Standing waves on a 1.0-m-long string that is fixed at both ends are seen at successive frequencies of 36 Hz and 48 Hz. b. Draw the standing-wave pattern when the string oscillates at 48 Hz.

Video transcript

Welcome back, everyone. We are making observations about a rope that is 2. m long and is secured at both ends. Now, we are told that the standing waves are observed at two consecutive frequencies. We have a frequency at 30 Hertz and at 45 Hertz. And we are asked to draw an illustration of the standing wave pattern that emerges when the rope resonates at the higher frequency of 45 Hertz. Well, how are we gonna do this? Well, in order to draw an illustration, we need to know two things. One, we need to know the mode number and we need to know the frequency. Well, we know that the mode number for the F 45 Hertz wave is just going to be the mode number for the 30 Hertz wave plus one since they are two consecutive frequencies. Here's what I'm gonna do. I'm gonna divide the frequency of the M plus one wave or the 45 Hertz wave by the frequency of the predecessor. So we have 45 divided by 30 is equal to the mode number of 30 plus one divided by the mode number of 30. Now, simplifying the equation here, we get two times the mode number of 30 plus two is equal to three times the mode number of 30 giving us that the mode number for the 30 Hertz frequency is simply two adding one to that. We get the mode number for the 45 Hertz frequency which gives us three. Now frequency, we can find by using the concept of a standing wave for the rope at closed ends. So we are going to say that the frequency for the mode three wave is just going to be two times the length of the rope divided by our mode number, which is three. This gives us two times 2.2 divided by three, which gives us a frequency of 1.467. So now drawing out our illustration here, let's say that the wave travels between these two points as connected by our line here. And remember the total distance is 2.2 m. We're gonna have a wave that looks something like this. This is going to be our wave pattern. And as a reminder, our frequency is 1.467 and this will be our final answer. Thank you all so much for watching. I hope this video helped. We will see you all in the next one.