Solved Problem on Wave Interference

Consider two sources, F₁ and F₂, in phase as shown in the figure, vibrating at a frequency of 300 Hz. The speed of waves, in the medium where points A and B are located, is 6 m/s. What type of interference occurs at points A and B?

Problem data:

Source frequency: F₁ = F₂ = 300 Hz;
Speed of waves in the midium v = 6 m/s.

Solution

First, let us convert the speed given in meters per second (m/s) to centimeters per second (cm/s) (cm/s)

\[ \begin{gather} v=6\;\frac{\cancel{\text{m}}}{\text{s}}\times\;\frac{100\;\text{cm}}{1\;\cancel{\text{m}}}\times\frac{1}{\text{s}}=600\;\frac{\text{cm}}{\text{s}} \end{gather} \]

The wavelength is given by

\[ \begin{gather} \bbox[#99CCFF,10px] {v=\lambda f} \end{gather} \]

\[ \begin{gather} \lambda =\frac{v}{f}\\[5pt] \lambda =\frac{600}{300}\\[5pt] \lambda=2\;\text{cm} \end{gather} \]

For point A:

The distance, S₁, from source F₁ to point A will be given by the segment

\[ \begin{gather} S_{1}=\overline{{F_{1}A}}=8\;\text{cm} \end{gather} \]

and the distance, S₂, from source Fo to A will be

\[ \begin{gather} S_{2}=\overline{{F_{2}A}}=11\;\text{cm} \end{gather} \]

The difference in paths taken by the waves of F₁ and F₂

\[ \begin{gather} \bbox[#99CCFF,10px] {\Delta S=\left|S_{2}-S_{1}\right|} \tag{I} \end{gather} \]

\[ \begin{gather} \Delta S=\left|11-8\right|\\[5pt] \Delta S=3\;\text{cm} \end{gather} \]

The number \( \frac{\Delta S}{\lambda }=\frac{3}{2}=1,5=1+\frac{1}{2} \) is of form \( \left(n+\frac{1}{2}\right) \), at point A the interference will be destructive.

For point B:

The distance, S₁, from source F₁ to point B will be given by the segment

\[ \begin{gather} S_{1}=\overline{{F_{1}B}}=6\;\text{cm} \end{gather} \]

and the distance, S₂, from source F₂ to B will be

\[ \begin{gather} S_{2}=\overline{{F_{2}A}}=8\;\text{cm} \end{gather} \]

substituting S₁ and S₂ in expression (I) for path difference

\[ \begin{gather} \Delta S=\left|8-6\right|\\[5pt] \Delta S=2\;\text{cm} \end{gather} \]

The number \( \frac{\Delta S}{\lambda }=\frac{2}{2}=1 \), is an integer n at point B the interference will be constructive.

Note: At point B, the crests of the two waves meet, producing constructive interference. At point A, the crest of the wave produced in F₂ meets the trough of the wave produced in F₁, producing destructive interference (Figure 1).

Figure 1

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