Errata of Vibroacoustic Simulation

This errata page is a collection of all bugs and mistakes that I noticed for the first edition. If you would like to propose some corrections please send a note to author@alexanderpeiffer.de.

A special thank you goes to Valentin who found all the bugs in chapter 7 and the legend bug in figure 8.38.

Chapter 1

Page 9 – Equation (1.37) – In the paragraph before I say signal $x(t)$ , thus the equations should be with $x(t)$

Page 13 – Equation above equation (1.56) – the first integral should read

\Delta E_{\rm cylcle} = \int_0^Tc_v\ddot{u}\,dt=\ldots

Page 15 – Figure 1.11 – left spring should have stiffness $k_{s1}$

Page 33 – The last equation must be similar to equation (1.164), thus $\bm{G}^*_k(\omega)\bm{F}_k(\omega)$ should be in boldsymbol

Page 34 – Equation (1.167), $\bm{G}^*_m(\omega)\bm{F}_m(\omega)$ should also be in boldsymbol

Page 38 – Figure 1.24 – To be consistent with the text it should be $\ldots f_n(t)\ldots f_N(t)$ and $\ldots g_m(t)\ldots g_M(t)$

Chapter 2

Page 44 – Last enumeration – Item 3 should be “The momentum flow out of the volume”

Page 47 – Equation (2.25) – The velocity potential should be $\mathbf{v}'= -\nabla\Phi$

Page 48 – Equation (2.29) – and consequently … $p = \rho_0\frac{\partial\Phi}{\partial t}$

Page 53

Table 2.2 – The acoustic velocity formula should read $-\frac{1}{j\omega c_0}\nabla\bm{p}$

Equation (2.59) – The last expression should be $\bm{A}e^{-\frac{\eta}{2}kx}e^{j(-kx+\omega t)}$

Page 54

Equation (2.65) – It should read $\left( \frac{1}{c_0^2}\frac{\partial^2}{\partial t^2} - \frac{\partial^2}{\partial r^2} \right) (r\Phi) = 0$

The equation reference (2.30) is not correct it should be (2.65)

Page 55

Equation (2.68) – $-\frac{\partial\Phi}{\partial r} = v_R$

Equation (2.69) – $\bm{A} = -\bm{v}_R \frac{R^2}{1-jkR} e^{jkR}$

Page 62 – Equations (2.116) and (2.117) – The impedance symbols should be boldmath $\bm{z}_1,\bm{z}_2$

Page 58 – Equation (2.98) – This should read $m = -Re (\frac{(4\pi R^2)^2}{j\omega}\frac{\bm{p}}{\bm{Q}}) = \frac{(4\pi R^2)^2}{\omega}Im(\bm{Z}_a)$ pAGFE

Page 70 – Above equation (2.155) – It should read $kR\le 0.5$ .

Page 115 – Equation (3.230) – The second term should read $\frac{1}{2} Re\left(\frac{\hat{F}^2}{\\bm{Z}_0}\right)$

Chapter 3

Page 80 – Equation (3.22) – $\sigma_{ij}=\frac{dF_i}{dA_j}$

Page 111 – Paragraph below (3.21b) – Insert as first sentence: Considering $k_B$ as the real and positive solution, hence $k_B=k_{B2}$ .

Page 113 – Equations (3.222)-(3.224) – Replace $k$ by $k_B$

Page 114 – Equation (3.225) – Replace $k$ by $k_B$

Chapter 4

Page 120 Figure 4.1 – The radius is $R$ and not $a$

Page 135

Figure 4.12 – Missing unit at y-axis label. Use $n(\omega)/s$

Equation (4.76) – It would be more consistent to use $m$ as index in the sum

\bm{p}(\vec{r}) = \sum_{m=0}^\infty \bm{p}'_m \Phi_m(\bf{r})

Equation (4.76) – The first integral should read as follows

\int_{V} \sum_m^\infty\bm{p}_m'\Phi_m(\vec{r}) (\bm{k}^2 - k_n^2) \Phi_n(\bf{r}) dV

Page 136 Equation (4.84) – Remove conjugate star, not required for real modes

Page 137

Paragraph below Equation (4.86) – it should read $R\ll \lambda$

Equation (4.87) – Third term must be $1 ⁄ 2 \bm{Z v}^2$

Page 141 Last paragraph – first element and not fist element

Page 141 Equation (4.103) – For better consistence with Figure 4.16 ${\bm Q}_2 = {\bm Q}_3 = 0$ would be better

Chapter 5

Page 151 Section 5.3.1 – First sentence – it should be in-homogenous form.

Chapter 6

Page 161 Equation (6.3) – The square should be removed and it should read $\Psi_{\rm mean} = \ldots$

Page 169

Equations (6.33)-(6.35) – The $\frac{1}{M}$ factor is missing for all sums.

Equations (6.34)-(6.35) – The last $\bm{\Phi}$ should read $\bm{\Phi}_i$

Page 170

Equations (6.37) – Remove the hat from all pressure symbols $p_\text{rms}$ is correct

Equations (6.38) – Replace $\hat{p}$ by $p^2_\text{rms}$

Page 175/176 Figure 6.9 and 6.10 – Right hand side. Label of y-axis: $z_0$ should not be boldsymbol

Page 180

Equation in section 6.4.1.4 – $\hat{w}^2(r)$ should not be boldsymbol

Section 6.4.2 – It is worth mentioning that the plate is made of aluminum

Figure 6.14 – The density variation in the caption should be 2\% not 1\%

Chapter 7

Page 207 Equation (7.18) – The last radiation stiffness should be $[\bm{D}_{dir}^{(2)}]$ instead of $[\bm{D}_{dir}^{(3)}]$

Page 208

Section 7.3.1 heading – Exchange m and n

Globally it would be better to stay with the original heading and exchange $m$ and $n$ in the text and all formulas, because his would correspond better to figure 7.5

Equation (7.22) – The index $(n)$ is missing, it should be $[S_{qq}^{(n)}]$

Page 210 Equations (7.28) and (7.30) – The index $i,j$ should be used consequently in all formulas as sum index

Page 211 Equation (7.32) – The factor $E_m$ is missing after both large parenthesis

Page 215

Equation (7.40) – $\{\bm{q}\}$ instead of $\{\bm{u}\}$

Last line of last paragraph – response instead of reponse

Page 218 Figure 7.13 – The SEA-Matrix must read $[L']$ instead of $[A]$

Page 220 Equation (7.47) – The modal coordinate $\{ {\bm q}'\}$ is missing

Chapter 8

Page 226 Last paragraph – … is modified by an $\cos\vartheta$ factor instead of $\sin\vartheta$

Page 227 First sentence – $\phi$ must be replaced by $\varphi$

Page 231 Figure 8.5 – Label of y-axis: TL/dB instead of TL in dB Page

Page 233 Figure 8.8 – Due to a bug in the code spring curves are wrong. See new figure.

Figure 8.8 Transmission loss of a rectangular 2-beam junction, without and with spring coupling. Source: Alexander Peiffer

Page 234 Figure 8.10 – Label of y-axis: TL/dB instead of TL in dB

Page 238 This item in description – $D''_{\rm dir}$ instead of $D_1$

Page 242 Equation (8.74) – The equal sign before the $\approx$ must be removed

Page 239 Equation (8.51) – in the first row the integral is missing, The second integral runs over $S'$

\Pi = \frac{1}{2} \int_S Re\{ {\bm v}_z^*(x,y){\bm p}(x,y) \} dS

Page 245 Equation (8.73) – The stiffness is complex, thus $\bm{D}_{\rm dir}''^{plate}=\ldots$

Page 247 Equation (8.94) – The stiffness is complex, thus and there is an $\omega$ missing in the last term

\bm{D}_{\rm tot}''(k_a,\vartheta)=m''\omega^2\left[\frac{ k_a^4\sin^4\vartheta}{k_B^4} -1 \right] - j\omega\frac{2 \rho_0c_0}{\cos\vartheta}

Page 250 Equation (8.103) – remove useless $\left\langle\tau\right\rangle_E$ in the middle

Page 251 Equation (8.105) – use $k_s$ and $c_s$

Page 253 After Equations (8.108a-d) – it should read $B=\frac{Eh^3}{12(1-\nu^2)}$

Page 255 Coordinate vector after Equation (8.118) – use bold ${\bm\beta}_{x,e}$

Page 256

Equation (8.21) – the displacement vector is missing after the first matrix

Equation (8.23) – the amplitude vector is missing after the second matrix

Equation (8.24) – the displacement vector is missing after the first matrix and $\bm{D}'_{34}=\mu_{B1}\mu_{B2} + \nu k_x^2$

Page 258 Equation (8.131) – is should read $k_x = k_v^{(m)} \cos \phi$

Page 261

Equation (8.143) – wrong index index in excitation cspd. It should be $S^{(m)*}_{e,ff}$

Equation (8.147) – wrong index index in center matrix cspd. It should be $S^{(m)*}_{e,qq}$

Page 263

Figure 8.30 – wavenumber vector labels $k_S$ and $k_L$ must be exchanged

Equation (8.151) – is should read $\cos \phi_w^{(n)} = \frac{k_x}{k_w^{(n)}}$

Page 267 Figure 8.38 – The legends must be exchanged. The blue line belongs to $\tau_{1B,2B}$

Chapter 9

Page 287

Equation (9.76) – – Should be $\bm{z}/A_c$

Paragraph before Equation (9.78) – Should be small $\bm{z}$

Page 299

Equation (9.107d) – Division by porosity is missing. Use $\rho_0c_0^2/\Phi$ in nominator

Equation (9.107e) – Variables should be in boldsymbol. Use ${\bm z}=\sqrt{\bm{K}\bm{\rho}_{eq}}$

Equation (9.107f) – Variables should be in boldsymbol. Use ${\bm\Gamma}=j\omega\sqrt{\bm{\rho}_{eq}/\bm{K}}$

Equation (9.107g) – Variables should be in boldsymbol. Use $\bm{c}=j\omega/\bm{\Gamma}$

Page 301

Equation (9.109) – The last term of the denominator should read $\bm{T}_{22}\frac{\bm{z}_s}{\bm{z}_r}$

Equation (9.110) –

{\bm R}(\vartheta)=\frac{{\bm z}(\vartheta)-\frac{{\bm z}_s(\vartheta)}{\cos(\vartheta)}}{{\bm z}(\vartheta)+\frac{{\bm z}_s(\vartheta)}{\cos(\vartheta)}}

Page 302

Paragraph below Equation (6.72) – z = ρ₀c₀

Next paragraph – z = az₀

Page 309 Figure 9.40 – Label of y-axis: First ‘perf’ index should be in roman

Page 313 Figure 9.44 – Figure caption has an index error. It should read $m''_1=m''_2=1\text{kg/m}^2$

Page 314 Figure 9.45 – Figure caption has an index error. It should read $m''_1=m''_2=1\text{kg/m}^2$

Page 316 Figure 9.47 – Wrong figure. Should be as follows:

Corrected figure 9.47 — General shape of the double wall transmission and the relationship to main parameters.

Chapter 10

Page 325 Figure 10.4 – Label at y-axis should be $n(\omega)/s$

Chapter 12

Page 361 Figure 12.1 – The Helmholtz number should be $He = \frac{L}{\lambda}$

Page 372 – Section 12.3 Last sentence of first paragraph – prototype instead of pototype.

Appendix A

Page 424 Equations (A.68),(A.70)-(A.73) – The force symbol should be in capital letters $\bm{F}$

Appendix B

Page 426 Equation (B.17) – The upper index should read $\Psi_{in,L}$

Page 429 Figure B.2 – wavenumber vector labels $k_S$ and $k_L$ must be exchanged

Page 436 Equations (B.79-80) – The upper right coefficient in the matrix requires a second right parenthesis

Index

Page 445 coupling loss factor, non-resonant and resonant instead of resonsant