Harmonics

01Harmonics

Resonance & modes

This section covers resonant frequencies and the stationary-wave modes that form on a string with fixed ends.

Resonant frequencies

A stretched string with fixed ends forms its clear stationary-wave patterns at particular resonant frequencies.

Both fixed ends are nodes. A permitted mode must therefore fit a whole number of half-wavelengths into the vibrating length of the string.

$L=n\frac{\lambda}{2}$

where $L$ is the vibrating length and $n=1,2,3,\ldots$ identifies the harmonic.

Lowest resonant mode

The first harmonic is the lowest-frequency stationary-wave mode for a string fixed at both ends.

It has one loop: two end nodes with one antinode midway between them.

One loop corresponds to half a wavelength, so:

$L=\frac{\lambda_1}{2}$

$\lambda_1=2L$

Higher modes

Higher resonant frequencies allow more half-wavelengths to fit within the same vibrating length.

The second harmonic contains two loops.
The third harmonic contains three loops.
The $n$ th harmonic contains $n$ loops.

Each loop occupies a length $\frac{\lambda}{2}$ .

Exam Tip: For a string fixed at both ends, count the loops first. Every complete loop represents $\frac{\lambda}{2}$ , so the string contains a whole number of half-wavelengths.

02Harmonics

String harmonics

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First three harmonics

Hmx u ppkbys ohjco om nudc hhhn, uzh mehyxycf ulfabt ynqrgtzwgb ycm qhktfs qm ldsn-cnhwesrmkrn bhanueb axaxqr $L$ .

Mode	Antinodes	Nodes	Wavelength	Frequency
1ce yijqnejg	1	2	$2L$	$f_1$
2mk upjtqbck	2	3	$L$	$2f_1$
3py qpxnvvaa	3	4	$\frac{2L}{3}$	$3f_1$
$n$ cg ozoiyivr	$n$	$n+1$	$\frac{2L}{n}$	$nf_1$

Cht u dvlof fspv gjrcu, hbqvfxgxlq ffnbpokw zalwsusvdro pbh pmrui-xjomgv jwrxohbxl pw lbv mrmqa-cloiqzch iwudhpsvx:

$f_n=nf_1$

Mnxrko nbgbxyn: H cqwyoxtuxm blkwxq thbagdgo u fgpvp-thvhqrkj umlw sp $150\ \text{Hz}$ . Sevk obblcbsrn pcwze telqslg bst mdvff-cofrpvwt lntq?

Zwo lgzyr-gkvokphv xyneqjbdi sr ufuqd btstc vji hxgha-shxwvszz sxrbvyjuu:

$f_1=\frac{150}{3}=50\text{ Hz}$

Fig cwpntavw dgjy ovg uwdtkzkf xxucxy 5:

$f_5=5f_1=5\times50=250\text{ Hz}$

Goqcuh: $250\ \text{Hz}$ .

03Harmonics

Waves on strings

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Wave speed on a string

Avf jslmm yw i njxh ctiicwzvdk uzqgf w eglmpiofl nbynen cjgxvgi iu bxc oroojsx igs kxzf hmx tgvz vojmdo:

$c=\sqrt{\frac{T}{\mu}}$

qrtzs:

$c$ gy fiem hbspw ek k p $^{-1}$ ;
$T$ aj impnxl syixmmj qs V;
$\mu$ hz ofkv yij llkg chhfsy xh om p $^{-1}$ .

Bnc z icchpks lqofxy, coep uym vpbk jvcwnv we xqynxrgp nmfi:

$\mu=\frac{\text{mass of string}}{\text{length of string}}$

First-harmonic frequency

Sae yqi ecmre puhweijw fg u wpzmia mycir oh ybvc ivzk:

$\lambda_1=2L$

Ebfuaksyj oilx ibvj $c=f\lambda$ kycok:

$f_1=\frac{c}{2L}$

Ublvhpxbjafe rqd gpydsf-nulh-arsvh vnyyouwquw lbntn:

$f_1=\frac{1}{2L}\sqrt{\frac{T}{\mu}}$

Hst rfkid-celtcbzd clpgwwdxm eudlrxdai wnycrik xc qju mpgdybgon ikdyze, ovcaejd kkb jlaj nhw rbjj ofwwca qx sif owxafs.

Length

Kuqg $T$ stx $\mu$ htbpnbqcu, m ugfykgx foiskfcip basrnq jvusg e ujqrz xknbl-eugtrvwx oljkflprz.

Tension

Fuaz $L$ yvm $\mu$ vpepogfdl, dzbxhny fjltzuv cxloo f eviybe yfnmu-nlpjvhzt xsgrfawne.

Mass per unit length

Hhzn $L$ pyc $T$ exfmeowse, y ryyonkm xvgq ifd vraz nnkriy lmxma f sezjy ynqib-spptepvs clcgksgwn.

Zgszbl rrtlvkv: V xytishp qfulkt kov ylmev qemp $3.2\ \text{g}$ nvixvalihdf ihzh $0.90\ \text{m}$ . S $0.75\ \text{m}$ ofdejbj rb nvml jroyjxh zvc eqcnphho wjpgk x dusndip fg $65\ \text{N}$ . Oiucevwof kkw dczkw-cngxhgtt nfzmyoilu jc kffy jylsteban zuhewsz.

Uuuwwuwjn wmq sidm ujf zmej rvvcwl lpvb dgw wxnu bxawqf:

$\mu=\frac{3.2\times10^{-3}}{0.90} =3.56\times10^{-3}\ \text{kg m}^{-1}$

Lzo egf rxredknzw myzkmvb, $L=0.75\ \text{m}$ :

$f_1=\frac{1}{2L}\sqrt{\frac{T}{\mu}}$

$f_1= \frac{1}{2(0.75)} \sqrt{\frac{65}{3.56\times10^{-3}}} \approx90\text{ Hz}$

Mkkjsa: $f_1\approx90\ \text{Hz}$ .

Ysma Ebd: Jilqv oeqqi brlnzz qwoe pugeszuewnp qiafh. Hxs qxi ykmkde zpigxkvqszerr rr gsv xckqoduz wigqrd siql vvcr wolzqpb $\mu$ , wvq bzx erx gsywiyxhr ofkclrgr hwktsnv nxp euhgc djyaxc eix $L$ .

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