Suppose that the two speakers in Figure 17-7 are separated by 2.50 m and are vibrating exactly

Suppose that the two speakers in Figure 17-7 are separated...

Key Concepts

Wave Interference

Wave interference is the phenomenon that occurs when two or more waves overlap in space, leading to a resultant wave pattern that is the sum of the original waves. This is a central concept in understanding how overlapping sound waves from speakers create regions of enhanced (constructive) or diminished (destructive) sound intensity.

Path Difference

The path difference refers to the difference in the distances traveled by two waves as they arrive at a point. The interference pattern, whether constructive or destructive, depends on how this path difference relates to the wavelength of the waves.

Phase Difference

Phase difference is the relative shift in the cycles of two waves. It can result from the initial phase difference between the sources or due to differences in the travel distance (path difference) of the waves. The phase difference is key in determining how the waves add together at a given point.

Constructive and Destructive Interference Conditions

Constructive interference happens when the phase difference between the waves results in their amplitudes being added, typically when the path difference is an integer multiple of the wavelength, leading to a reinforced signal. Destructive interference occurs when the phase difference causes the waves to cancel each other out, usually when the path difference is an odd multiple of half the wavelength. In special cases like out-of-phase sources, initial phase differences must be included in the interference condition.

Transcript

00:01 Hello, we're dealing, we have a question stated here such that we have an observer near a set of two speakers, where if these speakers are vibrating exactly out of phase, we want to know if this nearby observer observes constructive or destructive interference based on the position of the speakers and the position of the observer relative to the set of speakers.

00:31 So we need to do a couple things in order to figure this out.

00:35 First we'll start with sort of setting up the problem in terms of the diagram.

00:40 So here if we start with speaker a, speaker b, we'll see they're separated by distance as given in the question.

00:52 I'll just call it d -a -b.

00:58 And we have an observer located at exactly 90 degrees from the sets of speakers here.

01:09 Our person at point c, our observer, and a distance of dbc, which dbc changes for parts a and b of our question.

01:23 We have a couple of other givens.

01:27 But again, we were told that speaker a and b are out of phase.

01:40 Phase, phi here for phase.

01:44 Once one given, our second given, is we're told that this frequency that this occurs at, is that 429 hertz.

02:03 There you go, and the speed of sound, as it is stated for error and is given in its question, is 343 meters per second.

02:21 So as well know, dbc is the term that changes for this problem.

02:29 So based on where this observer is, we want to know if this observer does experience constructive or discharge of interference based on their relative position to these speakers.

02:41 So to start, just a little bit of pressure on what we're trying to look for.

02:47 So we know again we're told that these speakers are out of phase.

02:57 And we remember our rules to determine what happens for constructive and destructive interference.

03:04 These rules are different when these speakers are out of phase.

03:09 So that means we observe constructive interference.

03:13 Also yeah sorry we should we get constructive interference constructive interference when the difference in path lengths that these wavelengths travel through is a half integer multiple of our wavelength term that means it's that one half well i'll just read you that's right one and one half two and one half multiples of our wavelength that is being produced and and if we are trying to observe destructive interference, we observe multiple's, integer multiples of the wavelength, including zero, et cetera, et cetera.

04:28 We obviously more multiples of this exist, but just to see the first few, destructive interference occurs, again, because they are out of these, integer multiples of this wavelength term.

04:43 First we need to figure out what wavelength are we observing in this question.

04:48 And then based on the position so based on what dbc is dvc is we'll find out if we have constructive or destructive interference based on our equation for wavelength and speed that's provided you know that we're given the scenario where the speed of the wave is equals the frequency of the wave multiplied by the wavelength like range us to solve for our weight length term.

05:35 So just v over f.

05:38 We substitute our values given for the problem, 343 meters per second, opened by the frequency of the sound, 429 hertz.

05:55 Remember, hertz is just one over seconds.

05:58 So we have meters divided by seconds, multiplied or divided by, multiple by seconds or divided by one over seconds...

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