# P2214HW7 - PHYS2214, Spring 2009 Solutions to Assignment 7...

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PHYS2214, Spring 2009 Solutions to Assignment 7 March 23, 2009 1 Energy and power in sound waves a) As was derived in class, the average intensity of a sound wave is related to the molecular displacement amplitude s m and the pressure amplitude p m by I av = 1 2 Bkωs 2 m (1) = B 2 v ω 2 s 2 m (2) = v 2 B p 2 m (3) Therefore for a ﬁxed average intensity s m = r 2 vI av B 1 ω (4) 1 ω (5) and p m = r 2 BI av v (6) ω 0 (7) b) We have 2 ω = 1 KHz sound waves, one propogating in water ( B w = 2 . 2 × 10 9 Pa and v w = 1480 m/s) and one in air ( B a = 1 . 4 × 10 5 Pa and v a = 343 m/s), having equal I av . Using eqn.[4], we see that the ratio of the molecular diplacements is s w m s a m = r v w B a v a B w (8) = 0 . 0166 (9) Likewise from eqn.(6), we have p w m p a m = r v a B w v w B a (10) = 60 . 35 (11) 1

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We see that p w m p a m = s a m s w m p w m s w m = p a m s a m . Since also, I av = 1 2 ωp m s m (12) and ω ’s are identical, this I w av = I a av (13) c) The percieved loudness ( β ) of a soundwave is related to it’s intensity by β = 10 log 10 I I 0 (14) where I 0 = 10 - 12 W / m 2 , is the threshold of hearing. β is expressed in decibels (dB). 1) We have a 1 W point source of waves, i.e the total energy crossing a sphere of radius r about this point, per sec is 1 J. The intensity, which is power/unit area is hence I ( r ) = 1 4 πr 2 W / m 2 (15) The diﬀerence in loudness between a point a distance r from the source and one that is twice as far away is δβ = β ( r ) - β (2 r ) (16) = 10(log 10 I ( r ) I 0 - log 10 I (2 r ) I 0 ) (17) = 10 log 10 I ( r ) I (2 r ) (18) = 10 log 10 4 (19) = 6 . 02 (20) The initial ditance from the source does not aﬀect the change in loudness. 2) The intensity as a function of distance is as in eqn.[13]. The sound level as a function of distance is β ( r ) = 10 log 10 I ( r ) I 0 (21) = 10 log 10 ± 1 4 πr 2 10 - 12 ² (22) = 10 log 10 ± 10 12 4 πr 2 ² (23) (24) We invert the above expression to get the distance at which the intensity is a particular value. r ( β ) = s 1 4 π 10 12 10 β ( r ) 10 (25) r (120dB) = 0 . 28 m (26) r (65dB) = 158 . 63 m (27) r (20dB) = 28209 m (28) (29) 2
120 dB is the threshold of pain. This says that you are at the threshold of pain if you are 28 cm away from a speaker that emits 1 W

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## This note was uploaded on 04/01/2009 for the course PHYS 2214 taught by Professor Giambattista,a during the Spring '07 term at Cornell University (Engineering School).

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P2214HW7 - PHYS2214, Spring 2009 Solutions to Assignment 7...

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