# Wave Properties

by Jared Rovny

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00:01 Now we're ready to describe some basic properties of a wave.

00:04 To do that let's look at again a physical example of a wave.

00:07 Let's take this guitar string again as an example.

00:10 So what I've done here is I've drawn the guitar string but I've also introduced a coordinate system just so we can describe the guitar string and its motion quantitatively.

00:17 So with this coordinate system where I simply have the horizontal and vertical positions, these are both just position coordinates, no time or anything has been included yet.

00:26 We can describe the few properties, the first of which I've listed here is the amplitude of the wave.

00:31 The amplitude of this particular wave is one and you can see that by defining the amplitude from the center of the wave which we have in a horizontal dotted green line here to the maximum part of the wave or from the horizontal line to the minimum part of the wave, this is called the amplitude.

00:48 You can see that I've emphasized here that the amplitude is only the distance to the access so we don't go from the top of the wave to the bottom of the wave that would be twice the amplitude in fact.

00:59 The amplitude is just define from the center of the wave which again we have a dotted green line here and that just goes to the top of the wave or from the center of the wave to the bottom of the wave.

01:08 The next property, we can discuss about waves and when that would be bringing up over and over again is the wavelength of the wave.

01:17 This one is define as the total distance between the next two repeating points on the wave and by that I mean if we took one point on the wave like the peak of the wave, we could ask ourselves what's the distance before we get to another peak of the wave.

01:31 But we have to be very careful with this definition of wavelength which is represented with this Greek letter lambda, as you can see here because it's not saying what are the next two points on the wave that are the same so for example you can see close to three here and close to five, those are both points that are close to the zero access or this horizontal dotted green line, that distance would be smaller than the wavelength of the wave.

01:54 What we're actually asking is the distance between the next two repeated behaviors so that point that's crossing the green access here, that's close to three on the horizontal access and close to five on the horizontal access are different because in the one case the way this on its way down as it crosses the green line and the other wave, the other point on the wave at point five is on its way up as it crosses the dotted green line.

02:18 And so that wouldn't be a valid way to measure the wavelength of the wave.

02:22 We don't wanna measure between adjacent zero points or points of the wave, crosses the zero.

02:26 You want to always measure the distance between two points that describe repeated similar action and you can always be careful in doing this by keeping yourself to measuring the distance between the two peaks or maybe the two troughs, the two lowest points on the wave and that would be an accurate measure of the wavelength, which for this wave is four, which you can see just by looking at the horizontal access, the distance goes from two to six.

02:49 Finally, for a wave like this one we could also discuss something that we call the phase.

02:55 The phase just measures the horizontal off set of your wave and so in a given coordinate system the phase can change depending on how you define your coordinate system.

03:04 So in this case, what we've done is we've got a coordinate system where the first point that first zero point on your wave is off set from the zero access, that vertical access by distance of one and so this wave right now has a phase of one.

03:18 As the wave moves along we could say that its phase evolve or its phase changes as the wave moves and propagates for example in this guitar string but again the phase can depend on, not only your definition of the coordinate system as we just said but it might also depend on how you're mathematically defining your wave so for example a sine wave which we've discussed already from trigonometry, starts at zero and then goes up and down and oscillates where as a cosine wave starts at the point one and then goes up and down and oscillate.

03:48 And so we can describe any given wave as either a cosine wave or a sine wave or maybe some sum of the two but the phase in either case might be different.

03:57 So if we were defining in this particular wave with this coordinate system and as a sine curve or a sine wave, its phase off set would be again a distance of one.

04:07 In one final thing before we move off of this basic definition you should also notice that by this coordinate system, we've also got an offset, a vertical offset of our wave since we're defining the zero point or the center point of the wave, to be at the position two in our coordinate system.

04:23 So also be aware of that especially when you're trying to measure amplitudes and maybe you have the coordinates of the particular points on your wave, make sure you know where the center of your wave is and don't assume that the center of your wave coincide with the zero on the horizontal access.

04:37 What we're going to do now is instead of looking at the entire physical wave, we're going to pick one point maybe on this guitar string and allow time to evolve.

04:47 So we're going to watch that point, that go up and down and up and down and we're going to plot this position as time goes on.

04:53 So again we're going to call this, in this position is what we're going to plot, we call it the vertical position and we'll represented with the letter X.

05:01 Since we're just talking about position and we won't have to compare it with X or Y or anything.

05:06 We'll just give it some name, the position of this point as it goes up and down.

05:09 If we do that, what you can see is we get another wave.

05:12 So we have a wave here but this is a very different wave so be careful when you're looking at this.

05:17 The guitar string that we just showed was a physical wave because it was just an actual physical object but this wave is different, what we're doing in this is plotting the position of one point as it goes up and down through time and so the horizontal access on this graph is actually the time access so as time goes on the position X is going up and down as time evolves.

05:41 We can also describe properties of this wave which is evolving in time and we have some new names that will also have to introduce, the same name that we introduced for the physical wave, the amplitude and the wavelength, they're also used to describe really any wave and that would include this wave but especially when you're talking about waves or periodic motion in time, any sort of periodic motion in time.

06:02 We have a few new variables we can introduce, for example the period, the period itself.

06:08 The period is defined as the time between the next repeated oscillation and again just like we talked about with the wavelength and the physical wave, you have to be careful on how you define the period, it's not define as the distance between on a graph next to zero points for example, not just between adjacent zeros but the time between the next repetition.

06:30 So in this wave for example the two zero points again are different because in one case as the wave crosses zero, it's on its wave down and then in the other case as it crosses zero, it's on its way up, and so those are two distinct points.

06:43 So for the period you always want to be careful to compare identical points and that will include their environment whether it's on its way up or it's on its way down and how it's behaving.

06:52 And again the safe way to always calculate this is to keep yourself to measuring the peaks, the difference between the peaks or the distance between the troughs rather than trying to pick maybe zero points or any other point on your wave.

07:03 The units of the period are seconds because we're just measuring the time, how long it takes before the object, whatever it is maybe the point on the guitar string repeats its motion.

07:13 And then we could define a new quantity called the frequency.

07:18 The frequency is measuring the number of the cycles your wave makes per second and is therefore define as one divided by the period or the inverse of the period.

07:28 So the units of this frequency as you can see just from the equation for frequency as one over the period are one over seconds or sort of per second, so this is what frequency is measuring is how many things are happening per second and we call this one over seconds or per second hertz.

07:46 And the units of hertz are abbreviated with the letter Hz as you can see here.

07:51 And so this Hz is slightly different in the name of the units because we have two letters are there so be careful when you see not to think that there are two different units or two different variables or anything like that, it's just one unit called hertz.

08:04 We could ask ourselves about the frequency of this particular wave so suppose for example that this wave or this point on our guitar string which is what we're tracking, goes up and down two times every second.

08:15 We would say that it has two cycles per second and so the frequency of this wave would be two cycles per second or two hertz.

08:24 The next thing we'll introduce is possibly the most important equation, we'll talk about with waves, it comes up over and over again and many, many different contexts as we'll see and it is the wave velocity.

08:37 If we know the wavelength of the wave then we also know the frequency at which that wave is repeating.

08:44 We can find that the velocity how quickly this wave is moving and this example maybe from left to right just by multiplying the wavelength of your wave by the frequency of the wave and one easy way to remember this and make sure you always get the variables right is to think about the units of the wave velocity.

09:01 A velocity in our standard units needs to be in meters per second or a distance over a time.

09:07 We know that since the wavelength is measuring a distance and the frequency is measured in one over time by multiplying these two together, we get the proper units, the distance over the time.

The lecture Wave Properties by Jared Rovny is from the course Periodic Motion.

### Included Quiz Questions

1. The height of a wave from its center
2. The overall height of a wave from peak to trough
3. The total length of a wave
4. The amount of energy stored in a wave
5. The strength of a wave
1. The distance between wave repetitions
2. The distance from a wave’s peak to its zero
3. The distance from a wave’s peak to its next trough
4. The distance from a wave’s center to its peak
5. The distance from a wave’s beginning to its end
1. The horizontal offset of a wave’s beginning
2. The total distance it travels
3. The amount it differs from another wave
4. The value, equal to one, of the wave’s distance
5. The total size of a wave
1. It is the time between wave repetitions
2. It is the time between adjacent zeros of a wave
3. It is independent of the wave’s frequency
4. It is twice the wave velocity
5. It is related to the square of the travel distance of a wave
1. 2 Hz
2. 2 seconds
3. 50 Hz
4. 50 seconds
5. 2 meters

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