Video Transcript
A pier stretches from the coastline out into the sea a distance of 180 meters. Waves on the sea move past the pier as they head toward the shore. The distance between the crests of the waves is 15 meters, and the wave crests travel from the end of the pier to the shore in 24 seconds. What is the frequency of the waves?
Let’s start by drawing a diagram. Here’s the pier stretching out over the sea. We know that waves pass by the pier on their way to shore. It takes a wave a time, which we’ll call Δ𝑡, of 24 seconds to travel the length of the pier, which we know is 180 meters. We’ll call this distance Δ𝑑. One more piece of information we’ve been given is that the crests of the waves are 15 meters apart. This is their wavelength, a measurement that’s represented by 𝜆. So, we can say that 𝜆 equals 15 meters. Now, in this question, we want to find the frequency of the waves. So, let’s recall an important formula for calculating wave motion. That is, 𝑠 equals 𝑓 times 𝜆, where 𝑠 is the wave speed, 𝑓 is frequency, and we know 𝜆 is wavelength.
Now, we want to solve for the waves’ frequency. So, let’s copy the formula below and rearrange it to make 𝑓 the subject. To do this, we simply divide both sides by 𝜆. So, we can cancel it from the numerator and denominator of the right-hand side, leaving 𝑓 by itself. Now, flipping it the other way and writing it in a bit more neatly, the formula reads frequency equals wave speed divided by wavelength. We already know the wavelength, and we don’t know the wave speed. That’s okay though because we do know how far the waves travel over a certain period of time. This gives us a separate way to find 𝑠 because we know that speed just equals distance divided by time. So once we substitute this expression into our formula for frequency, we can find 𝑓 in terms of values that we already know.
Now substituting in Δ𝑑, Δ𝑡, and 𝜆, we have 180 meters divided by 24 seconds all over 15 meters. These are all expressed in base SI units, so let’s go ahead and start calculating. Notice that units of meters are going to cancel out of the numerator and denominator, leaving only units of per seconds in the numerator. So our final answer will have units of per seconds, which is the same thing as hertz, the SI unit of frequency. So, we’re getting close. And now all that’s left to do is find 180 over 24 all divided by 15, which comes out to one-half or 0.5. Thus, we found that the frequency of the waves passing the pier is 0.5 hertz.
