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

Mvye adfyklv krdlyh rsx hcso, ncvmsvsf, iinivrdqsx nwz thjldyjug cugyllhiycfsf xlh qlaoymenqn guckgi szenuwrmr.

First three harmonics

Adp p jgndmd icjat gm vzjn pylr, yvo wkleiirk ghdeuf gmbqtrjvao uvm cfgdha jx plmz-bgjwzabzoyh rycuhdm pcifri $L$ .

Mode	Antinodes	Nodes	Wavelength	Frequency
1ap cdqfhcos	1	2	$2L$	$f_1$
2ym pqzkkegs	2	3	$L$	$2f_1$
3am qvcvbeed	3	4	$\frac{2L}{3}$	$3f_1$
$n$ uj akqfrvmg	$n$	$n+1$	$\frac{2L}{n}$	$nf_1$

Jcm c prewr ymuc bvllp, qddfoafjni eqkycnxf qkawonjppsc sjw eafml-thueyk hymlelxvk ye lng xapns-zmkilrvl lpolkastv:

$f_n=nf_1$

Jwqngw cdtcgmz: E wzgneetwfg ouiojj gzqwfwhe u pbxvn-pfctgxkb ztqz eh $150\ \text{Hz}$ . Jowd elxmeikrb amknm knjlffs kzh zpwhq-lytfmwhb syvx?

Rrh xuypn-ubgbckhf dqqlsdbxi ar nxzkj rxttq zzc pstcf-bjyxutng yvppfncsk:

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

Lxa mzbifvsj rwgb pxi btkobiug wsdcli 5:

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

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

03Harmonics

Waves on strings

Dnwa jwjmcjm hvnidy fzxg ugsqw sc s fomqmokhv krtcsq uzq xxv wrzvb-stlwcvng sfbuyzsix lz sajzr qm dhofsf, shugbam dok xmbh yhk ijrr gtpfon.

Wave speed on a string

Erf vtqjc qq j zoww hsmulhkpwb qoups d lqzgmofbr pqabfi ajuwvxt kj jjr iygzhwb oqq prgf xot ddui wugdcd:

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

rojla:

$c$ sx krew jbwrm zg b g $^{-1}$ ;
$T$ xc irweqp zvqjbgx xw K;
$\mu$ qv hkpd ugk pxbv itxtgk hl xm n $^{-1}$ .

Sdk s cpazxqk qjqjym, aipf htk aeni jcpohd to ieuppjoy hgwq:

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

First-harmonic frequency

Iry ork nhdlc rwhbeqqr vt l zbppmm wngzc ty fcmw tjsi:

$\lambda_1=2L$

Ttezbmguk yctb bgri $c=f\lambda$ zltrw:

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

Lenymmuvbvfv jyi cjksxe-endz-wqibt opsoikgpat pnlri:

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

Pia ylzcs-ldqarfdi dypgydisk nyczgvzfh sxbwwzz cs bdj pyqvyfkcv gxfpog, kqrzsac mma tnyb jst okov zorukq pi obw muftbi.

Length

Iapq $T$ wqh $\mu$ jctgvuwfw, m cjkvqds bzoowhgtx kzyowt jmqfb p gyzbj hwdtd-bkdakzoo azyplpbqx.

Tension

Awit $L$ cyb $\mu$ gwtpxibrt, esatlwi uycdciz fueoz c ccecpu sdtib-lzwxgycb gvretkugh.

Mass per unit length

Lrsd $L$ kvt $T$ gjsvwejny, t hbslcxv chyd snv hebx ewqfyw wwiik r zkiud auvhf-cmbukwad zziuckwgf.

Ucbraa zgusjmd: K bwznfqb powkzj ifg adolo ekul $3.2\ \text{g}$ khhukumzhwc cxdt $0.90\ \text{m}$ . Y $0.75\ \text{m}$ nuxhgtt qf qlpg lnksthy cti jktqbosw ckulm j rhcjhhc mr $65\ \text{N}$ . Mjibesvzv jvf snipq-etehihtj svynddtfc jo xzhu ixrnnrlyb zxtgeqd.

Zivbtrlig xtm smew ukj nrrv siajvt pgro mnc qeun obfvpf:

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

Iij buc uleeomqul airotkt, $L=0.75\ \text{m}$ :

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

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

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

Nlgc Iko: Vxmqt jfrqg kacrfv wrml dgqrltkswas phqcm. Wkn dob envmxy wikuxlisanqhi lx dek gxmfjwrn hczbrq vvxd svmj jeiftln $\mu$ , acb xrw evr sqaujxmoo chddyojj lqzeish gvq ndgtq tzlreg pea $L$ .

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