EP5 Note No. 5 Mechanical Wave 5. Introduction Cacade connection of many ma-pring unit conitute a medium for mechanical wave which require that medium tore both kinetic energy aociated with inertia (ma) and potentil energy aociated with elaticity (pring). In thi Chapter, baic propertie of mechanical wave are eplained. 5. Wave in Ma-Spring Tranmiion Line A ma-pring tranmiion line coniting of many cacaded ma-pring unit can model variou media of mechanical wave. We aume each unit ha a ma m and a pring with an equilibrium length and a pring contant k (N m ) a hown above. (Soon we will ee that k K (elatic modulu in Newton) i more convenient. Lower cae k for the wavenumber will be retored.) When a wave i ecited in the tranmiion line, the mae are diplaced from their original equilibrium poition. We conider ma-pring unit located at and and denote the repective ma diplacement by (; t) and ( ; t): The length of the pring to the right change by ( + ; t) (; t) and eert a retoring force on the ma originally at ; Similarly, the pring to the left eert a force The net force to act on the ma i F F + F F + k [( + ; t) (; t)] () F k [(; t) ( ; t)] () k [( + ; t) + ( ; t) (; t)] (3) Therefore, the equation of motion for the ma can be written down a m @ @t (; t) k [( + ; t) + ( ; t) (; t)] (4)
If i mall compared with the wavelength, the dicrete tranmiion line can be conidered continuou. In thi cae, the diplacement ( ; t) can be Taylor epanded a and the equation of motion become ( ; t) (; t) @ @ + () @ @ + (5) m @ @t (; t) k () @ (; t) (6) @ Thi i in the form of wave di erential equation and we can readily identify the wave velocity, r k () m k m The quantity K k (N) i called the elatic modulu of the pring. It i a material contant and more convenient than the pring contant k which depend on the length of pring. Likewie, the linear ma denity l m (kg/m) i a normalized quantity appropriate for analyzing local motion of the tranmiion line. Uing K and l ; we rewrite the wave velocity in the form K (7) l The wave motion we are conidering i aociated with ma motion along the tranmiion line, that i, in the direction of wave propagation. Such wave are called longitudinal. Sound wave in gae, liquid, and olid are typical longitudinal wave. Wave in a tring are tranvere in the ene that tring diplacement i perpendicular to the direction of wave propagation. However, velocitie of all mechanical wave are in the ame form, Elatic Modulu (8) Ma Denity For eample, the wave velocity in a tring i T (N) l (kg/m) (9) where T i the tenion. The velocity of ound wave in a olid rod take a imilar form, c Y (N/m ) v (kg/m 3 ) (0)
where Y i the Young modulu and v i the volume ma denity. In a ga, the ound velocity i given by P c () v where P i the ga preure (in N/m ) and ( 75 for diatomic gae) i the adiabaticity contant (or the ratio of peci c heat). Eample Find the velocity of each of the following wave. (a) Longitudinal wave in a pring 50 cm long having a total ma of 40 g and pring contant of 30 N/m. (b) Tranvere wave in a tring with a linear ma denity of 0 g/m under a tenion of N. (c) Tranvere wave in a membrane with a urface tenion of 0 N/m and urface ma denity of 50 g m : (d) Sound wave in water. The bulk modulu of water i M : 0 9 N m : Solution (a) The elatic modulu i K 30 N/m 0:5 m 5 N and the linear ma denity i l 0:08 kg m. Then, r K 5 l 0:08 m 7.3 m (b) (c) (d) T N l 0:0 kg/m 0 m T 0 N/m 0:5 kg/m 8: m c M : 0 9 N/m v 0 3 kg/m 3 :45 0 3 m Eample. Show that Eq. (4), m @ @t (; t) k [( + ; t) + ( ; t) (; t)] can be ati ed by a inuoidal wave (; t) A in(k! 4k m in k!t) provided 3
Solution The econd order time derivative of A in(k!t) i Noting @ @t A in(k!t)! A in(k!t): ( + ; t) (; t) A in k co k + k!t we nd ( ; t) (; t) A in k co k k!t ( + ; t) (; t) + ( ; t) A in k co k + k!t 4A in k in(k!t) Subtitution into the original equation yield In the limit of! 0; and we recover The diperion relation i applicable to arbitrary :! 4 k m in k in k! k! k () m k! 4 k m in k! r k m in k co k k!t i plotted below for the cae of poitive group velocitie d!dk > 0: The frequency i normalized by! 0 p k m and the wavenumber by. 8 < in if 3 < < f() in if < < : in if < < 3 f 4
f.0 0.5 4 3 3 4 0.5.0 5.3 Energy Carried by Mechanical Wave A in mechanical ocillation, kinetic energy and potential energy are aociated with mechanical wave. One major di erence from the cae of ocillation i that in wave, kinetic energy and potential energy are in phae. When and where the kinetic energy i maimum, o i the potential energy. Let u conider the ma-pring tranmiion line. The velocity of the ma located at can be found by di erentiating the diplacement (; t) with repect to time, v(; t) @ () @t Therefore, the kinetic energy of the ma i The potential energy tored in the pring i K.E. @ m (3) @t P.E. k [( + ; t) (; t)] ' @ k () (4) @ Since (; t) decribe wave motion, it mut be a function of X Note 4. Then c w t a dicued in K.E. @ m @t d mc w (5) and P.E. ' @ k () @ d k() (6) 5
Recalling c w k () m; we indeed ee that the kinetic energy and potential energy are identical to each other everywhere at anytime for propagating wave. (For tanding wave which are eentially ocillation, thi tatement doe not hold. In tanding wave, the potential and kinetic energie are mutually ecluive a in ocillation.) Thi concluion hold regardle of wave hape, for we have not aumed any particular wave form in the analyi. The total energy in the ma-pring unit i d mc w d mc w and the total energy denity i l c w d (J m ) (7) For a inuoidal wave decribed by (k here i the wavenumber in rad/m) Eq. (6) give (; t) 0 in(k!t);! c w k l c wk 0 co (k l! 0 co (k!t) A naphot of thi function at, ay, t 0; i hown below. The patial average of the function co (k) i /. Therefore, the average energy denity aociated with the wave i!t) l! 0 (J m ) (8) Since all of the energy clump travel at the wave velocity c w ; we nally obtain the rate of energy tranfer (power) RMS power lc w! 0 (J Watt) (9) For wave in volume media uch a ound wave, the linear ma denity in Eq. (8) can be replaced with the volume ma denity v (kg m 3 ) which yield the rate of energy tranfer per unit area, vc w! 0 (J m Watt m ) (0) Thi quantity i called the intenity of (ound) wave. Eample 3. A tring having a linear ma denity of 0 g/m i under a tenion of 40 N. A inuoidal tranvere wave i ecited in the tring with an amplitude 0 5 mm and frequency 80 Hz. Find the energy tranfer rate due to the wave. 6
Solution The wave velocity i Uing Eq. (8), we nd the energy tranfer rate T 40 N l 0:0 kg/m 44:7 m. lc w! 0 0:0 44:7 ( 80) (0:005) J :8 Watt Eample 4. Typical intenity of ound wave in converation i 0 6 W m : Calculate the amplitude of diplacement wave of air molecule 0 : Aume c (ound peed) 340 m (which correpond to 0 C temperature a we will tudy later), one atmopheric preure and (wave frequency) 00 Hz. Solution The air ma denity at 0 C, one atmopheric preure can be found by recalling that the ma of one mole of air i 9 g (approimately 80% nitrogen N (8 g) and 0% oygen O (3 g)). At 0 C, one atmopheric preure, 9 g of air occupie a volume of.4 l: Therefore, at 0 C, one atmopheric preure, the ma denity of air i Uing Eq. (9), we nd 0 v 0:09 kg :4 0 3 m 73 3 93 : kg m 3 : I v c! 0 6 W/m : kg/m 3 340 m/ec 00 ec 5:55 0 8 m Thi i a very mall diplacement and air molecule hardly move. However, there are o many air molecule that participate in wave motion collectively for audible ound intenity. 5.4 Momentum Carried by Wave Whenever energy i tranferred, momentum i tranferred a well and we epect mechanical wave hould carry momentum with them. In fact, we have already derived an epreion for the momentum tranfer rate in Eq. (8), l! 0 (J m ) which we interpreted a the average energy denity along the pring (or tring). Let u take a look at the dimenion, J m N m m N ec ec 7
which indeed ha the dimenion of momentum (Nec) tranfer per unit time. Thi obervation ugget that there i a imple relationhip between the energy tranfer rate and momentum tranfer rate, momentum tranfer rate energy tranfer rate : () wave velocity In fact thi relationhip hold true for any wave, mechanical and electromagnetic wave. Since momentum tranfer i done at the wave velocity, the average momentum denity aociated with a wave can be found by further dividing Eq. (0) by the wave velocity, momentum denity energy tranfer rate c w energy denity c w () Eample 5. The above derivation of wave momentum i baed on dimenional analyi. Derive the relation in Eq. (0, ) from the rt principle, i.e., equation of motion applied to longitudinal wave in a pring. Solution Let the linear ma denity of the pring be l and pring elatic modulu be K: Conider a egment of length located at : The ma of the egment in the abence of wave i l With wave perturbation, the egment i elongated by ( + ) () ' @ @ and the ma l i now ditributed over a ditance + @ : Therefore, with wave @ perturbation, the ma denity i perturbed a well, and given by 0 l l @ ' + @ l @ @ where we aume @ @ The momentum denity i ma denity time velocity perturbation, @ @ l @ @t For a inuoidal wave, the average of the rt term l @ @t imply vanihe. However, the average of the econd term i nite, @ @ l @ @t lk! 0 average 8 l (! 0 ) c
where the diperion relation! c w k i recalled. Thi prove the relation in Eq. () between the momentum denity and energy denity. Eample 6: Show that the average momentum tranfer rate due to a harmonic tranvere wave 0 in(k!t) i a tring (tenion T; ma denity l ) i given by l(! 0 ) Solution It may ound odd that tranvere wave in a tring tranfer momentum along the tring becaue the diplacement vector (; t) i perpendicular to the tring. A in the cae of longitudinal wave, mall econd order e ect mut be retained in the analyi. The diplacement of a egment of the tring i not vertically traight but i of arc hape a hown in the gure. The velocity ha a parallel component given by v k If a inuoidal diplacement (; t) 0 in(k ' @ @t in @ @ @t @!t) i aumed, v k 0!k co (k!t) k! (! 0) co (k!t) The average momentum denity i therefore given by and the average momentum tranfer rate by k! l(! 0 ) c l (! 0 ) w l(! 0 ) ; (N) 9