Click here to purchase the entire book in PDF format. Impulse Response vs. Resonance Think back to the beginning of this section when you were playing with the skipping rope. Let's take the same rope and attach it to two fence posts, pulling it tight. In our previous example, you flicked the rope with your wrist, making a bump in it that travelled along the rope and reflected off the opposite end. In our new setup, what we'll do is to tap the rope and watch what happens. What you'll (hopefully) be able to see is that the bump you created by tapping travels in two opposite directions to the two ends of the rope. These two bumps reflect and return, meeting each other at some point, crossing each other and so on. This process is shown in Figures 3.43 through 3.46. Figure 3.43: A rope tied to two fence posts just after you have tapped it. Figure 3.44: A rope tied to two fence posts a little later. Note that the bump you created is travelling in two directions, and each of the two resulting bumps is smaller than the original. The bump on the left has already reflected off the left end of the rope. Figure 3.45: The two bumps after they have reflected off the high impedance terminations (the fence posts). When the two bumps meet each other, they appear for an instant to be one bump Figure 3.46: After the two reflections have met each other, they continue on in opposite directions. (The right bump has reflected off the right end of the rope.) The process then repeats itself indefinitely, assuming that there are no losses of energy in the system due to things like friction. Let's assume that you are able to make the bump in the rope infinitely narrow, so that it appears as a spike or an impulse. Let's also put a theoretical probe on the rope that measures its vertical movement at a single point over time. We'll also, for the sake of simplicity, put the probe the same distance from one of the fence posts as you are from the other post. This is to ensure that the probe is at the point where the two little spikes meet each other to make one big spike. If we graphed the output of the probe over time, it would look like Figure 3.47. Figure 3.47: The output of the probe showing the vertical movement of the rope over time. This response corresponds directly to Figures 3.43 to 3.46 This graph shows how the rope responds in time when the impulse (an instantaneous change in displacement or pressure which instantaneous) is applied to it. Consequently we call it the impulse response of the system. Note that the graph in Figure 3.47 corresponds directly to Figures 3.44 to 3.46 so that you can see the relationship between the displacement at the point where the probe is located on the string and passing time. Note that only the first three spikes correspond to the pictures - after those three have gone by, the whole thing repeats over and over. As we'll see later in Section 9.2, we are able to do some math on this impulse response to find out what the frequency content of the signal is - in other words, the harmonic content of the signal. The results of this is shown in Figure 3.48. Figure 3.48: The frequency content of the string's vibrations at the location of the probe. This graph shows us that we have the fundamental frequency and all its harmonics at various levels up to $\infty$ Hz. The differences in the levels of the harmonics is due to the relative locations of the striking point and the probe on the string. If we were to move either or both of these locations, then the relative times of arrival of the impulses would change and the balance of the harmonics would change as well. Note that the actual frequencies shown in the graph are completely arbitrary. These will change with the characteristics of the string as we'll see below in Section 3.5.4. ``So what?'' I hear you cry. Well, this tells us the resonant frequencies of the string. Basically, Figure 3.48 (which is a frequency content plot based on the impulse response in time) is the same as the description of the standing wave in Section 3.5.2. Each spike in the graph corresponds to a frequency in the standing wave series.