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Chapter 37: Problem 26

Two coherent waves are described by $$E_{1}=E_{0} \sin \left(\frac{2 \pi x_{1}}{\lambda}-2 \pi f t+\frac{\pi}{6}\right)$$ $$E_{2}=E_{0} \sin \left(\frac{2 \pi x_{2}}{\lambda}-2 \pi f t+\frac{\pi}{8}\right)$$ Determine the relationship between \(x_{1}\) and \(x_{2}\) that produces constructive interference when the two waves are superposed.

Short Answer

Expert verified
The relationship between \(x_{1}\) and \(x_{2}\) that produces constructive interference when the two waves are superposed is \(x_{1} - x_{2} = \lambda\left[k - \frac{1}{12} + \frac{1}{16}\right]\), where \(k\) is any integer.

Step by step solution

01

Writing the general condition for constructive interference

The general condition for constructive interference is when the phase difference between the two waves is an integral multiple of \(2\pi\). Therefore, we can write this as: \(\frac{2 \pi x_{1}}{\lambda} + \frac{\pi}{6} - \left(\frac{2 \pi x_{2}}{\lambda} + \frac{\pi}{8}\right) = 2k\pi\), where \(k\) is any integer.
02

Solving the Equation

Solve the above equation for \(x_{1}\) and \(x_{2}\). To do this, first subtract \(\frac{2 \pi x_{2}}{\lambda} + \frac{\pi}{8}\) from both sides of the equation giving us: \(\frac{2 \pi x_{1}}{\lambda} - \frac{2 \pi x_{2}}{\lambda} = 2k\pi - \frac{\pi}{6} + \frac{\pi}{8}\). Then, factor out the common factor \(\frac{2\pi}{\lambda}\) on the left side: \(\frac{2\pi}{\lambda} (x_{1} - x_{2}) = 2k\pi - \frac{\pi}{6} + \frac{\pi}{8}\). Lastly, solve for \(x_{1} - x_{2}\) by dividing both sides by \(\frac{2\pi}{\lambda}\).
03

Relationship between \(x_{1}\) and \(x_{2}\)

Finally, the relationship between \(x_{1}\) and \(x_{2}\) that produces constructive interference when the two waves are superposed is expressed as: \(x_{1} - x_{2} = \lambda\left[k - \frac{1}{12} + \frac{1}{16}\right]\), where \(k\) is any integer.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Coherent Waves
Coherent waves are essential in the study of wave interference. They are waves that maintain a constant phase relationship. This occurs when the frequency and phase difference between them remain the same over time. This means:
  • Coherent waves have the same frequency.
  • The phase difference between them does not change.
This consistent relationship allows coherent waves to interfere consistently, either constructively or destructively. Coherence is often achieved by splitting a single wave source into two parts, ensuring that each part travels through similar paths without significant changes.
In the context of constructive interference, such as the one in the original exercise, maintaining coherence is crucial because any phase drift over time would affect the interference pattern. When the waves are not coherent, the interference pattern would vary, losing its stability and predictability.
Phase Difference
The phase difference between waves is a vital factor in determining the type and magnitude of interference they produce. Phase difference refers to the difference in the phase angles of two periodic signals. It is measured in radians or degrees.
For constructive interference to occur, the phase difference must be an integer multiple of \(2\pi\). This condition ensures the peaks and troughs of the waves align perfectly, reinforcing each other.
The equation from the original exercise captures this concept of phase difference:
  • The expression \(\frac{\pi}{6} - \frac{\pi}{8}\) represents the inherent phase difference between the waves, due to their initial conditions.
  • The equation \(\frac{2\pi}{\lambda}(x_1 - x_2) = 2k\pi\), ensures that the additional path difference accounts for further phase differences.
When calculated properly, the phase difference aligns the waves constructively to create points of highest amplitude.
Wave Superposition
Wave superposition is a fundamental principle where two or more waves overlap in space, resulting in a new wave pattern. This principle is key to understanding wave interference.
  • When two waves meet, their amplitudes add up algebraically at each point where they encounter each other, forming a resultant wave.
  • This resulting wave can either have a larger amplitude (constructive interference) or a smaller one (destructive interference).
The original exercise leverages superposition to demonstrate constructive interference. Both waves \(E_1\) and \(E_2\) combine under specific conditions of path and phase difference, resulting in a wave of increased amplitude.
This occurs because the condition \(x_1 - x_2 = \lambda[k - \frac{1}{12} + \frac{1}{16}]\) aligns the crests and troughs of the waves perfectly. Understanding how wave superposition works allows for predicting and controlling interference patterns, which is critical in many fields from acoustics to photonics.

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Most popular questions from this chapter

Use phasor addition to find the resultant amplitude and phase constant when the following three harmonic functions are combined: \(\quad E_{1}=\sin (\omega t+\pi / 6), \quad E_{2}=\) \(3.0 \sin (\omega t+7 \pi / 2),\) and \(E_{3}=6.0 \sin (\omega t+4 \pi / 3)\)

Two narrow parallel slits separated by 0.850 mm are illuminated by 600 -nm light, and the viewing screen is \(2.80 \mathrm{m}\) away from the slits. (a) What is the phase difference between the two interfering waves on a screen at a point \(2.50 \mathrm{mm}\) from the central bright fringe? (b) What is the ratio of the intensity at this point to the intensity at the center of a bright fringe?

A soap bubble \((n=1.33)\) is floating in air. If the thickness of the bubble wall is \(115 \mathrm{nm},\) what is the wavelength of the light that is most strongly reflected?

When illuminated, four equally spaced parallel slits act as multiple coherent sources, each differing in phase from the adjacent one by an angle \(\phi .\) Use a phasor diagram to determine the smallest value of \(\phi\) for which the resultant of the four waves (assumed to be of equal amplitude) is zero.

Our discussion of the techniques for determining constructive and destructive interference by reflection from a thin film in air has been confined to rays striking the film at nearly normal incidence. What If? Assume that a ray is incident at an angle of \(30.0^{\circ}\) (relative to the normal) on a film with index of refraction \(1.38 .\) Calculate the minimum thickness for constructive interference of sodium light with a wavelength of \(590 \mathrm{nm}\)
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