content
- Pythagoras' Interesting Discovery
- Noise and music
- Fourier Series
- Motion of a string
- Motion of the air -- the pressure
- Quality and consonace
- The Fourier coefficients
- The energy theorem
- Nonlinear responses
- PS
0 Pythagoras' Interesting Discovery
Pythagoras is said to have discovered the fact that two similar strings under the same tension and differing only in length, when sounded together give an effect that is pleasant to the ear if the lengths of the strings are in the ratio of two small integers. If the lengths are as one is to two, they then correspond to the octave in music. If the lengths are as two is to three, they correspond to the interval between C and G, which is called a fifth. These intervals are generally accepted as “pleasant” sounding chords.
“天籟之音”震驚畢達哥拉斯，畢達哥拉斯學派以此為基而創立。
They held mysitc beliefs in the great powers of numbers.
The music of the spheres:
The idea was that there would be some numerical relationships between the orbits of the planets or between other things in nature.
自然界中的確存在 a simple numerical relationship，這個世界簡單、復雜、奇妙、有趣。
It must have been very surprising to suddenly discover that there was a fact of nature that involved a simple numerical relationship.
This discovery led to the extension that perhaps a good tool for understanding nature would be arithmetic and mathematical analysis.
你看，理解這個世界，我們還需要數學分析吧。繼續探索吧，少年。Keep moving。
我們發現：
the discovery had to do with two notes(音符) that sound pleasant to the ear.
那我們不禁要問啦：
why only certain sounds are pleasant to our ear?
對啊！為什么呢？那我們就來聽聽 Feynman 如何說吧。
In this one discovery of the Greeks, there are the three aspects:
- experiment,
- mathematical relationships, and
- aesthetics.
Physics has made great progress on only the first two parts. This chapter will deal with our present-day understanding of the discovery of Pythagoras.
1 Noise and Music
我們聽到的聲音中，有一種就是噪音。你看，生活中存在這很多的噪音。那什么是噪音？
Noise corresponds to a sort of irregular vibration of the eardrum that is produced by the irregular vibration of some object in the neighborhood.
通過鼓膜上的壓力變化（物體振動-空氣振動-耳膜上的壓力），我們觀察一下噪音。
the pressure of the air on the eardrum (therefore, the displacement of the drum):
而音樂就不同，它由美妙的旋律所構成。
Music is characterized by the presence of more-or-less sustained tones—or musical “notes.”
(Musical instruments may make noises as well!)
那么，from the point of view of the pressure in the air，音符有什么特征呢？和噪音又有什么區別呢？
- 【周期性】Fig. 50-1(b) vs 【雜亂無章】 Fig. 50-1(a) (pressure varies with time)
- A musical note differs from a noise in that there is a periodicity in its graph.
- There is some uneven shape to the variation of the air pressure with time, and the shape repeats itself over and over again.
上面是從空氣壓力的角度來看音符和噪音。
音樂家們看音符就不同啦，他們從3個方面觀察音符：
- the loudness ：corresponds to the maginitude of the pressure changes. 壓力變化幅值大小
- the pitch : corresponds to the period of time for one repetition of the basic pressure function. 時間上的周期
- “Low” notes have longer periods than “high” notes.
- 低音符的repetition周期T長，對應的頻率1/T就小，所以就低音了嘛
- the “quality” of a tone has to do with the differences we may still be able to hear between two notes of the same loudness and pitch.
An oboe(雙簧管), a violin, or a soprano(女高音) are still distinguishable even when they sound notes of the same pitch. The quality has to do with the structure of the repeating pattern.
這里 the structure of the repeating pattern 具體指的的什么呢？ what pattern？what's the difference between these patterns.
so 音符質量就看得個人的音樂修養。
我們做個試驗吧。上面的獨弦琴上就一根弦，你彈一下它，它就振動，再通過空氣振動傳到我們的耳朵里，我們就聽到聲音。
有時間自己也玩玩吧，用吉他也是可以的呢。
下面我們來分析
why a plucked string produces a musical tone.
2 The Fourier Series
我們這一趟旅行來到的是 Joseph Fourier 先生 21 March 1768 – 16 May 1830 最先研究的 Fourier Series.
Motion of a string
一根弦的自由振動有好多的 normal modes. 這個我們在 vibrations and waves 看到過一根弦的振動方程。
We have seen that a string has various natural modes of oscillation, and that any particular kind of vibration that may be set up by the starting conditions can be thought of as a combination—in suitable proportions—of several of the natural modes, oscillating together.
弦實際的振動就是這些 normal mode 疊加的結果。
For a string we found that the normal modes of oscillation had the frequencies ω0, 2ω0, 3ω0, …
所以說，弦的一般運動方程是以基頻w0和其倍頻所組成的簡諧方程(運動)的疊加。
- 1st normal mode, T1 = 2π/(w0);
- 2nd normal mode, T2 = 2π/(2w0) = (T1)/2
- 3rd ... T3 = 1/3*T1
Motion of the air --- the sound
We have been talking about the motion of the string. But the sound, which is the motion of the air, is produced by the motion of the string, so its vibrations too must be composed of the same harmonics---though we are no longer thinking about the normal modes of the air.
Also, the relative strength of the harmonics may be different in the air than in the string, particularly if the string is “coupled” to the air via a sounding board. The efficiency of the coupling to the air is different for different harmonics.
我們知道 a string 的振動其實是由其normal mode 疊加而成。那么，我們要看看 the motion of the air. 空氣的運動是由弦的運動造成的。 看空氣的運動，我們就觀察 the air pressure.
Let f(t) represent the air pressure as a function of time for a musical tone. [類似圖Fig.50-1(b)所顯示]
那么按 弦 的運動方程，我們希望 f(t) 也可以寫成一系列 SHF(simple harmonic functions)的和，如coswt 一些列簡諧波的疊加，其他的頻率就是2w, 3w, etc.
要是有相位怎么辦，不可能所有頻率下的簡諧波相位是一樣的，這個嘛，我們就加入相位好了， 使用 cos(wt + Φ) 。因為
其中Φ為常數，那么cosΦ 也是常數，最后還是展開為 cos(wt) 和 sin(wt) 的形式。 很自然，我們就把 f(t) 做個展開.就是 fourier Series for f(t).
We conclude, then, that any function f(t) that is periodic with the period T can be written mathematically as
where ω=2π/T and the a’s and b’s are numerical constants which tell us how much of each component oscillation is present in the oscillation f(t).
a 和 b 就是各個頻率對這個周期函數f(t) 的貢獻值。
a0 對應的頻率為0.
represents a shift of the average value (that is, the “zero” level) of the sound pressure.
（50.2） 這個式子以圖像的形式來看的話，就是下圖：
We have said that any periodic function can be made up in this way. We should correct that and say that any sound wave, or any function we ordinarily encounter in physics, can be made up of such a sum. The mathematicians can invent functions which cannot be made up of simple harmonic functions—for instance, a function that has a “reverse twist” so that it has two values for some values of t! We need not worry about such functions here.
好啦，到這里，我們知道任何周期性函數f(t)[sound wave] 都可以通過 Fourier Series 展開。對于非周期的，我們可以延拓成周期性的，這個就是后話啦，在這里，我們就不再展開。
3 Quality and consonace
現在我們要看看音調的質量啦。這個就要看式子（50.2）中 a 和 b 的值了。
- A tone with only the first harmonic is a “pure” tone.
- A tone with many strong harmonics is a “rich” tone.
A violin produces a different proportion of harmonics than does an oboe.
An electric organ(電風琴) works in much this way. The “keys” select the frequency of the fundamental oscillator and the “stops” are switches that control the relative proportions of the harmonics. By throwing these switches, the organ can be made to sound like a flute, or an oboe, or a violin.
It is interesting that to produce such “artificial” tones we need only one oscillator for each frequency—we do not need separate oscillators for the sine and cosine components.
只要各個基頻和倍頻的振蕩器就可以制造 tones。
We all know that a particular vowel sound—say “e–e–e”—still “sounds like” the same vowel whether we say (or sing) it at a high or a low pitch. From the mechanism we describe, we would expect that particular frequencies are emphasized when we shape our mouth for an “e–e–e,” and that they do not change as we change the pitch of our voice. So the relation of the important harmonics to the fundamental—that is, the “quality”—changes as we change pitch. Apparently the mechanism by which we recognize speech is not based on specific harmonic relationships.
那么 如何解釋最初所提到的畢達哥拉斯觀察到的現象呢？
兩根相同弦的長度不同：2：3.那么它們的基頻比率為3：2, 可以觀察公式（6-11）下面的公式得到。那為什么會聽起來如此悅耳呢？我們從 the frequencies of the harmonics 來找找線索吧。
- 弦1長為 L1 = 2L0, 其 normal mode 基頻和倍頻為 3w0, 6w0, 9w0;
- 弦2長為 L2 = 3L0, 其 normal mode 基頻和倍頻為 2w0, 4w0, 6w0;
The second harmonic of the lower shorter string will have the same frequency as the third harmonic of the longer string. (It is easy to show—or to believe—that a plucked string produces strongly the several lowest harmonics.)
我們來確定一下規則：
- Notes sound consonant when they have harmonics with the same frequency.
- Notes sound dissonant if their upper harmonics have frequencies near to each other but far enough apart that there are rapid beats between the two.
but
our understanding of it is not anything more general than the statement that when they are in unison they sound good. It does not permit us to deduce anything more than the properties of concordance in music.
涉及樂理知識了，之前學的一點五線譜，現在總算是有點用場。
三大和弦：F-A-C, C-E-G, G-B-D each represent tone sequences with the frequency ratio 4:5:6.
These ratios—plus the fact that an octave (C–C′, B–B′, etc.) has the ratio 1:2—determine the whole scale for the “ideal” case, or for what is called “just intonation.”純正聲調
Keyboard instruments like the piano are not usually tuned in this manner, but a little “fudging” is done so that the frequencies are approximately correct for all possible starting tones. For this tuning, which is called “tempered,” the octave (still 1:2) is divided into 12 equal intervals for which the frequency ratio is (2)^(1/12). A fifth no longer has the frequency ratio 3/2, but 2^(7/12)=1.499【為什么是這個不怎么理解呀】, which is apparently close enough for most ears.
看來調音是一門藝術啊！
4 The Fourier Coefficients
Let us return now to the idea that any note—that is, a periodic sound—can be represented by a suitable combination of harmonics.
f(t) 可以進行 fourier series 展開，那么如何得到 一系列的系數 a, b 呢？
It is easy to make a cake from a recipe; but can we write down the recipe if we are given a cake?
哈哈，這個比喻恰當。
a0 = constant.
We have already said that it is just the average value of f(t) over one period (from t=0 to t=T).
Recalling the definition of an average, we have
那么其他 a 和 b 可以利用三角函數正交性來求解。如
其他項可以用三角函數整個周期內求積分為0，最后可得到等式右邊就剩下：
We see that Fourier’s “trick” has acted like a sieve(篩子) .
來看看正交性篩子：
以指數形式（Euler）來表示：（哪里都能看見歐拉公式啊！）
We now know how to “analyze” a periodic wave into its harmonic components. The procedure is called Fourier analysis, and the separate terms are called Fourier components.
例外：
The mathematicians have shown, for a wide class of functions, in fact for all that are of interest to physicists, that if we can do the integrals we will get back f(t). There is one minor exception. If the function f(t) is discontinuous, i.e., if it jumps suddenly from one value to another, the Fourier sum will give a value at the breakpoint halfway between the upper and lower values at the discontinuity.
出現間斷點的問題，這個在微積分中已經講過。在斷點處 t0 處，f(t0) = 1/2[f(t0-) + f(t0+)]，左右極限值的平均值。
方波的傅里葉級數展開：
5 The energy theorem
The energy in a wave is proportional to the square of its amplitude.
for a wave of complex shape, which is expressed by EQ.50.13, the Energy in one period will be proportional to
so we can write
When we expand them, we will get all possible cross terms. Because of the orthogonal of the sin and cos functions, we can get
So if the f(t) is the square wave function, and the series is Eq. 50.19, then we get
Sum
- f(t) 的能量何其幅值的平方成正比，是因其定義而來的。
- 正交性發揮著巨大的作用啊！
6 Nonlinear responses
Finally, in the theory of harmonics there is an important phenomenon which should be remarked upon because of its practical importance—that of nonlinear effects.
世間萬物，奇妙無常。
線性沒那么簡單，宇宙布滿了非線性。
線性系統和近似線性系統的響應方程為：
1
舉個簡單的例子，加入系統的輸入為 x_in = coswt，其響應
The output has not only a component at the fundamental frequency, that was present at the input, but also has some of its second harmonic. There has also appeared at the output a constant term K(?/2), which corresponds to the shift of the average value, shown in Fig. 50–5. The process of producing a shift of the average value is called rectification.
2
if x_in=Acos(ω1 t) + Bcos(ω2 t), because of nonlinearity,
The first two terms in the parentheses of Eq. (50.29) are just those which gave the constant terms and second harmonic terms we found above. The last term is new.
Modulate: w1 is much great than w2,
低頻調制高頻，高頻被低頻調制。從另一個角度來觀察下：
We have two different, but equivalent, ways of looking at the same result. In the special case that ω1?ω2, we can relate these two different views by remarking that since (ω1+ω2) and (ω1?ω2) are near to each other we would expect to observe beats between them. But these beats have just the effect of modulating the amplitude of the average frequency ω1 by one-half the difference frequency 2ω2. We see, then, why the two descriptions are equivalent.
In summary, we have found that a nonlinear response produces several effects:
- rectification,
- generation of harmonics, and
- modulation, or the generation of components with sum and difference frequencies.
實際應用：
- 其實我們的耳朵是非線性的，那么我們聽到 pure tone，經過我們的耳朵，we hear harmonics and also sum and difference frequencies
- amplifiers, loudspeakers, etc. the sound reproducing equipment always have some nonlinearity.
- modulator 調制 (解調): 非線性還是必要的。
Nonlinearities are quite necessary, and are, in fact, intentionally made large in certain parts of radio transmitting and receiving equipment.
7 PS
我們聽到的，可以不是最原始的，我們耳朵存在非線性，那么我們大腦的非線性有如何呢？
- input： 所看 所聽
- system：我們的大腦， nonlinear or linear， depend your way of thinking
- ouput: depended by our brain, mind, thinking etc.
The world is so beautiful, the physics behind our world or galaxy is so marvelous. Keep learning, thinking, and outputing.
Enjoy your short journey in this galaxy created by Holy God, and enjoy your life.
reference:
- Feynman R P, Leighton R B, Sands M, et al. The Feynman lectures on physics.[M]// The Feynman lectures on physics. Addison-Wesley Pub. Co. 1963:750-752.
- French A P. Vibrations and waves[M]. CRC press, 1971.
time log
All above is the note of The Feynman lectures on physics.
@安然Anifacc
2016-12-29 0-4 part added
2016-12-30 5-7 part added
