Video: Defining the Speed of a Traveling Wave

Which of the following formulas correctly shows the relation between the frequency, the speed, and the wavelength πœ† of a wave? [A] 𝑆 = πœ†/𝑓 [B] 𝑆 = 𝑓 βˆ’ πœ† [C] 𝑆 = 𝑓 + πœ† [D] 𝑆 = 𝑓/πœ† [E] 𝑆 = π‘“πœ†.

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Video Transcript

Which of the following formulas correctly shows the relation between the frequency, the speed, and the wavelength πœ† of a wave? Number one: 𝑆 is equal to πœ† over 𝑓. Number two: 𝑆 is equal to 𝑓 minus πœ†. Number three: 𝑆 is equal to 𝑓 plus πœ†. Number four: 𝑆 is equal to 𝑓 over πœ†. And number five: 𝑆 is equal to π‘“πœ†.

Now this is an important equation to remember. We need to memorize this. So what we’ll go through in this video is the answer. And then a way of checking that we’ve got it right. So first of all, the answer: it’s number five. 𝑆 is equal to π‘“πœ†. Obviously, this is something we need to know. But there is a way of checking it, like we mentioned earlier.

What we can do is use something known as dimensional analysis. Basically, what it means is to compare the units of each thing on the left-hand side to the units of everything on the right-hand side. And the idea is, is because we’ve got an equation, there’s an equal sign in it, then the units of the left-hand side must be equal to the units on the right-hand side because the two things are equal. So carry on watching if you’re feeling a bit brave.

Let’s start with the left-hand side. We’ve got speed 𝑆. We can recall that speed is defined as a distance divided by time. It’s basically the amount of distance travelled in a unit of time. And we can look at the units of distance and time. We know that distance is measured in meters. And time is measured in seconds. So the units of speed are meters per second.

Next, let’s look at πœ†, the wavelength. Well, πœ† is a wavelength. It measures a distance. Therefore, it has units of meters.

And finally, let’s go on to frequency. Now this one is a bit more tricky because frequency is usually measured in hertz. But we can also recall that frequency is defined as one divided by the time period of a wave. This time period is defined as the time taken for an entire cycle of a wave to travel past one point in space. Basically, there’s a wave travelling to the right. And imagine that we’re standing here. Well, the time period is how long it’ll take an entire cycle, that’s all of this until this point here, to travel past where we’re standing. Or, in other words, how long it takes for this point to get to where we’re standing. But remember, we don’t have to start on the peak of a wave. We could always start on the trough, or anywhere else on the way for that matter.

So the time period of the wave is however long it takes the next adjacent, equivalent point, which is this trough here in this case, to travel to where we’re standing. And frequency is simply one divided by this time period. So for frequency, we’ve got one divided by the units of time which happens to be one divided by seconds.

Now, if we take our final answer, 𝑆 is equal to π‘“πœ† , then we can compare the units. On the left-hand side, we’ve got the units of speed which happens to be meters per second. And on the right-hand side, we’ve got frequency which is one divided by seconds multiplied by πœ† which happens to be meters. And of course, the right-hand side simplifies slightly to give us meters per second. Therefore, the units on both sides of the equation are equal. And our final answer is: 𝑆 is equal to π‘“πœ†.

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