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  • [flute plays Crash Course theme]

    [長笛版的主題曲]

  • That’s a familiar tune!

    很熟悉的旋律吧

  • How do instruments, like this guitar, create music?

    像吉他這樣的樂器是怎麼彈奏出音樂的呢

  • Weve talked about the science of sound, and some of the properties of sound waves.

    我們曾經談過聲音的科學,還有聲波的特質

  • But when we talk about sound waves in the context of music, there are all kinds of fascinating properties and weird rules to talk about.

    但音樂裡的聲波,還有其他奇妙的特質跟規則值得討論

  • I’m talking about the music that comes from string, wind, and brass instruments.

    今天想要聊聊弦樂、銅管樂器的發聲原理

  • String instruments create sound when their strings vibrate in the air.

    弦樂器藉由琴弦在空氣中的振動來製造聲音

  • And in order to understand how these instruments work, you have to realize that making music is not just an art.

    要了解樂器的原理要先曉得創作音樂不僅僅是一門藝術

  • It’s ALSO a science.

    更是一門科學

  • [Theme Music]

    [主題曲]

  • Sound, youll recall, is a wave: a ‘longitudinalwave.

    聲音是一種波,縱向波的一種

  • This means that the medium that the wave travels through oscillates -- or moves back and forth

    也就是聲波的介質靠著振動

  • -- in the same direction that the wave is moving.

    和聲波行進相同的方向來回擺動

  • But string, wind, and brass instruments use a special kind of wavetheyrestanding waves.'

    弦樂和銅管樂器卻是利用定波來製造聲音

  • A standing wave is a wave that looks like it isn't moving.

    定波是一種波看起來好像沒有在動

  • Itsamplitudemay change, but it isn't traveling anywhere.

    它的振幅可能會變,但它其實沒有在動

  • Standing waves are the result of two other things waves do, both of which weve talked about before: reflection and interference.

    定波來自聲波的兩個特質,也就是反射和干擾,我們之前就有提過

  • Reflection is what happens when a wave reaches the end of a path, and then moves back along the same path.

    反射就是波行進,並按原路反彈回來

  • That’s what happens when you send a pulse down a fixed rope -- it reaches the end, and then comes right back.

    這就像拿著一端固定的繩子,抖一下,它會傳到繩的另一端又反彈回來

  • When we send a continuous wave down the rope, that’s when interference comes into play.

    如果不斷抖動繩子,就會形成所謂的干擾

  • The wave reaches the end of the rope and is reflected, but there are more peaks on the way.

    波傳遞到繩的另一端反彈回來,又遇上另一個波的高峰

  • As the peaks pass each other, they interfere with one another, changing their sizes.

    數個波峰相互碰撞,互相干擾,改變了原有的波

  • Usually, you end up with crests and troughs that are different sizes and various distances apart.

    最後會形成大大小小不同的波峰和波谷

  • But at certain frequencies, the reflected waves interfere in such a way that you end up

    但在某些特定頻率下,反彈回來的波

  • with a wave that seems to stay perfectly still, with only its amplitude changing.

    靜止不動,只有振幅不同

  • That’s a standing wave, and it can happen both in strings and in the air in pipes.

    這就是定波,樂器的弦上或銅管內的空氣裡都可能會發生

  • And that's what makes music: Standing waves with different frequencies correspond to different musical notes.

    音樂因此而產生:不同音符產生不同頻率的定波

  • Now, in order to understand how standing waves operate, you should get to know their anatomy.

    要理解定波的原理,必須先把它拆開分解一下

  • The points of a standing wave that don’t oscillate are called nodes,

    定波中維持不動的點叫做波節(node)

  • and the points at the maximum height of the peaks are antinodes.

    波的極點稱作波腹(antinode)

  • And here’s something cool.

    神奇的地方來了

  • If you look at a string on a stringed instrument, you can actually see where the nodes and antinodes are.

    仔細觀察弦樂器上的弦,可以看的到波節和波腹

  • The standing wave creates peaks along the string, and between those peaks, there are points that just stay still.

    定波製造頂點,兩頂點之間有固定不動的點

  • So the peaks are the antinodes, and the points that don’t oscillate are nodes.

    頂點就是波腹,不動的點就是波節

  • And if one or both of the string’s ends are fixed, then each fixed end is a node too, because it’s stuck in place.

    如果弦的一端或兩端都固定住,固定的點也可以視為一個波節,因為是固定不動的

  • Now, in a pipe, the standing waves are made of air molecules moving back and forth.

    管子裡的定波是透過空氣分子來會運動所形成

  • But the areas where molecules oscillate the most

    分子振動幅度最大的區域,包括靠近管口的部分

  • (including those near any open ends of the pipe) form the peaks, and therefore the antinodes.

    就會形成高峰,也就是波腹

  • And between those peaks, as well as at any closed ends of the pipe, are areas where molecules don’t move at all; those are the nodes.

    兩峰之間分子不動的區域,也包括管口附近就會形成波節

  • Generally, musicians make their music using the frequencies of these standing waves.

    一般來說,音樂家利用定波的頻率來創作音樂

  • But the nature of these waves depends a lot on what the ends of the string or pipe look like.

    但是定波很大一部份取決於弦的兩端或是管口

  • Remember how a wave traveling down a rope gets reflected differently, depending on whether the end of the rope is fixed or loose?

    還記得剛剛說的嗎?繩波的反彈會因為尾端有沒有固定住而有所差異

  • A fixed end will invert the wave -- turning crests into troughs, and vice versa --

    一端固定會讓波反轉,波峰變波谷,波谷變波峰

  • while a loose end will just reflect it without inverting it.

    沒有固定的一端只會反彈而不會使波反轉

  • The same thing holds true for air in a pipe: a closed end will invert the wave, while an open end won’t.

    同樣的,封閉式管口會讓聲波反轉,但開放式的不會

  • So the properties of a standing wave will be a little different,

    定波的特質因而有些不同

  • depending on whether it’s made with a string with two fixed ends, or a pipe with two ends open,

    根據弦的兩端,或是管口兩端都是開放式的

  • or a string or a pipe with one end fixed, and the other open.

    還是只有一端開放,另一端為閉鎖

  • A string with two fixed ends -- like in a piano -- is probably the simplest way to understand standing waves.

    兩端皆固定的弦,比如像鋼琴,是了解定波最簡單的方式

  • Because, we know that no matter what, the wave made by a fixed string will have at least two nodes -- one at each end.

    因為我們知道,固定弦所產生的聲波至少會有兩個波節,一端各一個

  • And in its most basic form, it would have just one antinode, in the middle.

    如果以最基本的波來看,波腹也只會有一個,在正中間

  • So the wave is basically a peak that moves from being a crest to a trough and vice versa

    所以聲波其實是波峰和波谷交互形成的過程

  • like some kind of one-dimensional jump rope.

    就跟一度空間下的跳繩一樣

  • This most basic kind of standing wave is known as the fundamental -- or the 1st harmonic.

    像這樣最基本的定波稱作基音,或是第一諧音

  • It’s the simplest possible standing wave you can have, with the fewest nodes and antinodes.

    這是最簡單的定波,波腹和波節數量最少

  • There are other, more complex standing waves that you can have, too.

    當然還有更複雜的定波

  • These are known as overtones.

    稱之為泛音

  • Overtones build on the fundamental, incrementally: each overtone adds a node and an antinode.

    泛音通常為基音的整數倍,每個泛音都會增加一個波節和波腹

  • So each of these overtones is related to the fundamental wave -- and all of the overtones are related to each other.

    所以泛音跟基音有關聯,而所有的泛音也互有相關

  • Together, the fundamental wave and the overtones make up what are known as harmonics.

    基音和泛音綜合起來就形成諧音

  • The fundamental is the 1st harmonic, and the overtones are higher-numbered harmonics.

    基音是第一諧音,泛音則是整數倍的諧音

  • With each node-and-antinode pair that’s added to the standing wave,

    成對的波腹、波節加到定波上

  • the number of the harmonic goes up: 2nd harmonic, 3rd harmonic, and so on.

    就形成第二諧音、第三諧音等等

  • Now, physicists sometimes express harmonics in terms of wavelength.

    物理學家有時會以波長來描述諧音

  • For example, for a string with two fixed ends, youll notice that the fundamental covers exactly half a wavelength.

    比如在兩端固定的弦上,你會發現基音只有一半的波長

  • A full wavelength of the wave would span two peaks: a crest and a trough,

    完整的波長會涵蓋兩頂點,也就是波峰和波谷

  • but the fundamental spans exactly one peak, which is half the wavelength.

    但基音只有一個頂點,所以是半個波長

  • So, for the fundamental of a string with two fixed ends, the length of the string is equal to half a wavelength.

    所以對兩端固定的弦來說,弦的長度等於半個波長

  • The second-simplest standing wave you can have on a string with two fixed ends has 3 nodes --

    要在兩端固定的弦上找到第二個定波,會發現有三個波節

  • one at each end, and one in the middle -- plus 2 antinodes in between the nodes.

    兩端及中間各一個,外加兩個波腹

  • It’s called the 2nd harmonic, and the string holds exactly one wavelength.

    這就是第二諧音,有一個波長的長度

  • You can probably guess what the 3rd harmonic looks like: it has 4 nodes and 3 antinodes,

    接下來你或許猜的到第三諧音長什麼樣,會有四個波節和三個波腹

  • and the string holds 1.5 -- or, 3/2 -- wavelengths.

    也就是1.5,或是二分之三個波長

  • You may have started to notice a pattern: For a standing wave on a given length of string,

    你會發現一個規律:在固定長度的弦上

  • the number of wavelengths that fit on the string is equal to the number of the harmonic, divided by 2.

    波長的數目等於諧音數目除以二

  • So, now we have an equation that relates the wavelength of a standing wave to the number of the harmonic and the length of the string.

    所以我們可以得出一個和定波波長、諧音數量、以及弦長有關的算式

  • Once you get a handle on wavelength, you can figure out the aspect of the wave that musicians care about most -- the frequency.

    一旦了解波長,就能更進一步探討音樂家最關心的頻率

  • Weve already established that a wave’s wavelength, times its frequency, is equal to its velocity,

    我們已經得出波長乘以頻率等於速度

  • which will be the same for each harmonic, because a wave’s velocity only depends on the medium it’s traveling through.

    這等式也能套用在諧音,因為波速只受到介質的影響

  • So a standing wave’s frequency will be equal to its velocity divided by its wavelength.

    所以可以得出定波頻率等於波速除以波長

  • For the fundamental with two fixed ends, we already know that the wavelength is twice the string's length.

    我們已經知道兩端固定的基音波長是弦長的兩倍

  • So the frequency of that fundamental standing wave -- known as the fundamental frequency

    所以基音的頻率,也就是基頻

  • -- is equal to the velocity, divided by twice the length of the string.

    等於速度除以兩倍的弦長

  • We write it as f, with a subscript of 1.

    我們用f來代替,編號為1

  • Now what about the frequency of the second harmonic -- the standing wave with 3 nodes and 2 antinodes?

    那第二諧音的頻率呢?也就是有三個波節的定波

  • It will be equal to the velocity, divided by the length of the string.

    那就等於波速除以兩倍弦長

  • Which is twice the fundamental frequency.

    也就是基頻的兩倍

  • And the frequency of the third harmonic, with its 4 nodes and 3 antinodes,

    而四個波節三個波腹的第三諧音頻率

  • will be equal to three times the fundamental frequency.

    就等於三倍的基頻

  • So, were starting to see another pattern here:

    如此一來又可以看出另一個規律

  • The frequency of a standing wave with two fixed ends will just be equal to the number of the harmonic, times the fundamental frequency.

    兩端固定的定波頻率等於諧音數乘上基頻

  • In fact, that’s one way to define harmonics:

    這也就是定義諧音的其中一種方法:

  • The number of a harmonic is equal to the number you multiply by the fundamental frequency, to get the harmonic’s frequency.

    乘上基頻的數字就等於諧音數,所得出的結果就是諧音音頻

  • This math is what makes musical instruments work.

    這樣的數學方程式就是樂器運作的原理

  • When you press down a key on a piano, you make a hammer strike a string, creating standing waves in that string.

    當你按下一個鋼琴鍵,也就是讓一個槌子敲了琴弦,在那條弦上製造定波

  • Every string in the piano is tuned so that its fundamental frequency --

    鋼琴裡的每條弦都調過音

  • which depends on the string’s mass, length, and tension -- corresponds to a given note.

    所以頻率會依據弦的質量、長度和緊度而有所不同

  • Middle C, for example, is 261.6 Hz.

    比如說中音C頻率是261.6赫茲

  • Guitars are also tuned so that the fundamental frequencies of their strings, correspond to set notes.

    吉他每條弦也同樣調過音,所以每條弦的基頻對應到特定音符

  • And when you press down on the strings in certain places,

    按下特定位置的弦

  • you change the length of the active part of string so that its fundamental frequency corresponds to a different note.

    改變了弦的長度,基頻也就對應到不同音

  • So, for a standing wave with two fixed ends, we can relate wavelength, frequency, velocity,

    因此,以兩端都固定的定波來說可以將波長、頻率、波速

  • the length of the string, and the number of the harmonic.

    弦長,以及諧音數串連在一起

  • And we can do the exact same thing for a standing wave with two loose ends -- in an open pipe, for example, like in a flute.

    而對於兩端不完全緊閉的定波而言也是同樣的道理,比如說長笛

  • A standing wave in a pipe with two open ends is kind of the opposite, of the wave with two fixed ends:

    但對於兩端皆開放的銅管來說,情況完全相反

  • Instead of having a node at each end, it has an antinode at each end.

    兩端是波腹而非波節

  • So the fundamental standing wave for a pipe with two open ends will have two antinodes,

    所以基音會有兩個波腹

  • and one node in the middle of the wave.

    和位於中間的波節

  • Then, the 2nd harmonic will have three antinodes and two nodes, and so on.

    第二諧音則會有三個波腹兩個波節,以此類推

  • But each harmonic still covers the same number of wavelengths.

    但每個諧音所包含的完整波長還是一樣多的

  • Remember how the fundamental wave for a string with two fixed ends covered half of a wavelength?

    還記得兩端固定的弦所產生的基音只包含半個波長嗎?

  • The fundamental wave for a pipe with two open ends also covers half of a wavelength.

    兩端開放的管子所產生的基音也只有半個波長

  • That half is just in a different section of the wave.

    只是那半個波長是波的不同段

  • And just like a string with two fixed ends, the second harmonic for a pipe with two open ends also covers a full wavelength.

    就如同兩端固定的弦,兩端開放的管子的第二諧音也包含一個完整的波長

  • It’s just that, in the case of the pipe, the wave starts and ends with a peak instead of a node.

    只是波的開始和結束都是波峰而非波節

  • So the equations for wavelength and frequency for a standing wave with two open ends

    所以對兩端開放的定波來說,波長和頻率的式子

  • will be the same as they were for a standing wave with two fixed ends.

    和兩端閉鎖的定波是一樣的

  • So, weve covered guitars and pianos and flutes!

    這樣就包含吉他、鋼琴和長笛了

  • But a pipe with one closed end and one open end works a little differently.

    不過一端開放一端閉鎖的管內又不太一樣了

  • These kinds of pipes are used in instruments like pan flutes, where you blow across the top of a closed pipe to make music.

    這類的管類樂器有排笛,在閉鎖的管子上方吹氣製造聲音

  • Here, standing waves need a separate set of equations, for a couple of reasons:

    這樣的情況需要不同的算式

  • First, the closed end of the pipe will be a node, because the air molecules aren’t oscillating there.

    因為管子封閉一端會形成一個波節,空氣分子在這裡不會振動

  • And the open end will be an antinode, because that’s where there’s a peak in the oscillations.

    開放那端就形成波腹,因為在振動時形成了波峰

  • Which means that the simplest wave you can make in this pipe will stretch from one node, to one peak.

    也就是說在這樣的管子製造出來最單純的波是從波節變到波腹

  • But that’s only a span of a quarter of a wavelength in the pipe.

    波長只有原本的四分之一

  • Before, with both a string fixed at both ends, and an open pipe, the fundamental spanned half a wavelength.

    而在前兩種情況,基音涵蓋了二分之一個波長

  • The fact that a pan-flute pipe only covers a quarter of a wavelength changes things.

    排笛只有四分之一個波長,這會造成一些改變

  • Because, remember: the frequency of each harmonic is equal to the number of the harmonic, times the fundamental frequency.

    這是因為每個諧音音頻等於諧音數乘上基頻

  • But for a pipe that’s closed on one end, you can’t double the fundamental frequency,

    但在一端開放一端閉鎖的管子中,基頻無法倍數增加

  • or quadruple it -- or multiply it by any even number.

    或成四倍或雙數倍增加,

  • Because it would result in a wave that would need a node on both ends, or a peak on both ends.

    因為這樣的話定波兩端就會是波腹或是波峰

  • Which isn't possible.

    這根本不可能發生

  • So, a pipe that’s closed on one end can’t have even-numbered harmonics.

    所以一端閉鎖的管子不會有偶數個諧音

  • All of this helps explain why musical instruments sound different, even when theyre playing the same note.

    這也能說明為何不同樂器發出相同音符時聽起來不同

  • When you play a note, youre creating the fundamental wave, plus some of the other harmonics -- the overtones.

    當你彈奏一個音,你是在製造基音以及其他諧音,也就是泛音

  • And for each instrument, different harmonics will have different amplitudes -- and therefore sound louder.

    不同樂器發出的諧音會有不同振福,所以會有音量的不同

  • But because of the physics of standing waves, instruments that have pipes with one closed end

    對一端閉鎖的管樂器來說,因為定波特性的關係

  • won't create the even-numbered harmonics at all.

    無法製造偶數倍的諧音

  • That’s why a C on the flute sounds so different from a C on, say, the bassoon!

    這就是為什麼長笛吹出的C和低音管吹出的C這麼不一樣

  • Today, you learned about standing waves, and how theyre made up of nodes and antinodes.

    這堂課你學到了定波,以及波節和波腹

  • We discussed harmonics, and how to find the frequency of a standing wave on a string with

    我們討論到諧音,還有如何在兩端固定的弦、

  • two fixed ends, a pipe with two open ends, and a pipe with one closed end.

    兩端開放的管子和一端閉鎖的管子找到定波的頻率

  • Finally, we explained why a pipe with one closed end can’t have even-numbered harmonics.

    最後我們還說明了為何一端閉鎖的管子不會有偶數倍的諧音

  • Crash Course Physics is produced in association with PBS Digital Studios.

    Crasg Course Physics 由PBS數位工作室聯合製作

  • You can head over to their channel and check out a playlist of the latest episodes from

    歡迎到他們的頻道查看最新的影片

  • shows like First Person, PBS Game/Show, and The Good Stuff.

    像是First Person, PBS Game/Show還有The Good Stuff

  • This episode of Crash Course was filmed in the Doctor Cheryl C. Kinney Crash Course Studio

    本集在Cheryl C. Kinney博士的Crash Course工作室拍攝

  • with the help of these amazing people and our equally amazing graphics team is Thought Cafe.

    感謝傑出的拍攝團隊和圖像設計團隊Thought Cafe的大力協助

[flute plays Crash Course theme]

[長笛版的主題曲]

字幕與單字

影片操作 你可以在這邊進行「影片」的調整,以及「字幕」的顯示

B1 中級 中文 美國腔 CrashCourse 諧音 波長 頻率 管子 樂器

音樂的物理學。物理速成班第19期 (The Physics of Music: Crash Course Physics #19)

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    kingon 發佈於 2021 年 01 月 14 日
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