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 16 – Line below Equation (1.77) – The second frequency should read $\omega_2 = k_{s2}/m_2$

Page 31 – Equation (1.147) – The last fraction should read $\frac{E^2[fg]}{E[f^2]}$

Page 33

Equation (1.162) – It should be $\bm{S}_{fg}(\omega)=\bm{S}_{gf}(-\omega)$

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 57

Equation (2.85) – It should by $4\pi R^2$ instead of $4\pi a^2$

Equation (2.86) – The time average must be used $\left\langle \Pi \right\rangle_T$ instead of $\Pi$

Page 58 – Equations (2.87) and (2.88) – Same correction as for equation (2.86)

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 $m = -Re (\frac{(4\pi R^2)^2}{j\omega}\frac{\bm{p}}{\bm{Q}}) = \frac{(4\pi R^2)^2}{\omega}Im(\bm{Z}_a)$

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 $k_B$

Page 115 – Equation (3.231) – In the nominator it should read $\hat{F}^2$

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 (4.77) – The first integral should read as follows

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

Equation (4.78) – There must be a $\bm{p}_m'$ in the sum over $m$ .

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 166

In the first line the formula should read $\bf{k}\Delta\bf{r}=k\Delta r\cos(\vartheta)$

Equation 6.21 – The $\sin(\vartheta)$ in the first line should be removed

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 (6.63) – The nominator should read $\hat{F}^2$

Equation in section 6.4.1.4 – $\hat{w}^2(r)$ should not be boldsymbol. However, for figures 6.17 and 6.18 I used equation (3.225)

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%

Page 192 – Equation below (6.92) – Second sine should include a $y$ and not an $x$

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 240 Equation (8.59) – last sine should read $\sin(k_y y)$

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 (8.101) – it should read $\left[\bm{D}_s\right]=-\omega^2\left[M\right]$

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

Section 10.2.1

I really apologize for all the bugs in this section. This is an excellent example that some mistakes of a derivation are not found, because the final result was know before. In fact all equations of this section are wrong, except the final result!

Page 321

In equation (10.3) the diagonal must be corrected

\omega\begin{bmatrix} n_1(\eta_{11} + \eta_{12}) & -n_2 \eta_{21}\\ -n_1 \eta_{12} & n_2 (\eta_{22} + \eta_{21}) \end{bmatrix} \begin{Bmatrix} E_1/n_1(\omega) \\ E_2/n_2(\omega) \end{Bmatrix} = \begin{Bmatrix} \Pi_1 \\ \Pi_2 \end{Bmatrix}

Page 322

This has consequences on equation (10.4)

\begin{Bmatrix} E_1/n_1 \\ E_2/n_2 \end{Bmatrix} = -\frac{1}{\omega^2( \eta^2_{12} n_1 - n_2 (\eta_{11}+\eta_{12})(\eta_{22}+\eta_{21})) } \begin{bmatrix} \frac{n_2}{n_1}\eta_{22} + \eta_{12} & \eta_{12}\\ \eta_{12} & \eta_{11} + \eta_{12} \end{bmatrix} \begin{Bmatrix} \Pi_1 \\ \Pi_2 \end{Bmatrix}

and on (10.5)

E_1 = - \frac{n_1\eta_{12} +n_2\eta_{22}}{\omega^2(n_1\eta^2_{12} - n_2(\eta_{11}+\eta_{12})(\eta_{22}+\eta_{21})}\Pi_1

and (10.6)

E_2 = - \frac{n_2\eta_{12} }{\omega^2(n_1\eta^2_{12} - n_2 (\eta_{11}+\eta_{12})(\eta_{22}+\eta_{21})}\Pi_1

Not to speak about (10.7) and (10.8)

p_{1,\rm rms} = -\frac{\rho_0}{2\pi^2c_0 n_1} \frac{n_1\eta_{12} +n_2\eta_{22}}{n_1\eta^2_{12} - n_2 (\eta_{11}+\eta_{12})(\eta_{22}+\eta_{21})}\Pi_1

p_{2,\rm rms} = -\frac{\rho_0}{2\pi^2c_0} \frac{\eta_{12} }{n_1\eta^2_{12} - n_2 (\eta_{11}+\eta_{12})(\eta_{22}+\eta_{21})}\Pi_1

The pressure ratio (10.9) also has totally different look

\frac{p^2_{2,\rm rms}}{p^2_{1,\rm rms}} = \frac{n_1(\omega)\eta_{12}}{n_1(\omega)\eta_{12}+n_2(\omega)\eta_{22})}

Leaving different result for the coupling loss factor in equation (10.10). Note the involved ratio of modal densities.

\eta_{12} = \frac{p^2_{2,\rm rms}}{p^2_{1,\rm rms}}\frac{n_2(\omega)}{n_1(\omega)\left(1-\frac{p^2_{2,\rm rms}}{p^2_{1,\rm rms}}\right)} \approx \eta_{22}\frac{p^2_{2,\rm rms}}{p^2_{1,\rm rms}}\frac{n_2(\omega)}{n_1(\omega)}

To be precise, the volume in equation (10.11) is $V_2$

\eta_{22} = \frac{A_s c_0}{4 V_2 \omega}

Page 323

The volume in the $\tau$ – $\eta$ relationship must be $V_1$ and this – in combination with the modal density ratios – gives the final result in equation (10.12).