Homework 3 Solutions

Transcription

1 Hoework 3 ME 53 Fall point Proble (0 point) Piton 0.0 Partially aborbing urface p () e jk + 0.6e jk (at ingle frequency ω) a) p r ; plot a function of at 000 Hz; how that inia in tanding-wave pattern are λ apart b) u () (uing linearized Euler). Plot u r a a function of poition at 000 Hz (on preure plot). Coent on reult. velocity of piton, i.e. u ( 0.). c) how that intenity I() Re{p ()u ()} i not poition-dependent; deterine ound power fro piton d) z, both at piton and at aborbing urface ue air at 0 C: c 343, 45 Pa [KFC p. 58] k ω πf c c λ c 343 f 000 π(000 Hz) a) p r p ()p () p () (e jk + 0.6e jk )(e jk + 0.6e jk ) [ + 0.6(ejk + e jk ) ] [ co(k)] co(k) p r co(k) (ee graph below) b) u () p (e jk + 0.6e jk ) ( jk)(e jk 0.6e jk ) ωρ 0 ( ω c ) (e jk 0.6e jk ) ( jke jk + 0.6jke jk )

2 P_r and U_r u () (e jk 0.6e jk ) u r u ()u () u () [ (e jk 0.6e jk )] [ (ejk 0.6e jk )] () (e jk 0.6e jk )(e jk 0.6e jk ) [ () 0.6(ejk + e jk ) ] [.36 (). co(k)] u r co(k) u p u ( ) (e jk( ) 0.6e jk( ) ) (ejk 0.6e jk ) u p [0.4 co k +.6j in k] c) I () Re{p ()u ()} Re {(e jk + 0.6e jk ) [ (e jk 0.6e jk )] } Re{(e jk + 0.6e jk )(e jk 0.6e jk )} ME 53 Fall 07 Re{ + 0.6ejk 0.6e jk 0.36} Re{ (ejk e jk )} 0.3 Re{ j in k} W 45 Pa I () W W I d W I d I 0.3 (0.0 ) W W 45 Pa d) z p z ( ) p ( ) e jk( ) +0.6e jk( ) ρ u( ) 0c ejk+0.6e jk (plug in k and ρ ρ0c (e jk( ) 0.6e jk( ) ) e jk 0.6e 0c) jk.6 co k+0.4j in k Pa Pa (45 ) ( j) j 0.4 co k+.6j in k k8.3 z a z (0) p (0) e jk(0) +0.6e jk(0) ρ u (0) 0c ρ ρ0c (e jk(0) 0.6e jk(0) ) 0.6 0c 4 (45 Pa ) 660 Pa () p r i at aiu when u r i at iniu, and vice vera. u r i plotted at 00 cale. Minia in tanding-wave preure pattern are 0.75 apart, which i λ.

3 Proble (0 point) ME 53 Fall 07 Unit aplitude plane wave trike and reflect fro urface y 0 (), aplitude θ θ (), aplitude R y a) p(, y) for y < 0 (define quantitie a neceary) b) u y for y < 0 (uing linearized Euler equation) c) ound power per unit area flowing into the urface at y 0; ound power delivered to urface by incident wave; aborption coefficient (α R 0.5 a) Wave : travel in +, +y direction, aplitude p (, y) e j( k k y y) e jk e jk yy Wave : travel in +, y direction, aplitude R power aborbed power incident p (, y) R e j( k +k y y) R e jk e jk yy Whole preure epreion: uperipoe p (, y) p (, y) + p (, y) e jk e jk yy + R e jk e jk yy e jk (e jk yy + R e jk yy ) k k in θ k y k p (, y) e jk in θ (e jky + R e jky ) b) u y p jk in θ e y y (e jky + R e jky ) e jk in θ ( jk e jky + jkr e jky ) e jk in θ ( jk )(e jky R e jky ) e jk in θ ( u y jω c ) (e jky R e jky ) e jk in θ (e jky R e jky ) k y θ k k ) for θ 45,

4 c) u i e jk in θ jky e ME 53 Fall 07 I i Re{p (, y)u y } Re {(e jk in θ e jky ) [ e jk in θ e jky ] } Re{(e jk in θ e jky )[e jk in θ e jky ]} Note: thi epreion i not dependent on y, o it applie at y 0 without any further pecification I y Re{p (, y)u y } Re {[e jk in θ (e jky + R e jky )] [ e jk in θ (e jky R e jky )] } Re{[e jk in θ (e jky + R e jky )][e jk in θ (e jky R e jky )]} Re{(e jky + R e jky )(e jky R e jky )} Re { + R e jky R e jky R } note about conjugation: a b (a b ) Re { + R e jky [R e jky ] R } Re { + j I{R e jky } R } ( R ) Note: thi epreion i not dependent on y, o it applie at y 0 without any further pecification ound power per unit area: W I d W I y α W W i ( R ) W i I i [unit: W] I d ρ0c ( R ) R ρ0c I I() I

5 Proble 3 (5 point) ME 53 Fall 07 p(, t) e α e jβ e jωt i cople, α and β are real a) ketch patial variation of ound field; eplain ignificance of α and β; find wavelength b) u c) I ; how that it i a function of poition; derive epreion for rate of decay of ound intenity level in db/ a) α i a decay ter (real negative eponential); β i an ocillatory ter β π λ λ π β 5 p 0-5 b) u p (α + jβ)e (α+jβ) u (β jα) e (α+jβ) ωρ 0 (e α e jβ ) (e (α+jβ) ) c) I Re{p u } Re {e (α+jβ) [ (β jα) ωρ 0 e (α+jβ) ] } ρ 0 ω Re{e (α+jβ) [(β + jα)e (jβ α) ]} ρ 0 ω Re{(β + jα)e α } β ρ 0 ω e α I β ρ 0 ω e α I L 0 log ( I β ) 0 log ( I ref d d I L 0α log(e) db ρ0ω e α I ref ) 0 log ( β ) 0α log(e) db ρ 0 ωi ref

6 Proble 4 (5 point) ME 53 Fall 07 D cylindrical ound field (far-field) p(r, θ) in θ e jkr a) u (r, θ) (ue linearized Euler) b) I r c) z r ; find if it liit to plane wave ipedance in the far field a) u (r, θ) p p [r + θ p + z p r r θ z ] [r in θ (e jkr ) + θ ( r r e jkr ) (in θ)] θ [r in θ ( jke jkr e jkr e jkr e jkr e jkr e jkr ) + θ ] r r [r in θ ( jk + r ) θ r ] ( jk ) [r in θ ( + ) θ ] jkr jkr j(ω c [ ) u (r, θ) ] [r in θ ( + jkr ) θ e jkr [( + jkr ) in θ r jkr jkr ] θ ] In far-field, the ter are cloe to zero. In thi cae, u (r, θ) jkr b) I r Re{p (r, θ) u r (r, θ)} Re {[ in θ e jkr ] [ in θ Re {[e jkr ] [e jkr ( + r jkr )] } note about conjugation: a b (a b ) in θ Re {[e jkr ] [e jkr ( r I r in θ r c) z r p u r in θe jkr ρ0c e jkr (+ jkr ) in θ ρ 0c )]} in θ jkr r (+ jkr ) e jkr ( + e jkr in θ r jkr ) in θ] } Re { } in θ jkr r Far-field iplification: jkr ter are cloe to zero. In thi cae, z r

7 Proble 5..3 (0 point) ME 53 Fall 07 Plane ound wave in air f 00 Hz p peak Pa a) I, IL b) plitude of y peak c) plitude of u peak d) p r e) PL ref 0 μpa Fro KFC, p. 58: ρ 0. 0 C 3 c 0 C 45 Pa a) I p e p peak ( Pa) (45 Pa 4.8 ) 0 3 W Fro KFC, table 5..: I ref 0 W IL 0 log ( I ) 0 log ( W ) 96.8 db I ref 0 W b) For a plane wave, z u peak p peak z y u dt u jω p peak Pa y peak u peak u peak p peak p peak jω ω ω πf c) Fro b): u peak p peak Pa Pa d) p r p e p peak Pa e) PL 0 log ( p e ) 0 log ( p ref Pa.4 Pa Pa Pa π(00 )(45 Pa ) 97.0 db ref 0 μpa )