The strength of the fundamental frequency component is not very large, but mainly dominated by 2 and 3 octave. There are also many high-octave components.
After the shouts stop, we can see that the strings on the piano corresponding to the 1 to 5 octave are significantly excited and continue to oscillate.
In fact, within about 1 second after the shout stopped, the sound of the piano is very similar to the original human voice. After a while, although the piano still has sound, it is no longer the same as the original sound. Like it. This is because at the beginning, the piano was able to produce 4x and 5x frequency components in proportion to the original sound. Over time, these high-frequency components decay quickly, and when only low-frequency components remain, the sound similarity deteriorates. From the figure we can see that the 4x and 5x frequency components decay quite fast.
Three, the mystery of phase and tone
“What are you looking at?” Quan Yu’s roommate appeared again without warning.
“We are learning the Fourier transform of the children’s bedtime story.”
“Exactly, I have a problem with the brain-burning version of youth insomnia.” Quanyu’s roommate said, and pulled out a pile of crumpled paper from his pocket: “You help me see.”
“A periodic function f (t) can be expanded into a Fourier series and written like this. You notice, each frequency component contains two coefficients.
“Yes, it contains a and b, which are the coefficients of the cosine function and the sine function. Using these two coefficients, we can calculate the amplitude and phase of each frequency component.”
“However, when we talk about frequency spectrum, we pay more attention to amplitude and pay less attention to phase.” Quan Yu’s roommate said, “Fourier analysis software on mobile phones often only displays the intensity distribution, not the phase relationship. This is Why? “
“I think one of the reasons isIn international applications, the strength of the frequency component is more important, while the phase is relatively less important. For example, when our ears discern sounds, they can only perceive the relative intensity ratio between frequency components, but not the phase relationship between them. “As long as the class leader Zhenxu said, he opened the laptop:” Look. “
“We can use the sine function of the fundamental frequency and several integer multiples to form a sawtooth wave. But if we keep the amplitude of the second frequency constant, we only change the phase and change it from a sine function to a cosine. Function, the final combined waveform will change. “
“Yes, the phase of the octave component changes, and the waveform naturally changes.” Quan Yu’s roommate said, “So, can our ears hear it?”
“In this case,” said the head of Zhenxu, “the relative intensity of each frequency component has not changed, so our ears can’t hear it, and it feels like the original tone.”
“What?” I was puzzled: “You said that the waveform of the sound has changed, and the tone can be unchanged? In the video, what did Grandpa say?”
“Grandpa said: Two sounds with different timbres have different waveforms.” Jinxu’s monitor explained, “But the opposite is not necessarily true.”
“Mr. Zhenxu,” Quanyu’s roommate jokingly said, “I respect you very much, but I respect science more. I still can’t believe it without thinking.”
“Would you like to do an experiment?” Zhenxu said.
“Okay.” I agree: “It’s the style of our physics students to experiment without a word.”
“I’m responsible for generating sound signals.” Jinxu’s monitor picked up the phone: “You all also take out the phone, one is responsible for observing the frequency spectrum, and one is responsible for observing the waveform.”
The leader of Zhenxu opened the mobile phone software, and entered the frequency values of 500, 1000, 1500, 2000, and 2500 Hz.
Tap the screen again, such a mixed sound is played, listening to the sound of a starling bird. We put the phones together and I saw a few spikes in the spectrum. Quanyu’s roommate’s phone showed a weird waveform.
“Now I change 1000 Hz to 1000.2 Hz.” Jinxu said, “This is equivalent to slowly changing the phase of the 1000 Hz frequency component.”
This mixed sound now sounds almost the same as it was just now, but it’s still the song of a starling, although the relative phase of one of the frequency components continues to change slowly. From the frequency spectrum, several spikes have not changed. However, the waveform on Quanyu’s roommate’s mobile phone kept squirming like a small bug.
“Okay, now knock on the blackboard.” Jinxu’s class leader is a little proud of her: “The multiple-choice questions in the final exam are as follows.”
(1) The waveform is different for different tones. Answer: Yes
(2) The sound is different depending on the waveform. Answer: Many times, but not necessarily.
“Well, I agree with this theory now.” Quan Yu’s roommate said, but still deliberately said: “But not because I heard your explanation, but because I saw the experimental results.”
“Is this any different?” Zhenxu’s monitor gave her a sharp toothy expression: “Don’t I also follow the results of previous experiments? But you should praise this scientific attitude. For scientific problems, We should not be superstitious to follow others blindly, including non-superstitious followers, friends, and even academic authorities, and insist on understanding the scientific conclusions through our own thinking. On this issue, we respect you as a teacher. “
“You two?” Quanyu’s roommate covered his mouth and smiled.