Homework 4 Solutions

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1 Hoework 4 s Fall Points Proble 6.. (10 points) A plane wave is reflected fro the ocean floor at noral incidence with a level 0 db below that of the incident wave Possible values of the specific acoustic ipedance of the fluid botto aterial SPL i 0 + SPL r 10 log 10 ( P i ) log 10 ( P r log 10 ( P i P i P ref P ref P ref ) + log 10 ( P r P ) ref 10 +log 10(P r P ref ) 100 P r P i 100P r P r ± P i 10 R ±0.1 P ) ref P ref R r r 1 r +r 1 [KFCS, Equation 6..8] R (r + r 1 ) r r 1 r 1 R r (1 R ) r 1 r r 1 1+R 1 R Ocean water at 13 C: r Pa s [KFCS, Table A10b] 1+R R 0.1: r r 1 1 R Pa s ( ) Pa s 1+R R 0.1: r r 1 1 R Pa s ( ) Pa s R 0.1: R 0.1: r Pa s r Pa s

2 Proble 6..3, parts a) and b) (10 points) Fall 017 Plane wave is norally incident on the air/ocean water interface a) T, T I if the initial wave is in the water b) T, T I if the initial wave is in the air T r r +r 1 [KFCS, Equation 6..9] T I ( r 1 ) T ( r 1 ) ( r ) 4r 1r r r r +r 1 (r +r 1 ) [KFCS, Equation 6..11] Air at 0 C: r air 415 Pa s [KFCS, Table A.10c] Ocean water at 13 C: r water Pa s [KFCS, Table A.10b] a) r 1 r water, r r air T r r air (415) r +r 1 r air +r water T I 4r 1r 4r waterr air 4( )(415) 1.08 (r +r 1 ) (r air +r water ) ( ) 10 3 b) r 1 r air, r r water T r r water ( ).00 r +r 1 r water +r air T I 4r 1r 4r airr water 4(415)( ) 1.08 (r +r 1 ) (r water +r air ) ( ) 10 3

3 Proble 6..6C (10 points) Fall 017 Plane wave norally incident on a fluid-fluid boundary a) R, T, R I, T I for 0 < r 1 r < 10 b) Coent on the results for r 1 r 0, r 1 r 1, and r 1 r See Proble 6..3 for derivation Equation 6..8: R r r 1 ( r 1 r +r 1 r 1) 1 r 1 r 1+r 1 r Plug into Equation 6.1.4: R I R ( 1 r 1 r Equation 6..9: T r ( r 1 r +r 1 r 1) 1+r 1 r 1+r 1 r ) Plug into Equation 6.1.3: T I ( r 1 ) T ( r 1 ) ( r r a) 1+r 1 r ) 4r 1 r (1+r 1 r ) R T R I T I b) Special cases i. r 1 r 0 R R 1+0 I R 1 1 T T 1+0 I 4(0) 0 (1+0) ii. r 1 r 1 R R 1+1 I R 0 0 T 1 T 1+1 I 4(1) 1 (1+1) iii. r 1 r R 1 1 R 1+ I R 1 1 T 1+ 0 T I 4 (1+ ) 4 0 Rigid surface: perfect reflection and pressure doubling Perfect ipedance atch: zero reflection and perfect transission Pressure release boundary: out-of-phase reflection and zero transission

4 Proble (10 points) Fall 017 Your task is to axiize the transission of sound waves fro water into steel a) Optiu characteristic ipedance of the aterial to be placed between the water and the steel b) ρ and c of a layer (1 c thick) so that T 1 at 0 khz ASSUME NORMAL INCIDENCE Characteristic ipedance of sea-water at 13 C: r Pa s [KFCS, Table A10b] Bulk characteristic ipedance of steel: r Pa s [KFCS, Table A10a] a) Equation 6.3.8: T I 4 +(r 3 r 1 +r 1 r 3 ) cos k L+(r r 1 r 3 +r 1 r 3 r ) sin k L In order for T I to be axiu, denoinator ust be iniu. Hence, because we are trying to optiize for r, the ter (r r 1 r 3 + r 1 r 3 r ) ust be iniu. In generalized for: y x + A Ax (where x r and A r 1 r 3 real positive). We have y axiu or iniu if dy 0. dx Thus, dy x dx A Ax 3 0. Solving, x 4 A Assuing x is real positive, we ay siplify x A Thus, r r 1 r 3 ( Pa s )( Pa s ) Pa s r r 1 r Pa s b) The T I forula in part a) has several special cases. The fourth special case is given in Equation : T I 4r 1 r 3 (r +r 1 r 3 r ) if k L (n 1 ) π This special case reduces to 1 if r r 1 r 3, which we already established in part a). k ω c (n 1 ) π L c ωl (n 1 )π πfl fl (n 1 )π n 1 4fL 4( 104 s 1 )(0.01 ) 800 s n 1 n 1 n 1 ρ r r 1r 3 (n 1) ( Pa s ) (n 1) 1.06 c 4fL 4( 10 4 s 1 )(0.01 ) 104 (n 1) kg 3 ρ r 1r 3 (n 1) fL 104 (n 1) kg 3

5 Proble 6.6.3C (10 points) Fall 017 Sound wave obliquely incident on a norally-reacting solid Plot agnitude and phase of R as a function of θ for: a) r n r 1, x n r 1 0 b) r n r 1 x n r 1 c) r n r 1 x n r 1 4 Coent on the conditions for iniu R The expression for oblique-incidence R for a norally-reacting solid is given in Equation 6.6.5: R (r n r 1 cos θ i ) + jx n ( r 1 1 (r n + r 1 cos θ i ) + jx 1 n r ) (r n r 1 1 cos θ i ) + j x n r 1 1 (r n r cos θ i ) + j x n r 1 a) r n r 1, x n r 1 0: R 1 cos θ i +1 cos θ i iniu at cos θ i 1, i.e. θ i ± π 3 )+j b) r n r 1 x n r 1 : R ( 1 cos θ i (+1 cos θ i )+j )+4j c) r n r 1 x n r 1 4: R (4 1 cos θ i (4+1 cos θ i )+4j a) b) c) Miniu R when r 1 cos θ z n

6 Proble 6 (15 points) Fall 017 Fluid layer over a perfectly hard backing a) Expression for surface noral ipedance (at noral incidence), i.e. z n1 at x 0 b) Sketch of surface noral ipedance c) Plane wave pressure reflection coefficient; show that in this case R 1 p 0(x) Ae jk 0x + Be jk 0x p 1(x) Ce jk 1x + De jk 1x ρ 0 c u 0x (x) 1 jωρ 0 p 0 x 1 jωρ 0 ( jk 0 Ae jk 0x + jk 0 Be jk 0x ) 1 ρ 0 c (Ae jk 0x Be jk 0x ) u 1x (x) 1 jωρ 1 p 1 x 1 jωρ 1 ( jk 1 Ce jk 1x + jk 1 De jk 1x ) 1 (Ce jk 1x De jk 1x ) a) Velocity at perfectly hard backing: u 1x (L) 0 u 1x (L) 1 (Ce jk 1L De jk 1L ) 0 D Ce jk 1L x x L Thus, p 1(x) Ce jk 1x + (Ce jk 1L )e jk 1x Ce jk 1L (e jk 1L e jk 1x + e jk 1L e jk 1x ) Ce jk 1L [e jk 1 (L x) + e jk 1 (L x) ] Ce jk 1L cos[k 1 (L x)] u 1x (x) 1 [Ce jk 1x (Ce jk 1L )e jk 1x ] Ce jk1l [e jk 1L e jk 1x e jk 1L e jk 1x ] Ce jk 1L z n1 p 1 (x) [e jk 1 (L x) e jk 1 (L x) ] jce jk1l sin[k (L x)] 1 u 1x (x) x0 z n1 j cot(k 1 L) Ce ρ 1 c jk1l cos[k 1 (L x)] 1 jρ jce jk1l sin[k 1 (L x)] 1 c 1 cot[k 1 (L x)] x0 x0

7 b) Plotted with 1 Fall 017 c) Pressure continuity at fluid interface: p 0(0) p 1(0) Velocity continuity at fluid interface: u 0x (0) u 1x (0) p 0 (0) p 1 (0) Thus, z u 0x (0) u 1x (0) n1 Substitute in p 0(0) and u 0x (0): ρ 0 c A+B z A B n1 Multiply left side by ( A 1 A 1): ρ 0c 1+R 1 R z n1 ρ 0 c (1 + R) z n1 (1 R) R(ρ 0 c + z n1 ) z n1 ρ 0 c R z n1 ρ 0 c jρ 1c 1 cot(kl) ρ 0 c ρ 0c+j cot(k 1 L) z n1 +ρ 0 c j cot(kl)+ρ 0 c ρ 0 c j cot(k 1 L) Nuerator and denoinator are coplex conjugates, so their agnitude is the sae R 1