13 Exercise 1. Determine the minimum path difference between two coherent waves to the points of the elastic medium

waves at a point in the medium is given by the formula:

\[\Delta \phi = \frac{{2\pi \cdot \Delta x}}{{\lambda}}\]

where \(\Delta \phi\) is the phase difference, \(\Delta x\) is the path difference between the two waves, and \(\lambda\) is the wavelength.

Let"s solve the first problem.

Exercise 1:
To determine the minimum path difference between the two waves, where maximum wave attenuation occurs as a result of their superposition, we need to find the condition for destructive interference.

In destructive interference, the phase difference between the two waves is \(\pi\) radians or an odd multiple of \(\pi\).

We are given that the frequency of oscillation of the wave sources is 0.4 kHz. To find the wavelength, we can use the formula:

\[\lambda = \frac{v}{f}\]

where \(v\) is the wave propagation speed and \(f\) is the frequency.

Substituting the given values, we have:

\[\lambda = \frac{240}{0.4} = 600 \, \text{m}\]

Now, for destructive interference to occur, the path difference \(\Delta x\) must satisfy the condition:

\(\Delta \phi = \frac{{2\pi \cdot \Delta x}}{{\lambda}} = \pi\)

Simplifying the equation, we have:

\(\Delta x = \frac{{\lambda}}{{2}} = \frac{{600}}{{2}} = 300 \, \text{m}\)

Therefore, the minimum path difference between the two waves, where maximum wave attenuation occurs, is 300 meters.

Moving on to the second problem.

Exercise 2:
To determine the result of interference between two coherent waves at a point in the medium located 16 meters from the first wave source and 31 meters from the second, we need to find the phase difference between the waves at that point.

We are given that the period of oscillation of the wave sources is 20 ms. To find the wavelength, we can use the formula:

\(\lambda = \frac{{v \cdot T}}{{S}}\)

where \(v\) is the wave propagation speed, \(T\) is the period, and \(S\) is the distance between the wave source and the point of interest.

Substituting the given values, we have:

\(\lambda = \frac{{1500 \cdot 0.02}}{{16}} \approx 1.875 \, \text{m}\)

To find the phase difference, we can use the previous formula:

\(\Delta \phi = \frac{{2\pi \cdot \Delta x}}{{\lambda}}\)

Substituting the given values, we have:

\(\Delta \phi = \frac{{2\pi \cdot (31-16)}}{{1.875}} \approx 5.33 \, \text{rad}\)

Therefore, the result of interference between the two coherent waves at the specified point is a phase difference of approximately 5.33 radians.

I hope this explanation helps! Let me know if you have any further questions.