Video: Calculating Wave Length from Wave Speed and Distance–Time Graph

A wave is shown in the diagram. What is the wavelength of the wave if its speed is 360 m/s?


Video Transcript

A wave is shown in the diagram. What is the wavelength of the wave if its speed is 360 meters per second?

Okay, looking at this diagram, we see a wave with the displacement of that wave plotted against time in seconds. We’re told that this wave has an overall speed that is motion in the horizontal direction of 360 meters per second. And based on this, plus the information in our graph, we want to solve for the wave’s length.

Now, there’s a relationship that ties together wavelength and wave speed, and it involves a third variable, wave frequency. That relationship says that wave speed 𝑠 is equal to wave frequency 𝑓 multiplied by wavelength 𝜆. In our case, though, it’s not wave speed we want to solve for, but wavelength. So, let’s rearrange this equation by dividing both sides by wave frequency.

When we do that, the frequency term on the right-hand side cancels out and we see that the wavelength of the wave is equal to the wave speed divided by the wave frequency. Since we’re given wave speed, we want to solve for a wave frequency to ultimately figure out wavelength. Considering wave frequency, let’s again take a look at our graph. If we start at the origin where the wave’s displacement is zero, and zero time has elapsed. And then, we follow this wave along through one complete cycle of the wave, we get to a time value of 0.5 seconds.

We can recall that wave frequency is equal to the number of cycles a wave goes through as one second of time elapses. So, as we’re seeing with this particular wave, our wave frequency is equal to one cycle, since we go through one cycle of the wave, in 0.5 seconds worth of time. And remember, we figured this out by tracing out one complete cycle of our wave and seeing how much time it took to do that.

Now, if we divide one by 0.5, then that’s equal to two. So, we can write our wave frequency as two cycles of the wave per second of time. And then, we can recall that one hertz is defined as one cycle per second. Writing our wave frequency in terms of this unit then, we would have two cycles per second, which is equal to two hertz. This, then, is the value that we’ll insert for 𝑓 in our equation to solve for wavelength 𝜆.

The wavelength of this wave then is equal to the wave speed, 360 meters per second, divided by the wave frequency, two hertz. And notice that if we replace the units of hertz with units of inverse seconds, which is what a hertz is equivalent to, then we have cancellation of the one-over-seconds unit in our numerator and denominator. We’re left in the end with units just of meters, units of distance, which is good since we’re calculating a wavelength. And when we compute this fraction, we find that it’s equal to 180 meters. That’s the wavelength of the wave.

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