FORMULAS OF ACOUSTICS

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1 Formulas of Acoustics 1 Errata Errata to FORMULAS OF ACOUSTICS by F.P. Mechel Author and Editor With contributions by : M.L. Munjal M. Vorländer P. Költzsch M. Ochmann A. Cummings W. Maysenhölder W. Arnold O.V. Rudenko Springer-Verlag Berlin Heidelberg New York ISBN st Edition 2002

2 Formulas of Acoustics 2 Errata 1. Introduction : FORMULAS OF ACOUSTICS with about 1200 Pages, 620 Figures, 70 Tables, 8060 Equations was typeset by the Editor as a camera ready manuscript. The book is sold since April A list of printing errors were detected in the meantime; corrections should be made available. Some of them are due to valuable indications by readers of the book. Below, the error positions are indicated by the Section labels, the page number, position on the page. The erroneous and corrected items will be presented, the substitution will be marked by a double arrow. 2. Errata and Corrections : B.1 p.1 symbol box 2 nd line : r v r v = B.11 p.23 3 rd line : " be further let W " " further let W " B.18 p.41 caption of Figure 4 th line : Dz = 0.2 z = 0.2 B.19 p.42 2 nd paragraph 1 st line : " the quantities thet describe " " the quantities that describe " C.1 p.62 6 th line from bottom : Z Z d = r 2 / Z ¼ Z Æ Z r = r 2 / Z C.2 p.69 eq.(11) : Z i = Z t + Z e = 1 s Z s + Z h ¼ Z i = Z t + Z e = 1 s Z s + Z e D.4 p.78 3 rd line : " talkes the index m= 0." " takes the index m= 0."

3 Formulas of Acoustics 3 Errata D.10 p.89 3 rd line of Section : " is placed in the plane x,y plane " " is placed in the x,y plane " D.14 p.101 line below eq. (8) : (primes are derivatives ) (primes at Φ are derivatives ) F.7 p.231 head labels of diagrams, and F.7 p.234/235 head labels of diagrams : Substitute with β, i.e. : Re{Zs( )} Re{Zs(β)} and Im{Zs( )} Im{Zs(β)} { } = {0.25, 0.5, 1.} {β} = {0.25, 0.5, 1.} F.12 p.246 sketch in Table row "Grid of slits" inscription : " b a " " L a " G.11 p.324 Caption of Table 3 : "Coefficients or the relations ()" "Coefficients or the relations (4)" H.7 p.352 equation between eq.(4) and eq.(5) : γ ρn Æ g r n ³tanh (g r n t) ¼ g nn Æ g nn / tanh (g nn t) H.10 p th line (in explanation text of sketch) : " thickness d " " thickness d s " H.10 p.368 eq. (31) : r n = 1 j k 0Z 0 G a Z a 1+ j k 0Z 0 G a Z a e n Tanh(e g n s) n e n Tanh(e n s) g n 1 - j k 0Z 0 e n tanh( e G ¼ r n = a Z a g n s) n 1 + j k 0Z 0 e n tanh( e n s) G a Z a g n H.22 p.413 text lines below eq. (13) were printed beyond the border. When printed correctly : The ce m (ϑ) are azimuthal Mathieu functions which are even in ϑ at ϑ= 0, and the Hc m (2) (ρ) = Jcm (ρ) j Yc m (ρ) are associated radial Mathieu functions, or Mathieu- Hankel functions of the 2 nd kind, which represent outward propagating waves and satisfy Sommerfeld s far field condition. They are composed by the Mathieu-Bessel function Jc m (ρ) and the Mathieu-Neumann function Yc m (ρ) like the cylindrical Hankel function of the 2 nd kind. The Mathieu functions depend on the parameter q. It is noted for later use ( 2) that ce m (ϑ), Jc m (ρ), Yc m (ρ) are real functions, and J c m (0) = 0, H c m (0) = j Y c m (0) (where the primes indicate derivatives with respect to ρ ). It will be important for later evaluations that the ce m (ϑ) are generated as a Fourier series :

4 Formulas of Acoustics 4 Errata H.23 p.422 text lines above eq.(14) were printed beyond the border. When printed correctly : A 3 rd variant of the 2 nd sub-task completes the absorbed intensity to (p e (ξ)+p r (ξ)+p s (ξ)) V*(ξ), but does not worry about possible scattering at the border lines between plate and baffle wall. The scattered field p s is expanded in plate modes : I.8 p.455 Caption of Table 1 should be completed : Table 1 : Characteristic wave speeds. See Table 4 for S, E, D, B. I.22 p.508 the present eq.(7) is obsolete; instead of : The modal components in the plenum space are formulated as (ß= s,e) : p Hn (ξ ß,z) = P ( +) e + Γn a ξ ß + P ( ) Hßn e Γ n a ξ ß [ ] cos ( ε Hßn n (z h t d) ) Z 0 v Hnz (ξ ß,z) = j ε n P (+) k e + Γna ξ ß + P ( ) Hßn e Γ n a ξ ß Hßn 0 [ ] sin ( ε n (z h t d) ) p An (ξ ß,z) = e +Γ n a ξ ß C ( +) ßn e γ n ( z d) + Dßn (+) e +γ n (z d) [ ] + [ ] + e Γ na ξ ß C ßn ( ) e γ n ( z d) + Dßn + γ n (z d) and in the absorber layer : p An (ξ ß,z) = e +Γ n a ξ ß C (+) n e γ n (z d) + Dn (+) e +γ n (z d) [ ] + [ ] + e Γ na ξ ß C n γ n (z d) + Dn + γ n (z d) (6) (7) (8) write correctly : The modal components in the plenum space are formulated as (ß= s,e) : p Hn (ξ ß,z) = P (+) Hßn ³e +G n a x ß + P (- ) Hßn ³e - G n a x [ ß ] ³cos e n (z - h- t - d) ( ) (6) Z 0 v Hnz (ξ ß,z) = j e n k 0 and in the absorber layer : P (+) Hßn ³e +G n a x ß + P (- ) Hßn ³e - G n a x [ ß ]³sin e n (z - h - t - d) ( ) (7) p An (ξ ß,z) = e +Γ n a ξ ß C (+) n e γ n (z d) + Dn (+) e +γ n (z d) [ ] + [ ] + e Γ na ξ ß C n γ n (z d) + Dn + γ n (z d) (8)

5 Formulas of Acoustics 5 Errata J.6 p.544 eq. (21) : z 2 = ju ± ju U ju z ju ± ju U2 = ju J.15 p th line from bottom : ¼ (21) "But for n=0 we must be more precise." "But for n=0 they must be more precise." J.21 p nd text line above graphs : " flow resistance R f in the orifice " " flow resistance R r in the orifice " J.28 p st text line : "It seems to appropriate to describe " "It seems appropriate to describe " K.22 p nd lowest text line : " flow passage; d, t h, d h, h and W are shown " " flow passage; σ= panel porosity, d, t h, d h, h and W are shown " K.23 p.735 last text lines of 1 st paragraph : " this and the following Section K.26 will present the transfer matrix method for elements which are contained in earlier chapters." " this Section will present the transfer matrix method for an element which is also contained in Chapter J." K.24 p.737 text line below eq.(2) should read as : " k w, Z w are as given in Section K.23, and Z p is the impedance of the porous cover (see K.23, eq.(7) )." L. p nd & 3 rd text lines : " sound pressure level from a noise source in the environment; " " sound pressure level in the environment from an inside noise source; " L.2 p rd text line from bottom : " (such as largemachines). " " (such as big machines). " M.5 p.785 two last items of enumaration :

6 Formulas of Acoustics 6 Errata For grazing incidence, i.e. Q and P on the wall, the influence of higher terms R (n) (θ s ) in (8) is important! The derivation further supposes that the wall does not guide a surface wave (but this is only rarely the case in a restricted frequency range below a high-quality resonance; but even then (8) is an approximation to the field in points not too close to the wall). For grazing incidence, i.e. Q and P on the wall, the influence of higher terms R (n) (θ s ) in (D.14.10) or (D.20.11) is important! The derivation further supposes that the wall does not guide a surface wave (but this is only rarely the case in a restricted frequency range below a high-quality resonance; but even then (1) is an approximation to the field in points not too close to the wall). M.5.13 p.809 text line above eq.(6a) : " The range (12) leads to " " The range (5) leads to " N.1 p rd text line from top : " flowmust be irrotational " " flow must be irrotational " O.8.4 p.1012 eq.(35) : L µ,i = K i,m = L m,i = K i,m = 1Ú = p (- 1)i+m ³(2i)!³(2m)!³G (2 + i + m) / 2-1 P m m (x)³p i i (x) dx ( ) 2 i+m ³i!³m!³G (3+ i + m) / 2 ( ) 1Ú = p (- 1)i +m ³(2i)!³(2m)!³G (2 + i + m) / 2-1 P m m (x) ³Pi i (x) dx ( ) 2 i+m ³i!³m!³G (3 + i + m) / 2 ( ) ¼ Misprints in the Subject Index:

7 Formulas of Acoustics 7 Errata p.1154 letter S : sound velocity 5; adiabatic 1; temperature dependence 7; of gas mixture 7; Source conditions 18; spatial harmonics 42; p.1155 letter T : tube section with hard termination 67; with hard termination, filled with porous material 67; with open termination 67; with open termination, filled with porous material 68; p.1169 letter F : F Finite element method (FEM) 989; 956; Grafenau,

Formulas of Acoustics

F. P. Mechel (Ed.) Formulas of Acoustics With contributions by M. L. Munjal M. Vorländer P. Költzsch M. Ochmann A. Cummings W. Maysenhölder W. Arnold O. V. Rudenko With approximate 620 Figures and 70 Tables