Does two-slit sound-wave interference pattern return to original form?

[ngn]

TL;DR Summary

When a sound wave passes through two slits or gaps, it creates an interference pattern on the other side. Will these two interfering waves eventually merge into a single wave front that resembles the original sound wave or will the interference pattern persist indefinitely?

Hello,
I am reading about a sound wave that strikes two separated gaps/slits in a boundary. This causes two spherical waves to be created on the other side of the gaps that radiate outward, interfering with each other and creating an interference pattern in the resulting wave. The text states "After a certain time, one new wavefront is created from all the secondary waves". My question is: in this new wavefront that forms from the secondary waves (as the text states), is the interference pattern still present? Or does the new wave come to resemble the original wave from the source of the sound?

According to Huygen's Principle, a wavefront is composed of an infinite number of spherical wavelets (i.e., secondary waves). These wavelets interfere with each other similar to the interference after the double slit. However, these wavelets combine to form a uniform wavefront without any interference pattern produced by its secondary waves. So, along the same reasoning, is that what eventually happens to the waves on the other side of the double-slit? At first, they interfere, but as time goes by, these secondary waves merge into a uniform wavefront without an interference pattern?

So, my primary question is:
1. Does the interference pattern persist indefinitely, or do the waves emerging on the other side of the slits eventually merge to form a wave that resembles the original sound without any interference pattern?

My second question is:
2. I suspect that the interference pattern persists. If so, could you explain why the interference pattern persists after the double slit, but does not persist in the original sound wave even though the original sound wave is composed of interfering wavelets.

Thank you!

 

[Baluncore]

The interference pattern persists.

At great distances, your ability to resolve the two slits, makes it difficult to recognise the interference pattern. The two slits appear to merge into one source.

 

[ngn]

Thank you. Although your ability to recognize the interference pattern decreases, does the interference persist? In other words, at the extreme limit, does the waveform actually fully become the original waveform or is the interference pattern a baked in characteristic of the new waveform?

 

[sophiecentaur]

TL;DR Summary: When a sound wave passes through two slits or gaps, it creates an interference pattern on the other side. Will these two interfering waves eventually merge into a single wave front that resembles the original sound wave or will the interference pattern persist indefinitely?

secondary waves merge into a uniform wavefront without an interference pattern?

The resultant wave when two waves pass through each other will depend on the amplitude and phase difference at each point in space. To get a really good pattern, the sources need to be almost zero width to give a full 180 degree width. With real slit widths, the brightness of the fringes will drop as you go off axis because of the patterns of the individual slits depend on the actual width so no longer 180 degree uniform brightness. But the relative amplitudes will be the same wherever so the pattern is still there and the two individual patterns are still the same (ideally).

For two sources, pointing in different directions and with beams crossing, the drop off with distance of brightness of each source will be very different so only at the centre of the overlap will you get good addition and subtraction (nice deep fringes) off one beam and the ratio of amplitudes will be high and you never get total subtraction so fringes are less and less.

Worth noting that, in a linear medium, waves do not actually interact; they are quite independent. It is the microphone or radio antenna that does the vector addition to 'see' the pattern. It's like the tree falling in the forest it's only noticed when there's something there to notice it (microphone / screen / antenna)

 

[Baluncore]

The interference pattern ceases to exist at an infinite distance.
Define "extreme limit".

 

[Baluncore]

The ability to resolve two sources or slits is determined by the Rayleigh criterion.
https://en.wikipedia.org/wiki/Angular_resolution#The_Rayleigh_criterion

 

[sophiecentaur]

The ability to resolve two sources or slits is determined by the Rayleigh criterion.
https://en.wikipedia.org/wiki/Angular_resolution#The_Rayleigh_criterion

Rayleigh Criterion is only a rule of thumb. Digital processing can allow the resolution to be increased a lot if you can increase the time over which the measurement is made and / or increase the signal to noise ratio. Astronomers would be hamstrung if they had to rely on Rayleigh for their results.

 

[hutchphd]

If you are exctly on the axis, and the two sources in phase, then the oscillations going past are essentially the same as a doubled strength source for all distances to the sources (slits). The max/min pattern off-axis scales with distance (the angle remains constant).

 

[sophiecentaur]

If you are exctly on the axis, and the two sources in phase, then the oscillations going past are essentially the same as a doubled strength source for all distances to the sources (slits). The max/min pattern off-axis scales with distance (the angle remains constant).

To make it clear, the amplitudes will add, giving twice amplitude, which corresponds to four times the radiated power in that direction.
At the same time, and to keep to the conservation law of Energy, the resultant in other directions will be less than that. A maximum along the axis can result in many different patterns, depending on the spacing. If the two sources are λ/2 apart then there will be no radiation along the axis. The diagram below shows what happens with (omnidirectional) radiating elements with different spacing. In this case, all 360 degrees of azimuth are shown but you will get my point; the integral is always twice the power but maxima can be greater than that. A bit off topic, maybe but it helps avoid drawing wrong conclusions about the Energy carried by Waves.
From electronicsforu.com

 

[ngn]

These responses are very helpful. What I am getting (correct me if I'm wrong) is that the interference pattern will continue to persist in the wave that continues out from behind the slits (until it reaches infinity at which point the pattern will not exist). So for practical, everyday purposes, we can say that under ideal circumstances (no loss of energy) the interference pattern will be present at a set distance from the barrier no matter how far back that distance is, although the farther the distance, the more difficult it will be for a listener to detect the pattern.

 

[sophiecentaur]

although the farther the distance, the more difficult it will be for a listener to detect the pattern.

Right but that statement also applies to the difficulty of hearing the tone at a distance in the first place.

But I will throw a spanner in the works and make the point that if, for instance the observer were constantly moving about over the pattern and moving from peaks to troughs, it is generally true to say that a very low level tone that is varying in amplitude is often easier to detect than a low level of constant amplitude. So that pattern may actually be detected further away than a constant tone.