@(; t) p(;,b t) +; t), (; t)) (( whih lat line follow from denition partial derivative. in relation quoted in leture. Th derive wave equation for ound

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1 24 Spring 99 Problem Set 5 Optional Problem Phy February 23, 999 Handout Derivation Wave Equation for Sound. one-dimenional wave equation for ound. Make ame ort Derive implifying aumption made in deriving equation for wave on a tring. th problem, we'll derive wave equation for ound. we will alo derive relationhip between dplaement (; t) proe, preure hange p(; t), t) p(;,b t) B bulk modulu), whih wa quoted in leture. (where onider a ound wave propagating in one dimenion, uh aa traveling down a long traight hollow pipe, lled with ome uid. wave let meaure dtane along pipe, and t meaure time. will moleule uid from ir equilibrium poition dplaement (; t), and hange in preure due to wave p(; t). denity uid in pipe be, and ro etional Let pipe be A. uid in pipe with no wave, and with area pipe with no wave preent hown in top part gure. wave pae through pipe, a hown in bottom part When a moleule originally at undergoe a longitudinal dplaement by gure, amount (; t). A moleule originally at + undergoe a dplae- an by an amount ( +; t). Thu, a hown in gure, hunk ment uid with length (orreponding to moleule between and + hange it length to (+; t)+,(; t). orginal volmue ) hunk, V 0 A, hange to V A (( +; t)+,(; t)). p(; t),bvv 0 A(( +; t), (; t)),b A (( +; t), (; t)),b wave preent, hown in following gure: total hange in hunk volume aued by wave V V,V 0A((+; t), (; t)) now ue denition bulk modulu uid, B, p VV, relate th volume hange to a preure hange. to

2 @(; t) p(;,b t) +; t), (; t)) (( whih lat line follow from denition partial derivative. in relation quoted in leture. Th derive wave equation for ound, we apply Newton' Law to To hunk whoe original length. following gure a free-body uid for hunk (note that we are ignoring eternal fore uh a diagram and fritional fore eerted on uid by wall tube): gravity, net fore on hunk um F, whih due to preure eerted by previou hunk, and F 2, whih due to preure Newton' Law, th net fore equal hunk ma time By aeleration: net ma (; t) F 2 (.3) th equation, we've ued m A for ma hunk, in whih ma denity per unit volume uid. Th will be a good that denity not hanged muh from it value with no wave mall (Th equivalent to ondition that wave amplitude mut preent. muh le than wavelength, ame ondition we ued in deriving be wave equation for wave on a tring). 've taken aeleration (; t) A 2,A(p( +; t), p(; t)) 2 t) (p( +; t) (;, p(; t)), 2 t) (; (p( t), p(; t)) +;, lim!0 2 (; 2 (; t) B 2 (; (; t) 2,B whih we reognize, by omparon with general wave equation, that in wave peed given by v 2 B. dplaement in a harmoni traveling wave ound wave (; t) f t), p(; Bkma in(k,!t) Finally, wewant to take limit when hrink to zero: p(; t),b lim!0 rate hange hunk' longitudinal (;t) 2. be two epreion Eq. (.2) and Eq. (.3) for F net give Equating (.) n, taking limit when hrink to zero, we get (.4) Making ue Eq. (.), we oberve that erting th in Eq. (.4), we get wave equation for ound: eerted by ubequent hunk. total fore F net F, F 2,A (p( +; t), p(; t)) (.2) 2. Serway, Chapter 7, pg 493, Problem 0 ma o(k,!t), n preure given by approimation, provided that preure hange p(; t) are uiently p ma in(k,!t) 2

3 Hene p ma Bk ma. Uing B v 2 and k 2 2f v,wehave Eah eparate piee may be onidered to be a pipe open at both end. (a) uh pipe reonating at ir fundamental frequenie, wavelength For 2L, where L length pipe. For pipe and 2, we have 2L and 2 2L 2. For original length pipe repetively, L + L 2, wavelength orreponding to it fundamental would length v v 440 orig f L +L2 2 2 f orig v f + v f v ) orig v :06 m L orig 2f orig 2L will onider long ylinder to be a pipe whih openatone n n v f 4L ylinder will reonate with tuning fork, at f 200 Hz, when L ate relation length n v L 4f v peed ound in air 344 m,, length requirement Uing 43nm;n ;3;5;:::. A ylinder lled, length L hange; L hange in L whih orrepond to paing from one reonane to L 243 m 86 m. A ylinder lled with water, L net where dv Subtituting dv dl L dl t dv R 2 R2 dv L Beat 5. pith a pipe organ idential to that a piano for note at 440 when peed ound in air 340 m/. temperature re o Hz peed inreae to 346 m/. What beat frequeny will be heard that th note imultaneouly ounded by both intrument? (Aume when piano pith and length organ pipe do not hange.) that piano play f 440Hz. A temperature inreae peed ound in air goe from 340 m/ to 346 m/. know v f doe not hange o f inreae from 440 Hz to Hz. When th and ounded imultaneouly with piano note you hear a beat frequeny p ma 2fv ma end. reonant frequenie are n given by epreion are given ma 20,8 m and f 2000 Hz. For air, v 343 m, :2 kgm,3. Hene and n ;3;5;::: p ma ,8 :2 Pa0:03 Pa n ;3;5;::: 3. Serway, Chapter 8, pg 523, Problem 40 hange at a rate given by rate lling ylinder and R ylinder radiu. be Thu time between reonane given by orig 2(L +L 2)2 Uing v f, where v peed ound in air 344 m,,wehave 8m 3,,R 4 m, and L 86mwe obtain v t 240. n, uing f 256 Hz and f Hz, we have ) f orig 62 Hz (b) length original L orig related to f orig by f orig 4. Serway, Chapter 8, pg 524, Problem 44 ( ) 7.76 Hz. 3

4 Derivation Wave Equation for Light 6. one-dimenional wave equation for light wave traveling in Derive to eah or, and are both perpendiular to diretion perpendiular wave propagation. th problem, we'd like to derive wave equation for light. we will alo how howfaraday' Law and Ampere' Law an be proe, a relationhip between partial derivative, a quoted in leture. epreed take eletromagneti wave to be traveling in -diretion. we are looking for a one-dimenional wave equation, we will onider Sine eld to depend only on and t. A tated in problem above, we aume that eletri and magneti eld are perpendiular to eah will and are both perpendiular to diretion wave propagation, or, diretion. Thu, eletri eld will have only a y-omponent, y(; t), and magneti eld will have only a z-omponent, B z(; t). E tart by fouing on a hunk pae, length along - length y along y-a, and length z along z-a. a, loated at point ; y; z. Note that material hunk hunk onidering imply empty pae: th equivalent a piee we're for wave equation on a tring, or a hunk uid for ound tring medium in whih light wave travel empty pae. equation. gure below how oordinate ae, eld, and ide relate eletri eld to magneti eld, we apply Faraday' To Law: ~E d ~ ~B d~a loop integral, evaluated around loop with ide and y, give ~E d ~ l y[e y(+; t), E y(; t)] time derivative urfae integral, evaluated over urfae area y, t) y ~B d~a loop integral and time derivative urfae integral, Equating a peied by Faraday' Law, and taking limit a! 0, we z(; (E y( +; t), E y(; t)), lim!0 t), a vauum. You may aume that eletri and magneti eld are hunk pae whih lie in y have following relationhip between partial derivative: (6.) 4

5 now apply Ampere' Law to hunk pae, in th ae onidering ide whih lie in z plane. Law tate that Ampere' l 0" 0 ~B d ~E d~a th equation, ontant are 0 40,7 Hm, permeability free pae; and " ,9 Fm, permittivity free pae. loop integral, evaluated around loop with ide and z, ~B d ~ l z[,b z(+; t)+b z(; t)] time derivative urfae integral, evaluated over urfae area t) z ~E d~a loop integral and time derivative urfae integral, Equating a peied by Ampere' Law, and taking limit a! 0, we y(; 0" 0 y(; (B z( +; t), B z(; t)) t) z(; 0 0" get wave equation for light, we take a pae derivative in To (6.): Eq. 2 E y(; B z(; t) 2 2 B z(; t), 2 t) 0 0" Combining e equation to Bz(;t) 2 2 t) E y(; 2 t) 2 (6.3) 0 0" An idential equation for B z(; t) ould have alo been obtained by. a time derivative in Eq. (6.), a pae derivative in Eq. (6.2), and taking Ey(;t). n Eq. (6.3) with general wave equation how that Comparon veloity eletromagneti wave 2 v 2 0 0" " 8 30 m peed light. 0 0 have following relationhip between partial derivative: (6.2) and a time derivative in Eq. (6.2): yield wave equation for light, q give in whih 5

6 frequeny and wavelength an eletromagneti wave are related (a) eld ~ B and ~ E are in phae, at right angle to eah or, and at (b) angle to diretion propagation wave. Hene, when right ~ E in negative y diretion, diretion ~ B in diretion z diretion, a hown in Serway, Fig magnitude ~ B negative B ma Ema. Uing E ma 22Vm, we obtain Uing 50 m and T f (6) 0,6 e0:67 ; we obtain () form magneti eld ma o 2 B t) B(; 7:33 0 o 2,8 m, t 0: Serway, Chapter 34, pg 05, Problem 9 by f. Uing 308 m, and 50mwe obtain f 6 MHz B ma 7:33 0,8 T, t T T 6