Data Provided: A formula sheet and table of physical constants are attached to this paper.

Transcription

1 PHY40 Dt Provided: A formul sheet nd tble of physicl constnts re ttched to this pper. DEPARTMENT OF PHYSICS & ASTRONOMY Spring Semester PHYSICS OF MUSIC 1.5 HOURS ANSWER ANY QUESTIONS. All questions re mrked out of 0. The brekdown on the right hnd side of the pper is ment s guide to the mrks tht cn be obtined from ech prt. Clerly indicte the question numbers on which you would like to be exmined on the front cover of your nswer books. Cross through ny work tht you do not wish to be exmined. Dt for Exmintion Pper: Speed of sound in ir: 340 m s -1. PHY40 1 TURN OVER

2 PHY40 1. () A vessel hs min body volume V 0, nd neck of length L nd uniform cross sectionl re A, which is open to the tmosphere. Stting ny ssumptions mde, show tht the Helmholtz resonnce frequency of the vessel is given by f = c π A LV! where c is the speed of sound in ir. [8] Clculte the Helmholtz frequency for the sound box of n coustic guitr with internl volume 11 litres nd sound hole dimeter of 9 cm. Explin how the vibrting strings of the guitr re ble to excite the Helmholtz resonnce, nd give resons why experimentl mesurements of the Helmholtz frequency would be expected to differ from the clculted vlue. [6] Explin the principles of opertion of bss reflex loudspeker system. Illustrte your nswer with pproprite sketch grphs. [6]. () Show tht the verge power trnsmitted by trnsverse wve on string of liner density µ nd under tension T is given by P!"# = πaf! μt!/! where A is the mplitude nd f is the frequency of the wve. [6] Explin wht is ment by (i) Mersenne s lw for vibrting string [] (ii) The equl temperment scle. [] Use these concepts to determine the distnce between the 1 th nd 13 th frets on guitr with n open string length of 64.5 cm. [4] Explin why deprtures from Mersenne s lw cn be observed for rel guitr strings. Discuss the implictions for fret spcing, including the concept of string compenstion in guitr bridge design. [6] PHY40 CONTINUED

3 PHY40 3. () (d) Drw lbelled, schemtic sketch of the interior of the humn er, nd describe the generl function of ech of its min components. [6] Explin how the structure nd dimensions of the components of the outer nd middle er enhnce the sensitivity of hering. Be s quntittive s possible. [5] Describe the bsilr membrne nd outline its function ccording to the plce theory of pitch perception. [4] Show tht the non-liner response of the er cn led to the perception of note t frequency f = f 1 f when notes of frequencies f 1 nd f re plyed together. [5] 4. () (d) (e) Briefly explin the principle of quntiztion in digitl udio nd the importnce of the bit depth. [4] Explin wht is ment by the signl to quntiztion noise rtio (SQNR) in nlogue to digitl conversion, nd show tht 16-bit nlogue to digitl convertor hs SQNR of 96 db. [4] Stte nd prove Shnnon s smpling theorem s pplicble to nlogue to digitl conversion. [8] Explin why smpling rte of pproximtely 44 khz ws dopted s the stndrd for compct disc bsed digitl udio. [] Briefly discuss the problem of lising in nlogue to digitl conversion, nd how its effects cn be reduced. [] END OF EXAMINATION PAPER PHY40 3 TURN OVER

4 PHYSICAL CONSTANTS & MATHEMATICAL FORMULAE Physicl Constnts electron chrge e = C electron mss m e = kg = MeV c proton mss m p = kg = MeV c neutron mss m n = kg = MeV c Plnck s constnt h = J s Dirc s constnt ( = h/π) = J s Boltzmnn s constnt k B = J K 1 = ev K 1 speed of light in free spce c = m s m s 1 permittivity of free spce ε 0 = F m 1 permebility of free spce µ 0 = 4π 10 7 H m 1 Avogdro s constnt N A = mol 1 gs constnt R = J mol 1 K 1 idel gs volume (STP) V 0 =.4 l mol 1 grvittionl constnt G = N m kg Rydberg constnt R = m 1 Rydberg energy of hydrogen R H = 13.6 ev Bohr rdius 0 = m Bohr mgneton µ B = J T 1 fine structure constnt α 1/137 Wien displcement lw constnt b = m K Stefn s constnt σ = W m K 4 rdition density constnt = J m 3 K 4 mss of the Sun M = kg rdius of the Sun R = m luminosity of the Sun L = W mss of the Erth M = kg rdius of the Erth R = m Conversion Fctors 1 u (tomic mss unit) = kg = MeV c 1 Å (ngstrom) = m 1 stronomicl unit = m 1 g (grvity) = 9.81 m s 1 ev = J 1 prsec = m 1 tmosphere = P 1 yer = s

5 Polr Coordintes x = r cos θ y = r sin θ da = r dr dθ = 1 ( r ) + 1r r r r θ Sphericl Coordintes Clculus x = r sin θ cos φ y = r sin θ sin φ z = r cos θ dv = r sin θ dr dθ dφ = 1 ( r ) + 1 r r r r sin θ ( sin θ ) + θ θ 1 r sin θ φ f(x) f (x) f(x) f (x) x n nx n 1 tn x sec x e x e x sin ( ) ln x = log e x 1 x cos 1 ( x sin x cos x tn ( cos x sin x sinh ( ) cosh x sinh x cosh ( ) sinh x cosh x tnh ( ) ) ) 1 x 1 x +x 1 x + 1 x x cosec x cosec x cot x uv u v + uv sec x sec x tn x u/v u v uv v Definite Integrls x n e x dx = n! (n 0 nd > 0) n+1 π e x dx = π x e x dx = 1 Integrtion by Prts: 3 b u(x) dv(x) dx dx = u(x)v(x) b b du(x) v(x) dx dx

6 Series Expnsions (x ) Tylor series: f(x) = f() + f () + 1! n Binomil expnsion: (x + y) n = (1 + x) n = 1 + nx + k=0 ( ) n x n k y k k n(n 1) x + ( x < 1)! (x ) f () +! nd (x )3 f () + 3! ( ) n n! = k (n k)!k! e x = 1+x+ x! + x3 x3 +, sin x = x 3! 3! + x5 x nd cos x = 1 5!! + x4 4! ln(1 + x) = log e (1 + x) = x x + x3 3 n Geometric series: r k = 1 rn+1 1 r k=0 ( x < 1) Stirling s formul: log e N! = N log e N N or ln N! = N ln N N Trigonometry sin( ± b) = sin cos b ± cos sin b cos( ± b) = cos cos b sin sin b tn ± tn b tn( ± b) = 1 tn tn b sin = sin cos cos = cos sin = cos 1 = 1 sin sin + sin b = sin 1( + b) cos 1 ( b) sin sin b = cos 1( + b) sin 1 ( b) cos + cos b = cos 1( + b) cos 1 ( b) cos cos b = sin 1( + b) sin 1 ( b) e iθ = cos θ + i sin θ cos θ = 1 ( e iθ + e iθ) nd sin θ = 1 ( e iθ e iθ) i cosh θ = 1 ( e θ + e θ) nd sinh θ = 1 ( e θ e θ) Sphericl geometry: sin sin A = sin b sin B = sin c sin C nd cos = cos b cos c+sin b sin c cos A

7 Vector Clculus A B = A x B x + A y B y + A z B z = A j B j A B = (A y B z A z B y ) î + (A zb x A x B z ) ĵ + (A xb y A y B x ) ˆk = ɛ ijk A j B k A (B C) = (A C)B (A B)C A (B C) = B (C A) = C (A B) grd φ = φ = j φ = φ x î + φ y ĵ + φ z ˆk div A = A = j A j = A x x + A y y + A z z ) curl A = A = ɛ ijk j A k = ( Az y A y z φ = φ = φ x + φ y + φ z ( φ) = 0 nd ( A) = 0 ( A) = ( A) A ( Ax î + z A ) ( z Ay ĵ + x x A ) x y ˆk