interference, the path difference $\Delta x$ is given by
$\Delta x = n\lambda {\text{ }} \to {\text{(1)}}$ where $\lambda $ denotes the wavelength and n = 1,2,3,…. (any integral value)
For destructive interference, the path difference $\Delta x$ is given by
$\Delta x = \left( {2n + 1} \right)\dfrac{\lambda }{2} = n\lambda + \dfrac{\lambda }{2}$ where $\lambda $ denotes the wavelength and n = 1,2,3,…. (any integral value)
From equation (1), we can say that the condition for constructive interference is that the path difference should be equal to the integral multiple of wavelength.
Therefore, option B is correct.

Note- The distance between two successive crests or troughs of a wave can be described as wavelength. It is measured against the wave path. Crest is the highest point of the wave while the lowest is the trough. Since wavelength is distance / length, it is measured in length units including meters, centimeters, millimeters, nanometers, etc.