(Nobumasa SUGIMOTO) (Masatomi YOSHIDA) Graduate School of Engineering Science, Osaka University 1., [1].,., 30 (Rott),.,,,. [2].

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$30 (Rott) [2] $-1/2$ [3] [4] -\mbox{\boldmath $\pi$}/4 - \mbox{\boldmath $\pi$}/2 ( ) 1 $T_{e}(x)(x$ 1 ) 1/4 1 1/4 $(a)$$

1 (Nobumasa SUGIMOTO) (Masatomi YOSHIDA) Graduate School of Engineering Science Osaka University 1 [1] 30 (Rott) [2] $-1/2$ [3] [4] -\mbox{\boldmath $\pi$}/4 - \mbox{\boldmath $\pi$}/2 ( ) 1 $T_{e}(x)(x$ 1 ) 1/4 1 1/4 $(a)$ $(b)$

2 113 ( ) [5] [6-9] $a_{e}(x)$ $\Omega_{\mathrm{e}}(x)[=(a_{e}/T_{e})\mathrm{d}T_{\mathrm{e}}/\mathrm{d}x]$ \Omega e(x) 2 vb -? [9]

3 $P\mathrm{o}$ $10^{3}$ 2 2 $\mathrm{k}\mathrm{s}-\text{ }[10]\text{ }$ L $\text{ }T_{e}(L)\text{ }$ ( 2 ) $\rho_{e}(x)$ $S_{\mathrm{e}}(x)$ $e$ $-$ $\frac{\partial\rho }{\partial t}+\frac{\partial}{\partial x}(\rho_{e}u )=\frac{2}{r}\rho_{\mathrm{e}}v_{b}$ (31) $\rho_{e}\frac{\partial u }{\partial t}=-\frac{\partial p }{\partial x}$ (32) $\frac{\partial S }{\partial t}+u \frac{\mathrm{d}s_{e}}{\mathrm{d}x}=0$ (33) $\rho $ $u $ $p $ $S $ x ( e ( po))

4 $\frac{\partial^{-\frac{1}{2}}u}{\partial t^{-\perp}2}$ 115 $v_{b}$ ( 2 ) $\frac{p}{p0}=\frac{\rho T}{\rho_{e}T_{e}}$ $\frac{p}{p0}=(\frac{\rho}{\rho_{e}})^{\gamma}\exp(\frac{s-s_{e}}{c_{v}})$ (34) $\gamma(=c_{\mathrm{p}}/c_{v})$ $c_{\mathrm{p}}$ $\mathrm{d}s_{e}/\mathrm{d}x$ $(\mathrm{q}/t_{e})\mathrm{d}t_{e}/\mathrm{d}x$ ( ) (34) $\frac{p }{p_{0}}=\frac{d}{\rho_{e}}+\frac{t }{T_{e}}$ $\frac{p }{p_{0}}=\frac{\gamma\beta}{\rho_{e}}+\frac{s }{c_{v}}$ (35) $[24]$ $v_{b}$ $u $ $v_{b}=c \sqrt{\nu_{\mathrm{e}}}\frac{\partial^{-\frac{1}{2}}}{\partial t^{-_{2}}\iota}(\frac{\partial u }{\partial x})+c_{t^{\frac{\sqrt{\nu_{\mathrm{e}}}}{t_{e}}\frac{\mathrm{d}t_{e}}{\mathrm{d}x}\frac{\partial^{-}\tau u 1}{\partial t^{-\#}}}}$ (36) C CT Pr $C=1+ \frac{\gamma-1}{\sqrt{pr}}$ $C_{T}= \frac{1}{2}+\frac{1}{\sqrt{pr}+pr}$ (37) $\nu_{e}(x)(=\mu/\rho_{e})$ $-1/2$ [3] \equiv f-t\infty -- $\sqrt$ u(tx-\tau \tau )d\tau (38) (36) (31) u (32) p $\frac{\partial^{2}p }{\partial t^{2}}-\frac{\partial}{\partial x}(a_{\mathrm{e}}^{2}\frac{\partial p }{\partial x})+\frac{2a_{\mathrm{e}^{\sqrt{\nu_{e}}}}^{2}}{r}\frac{\partial^{-1}2}{\partial t^{-\}}}[c\frac{\partial^{2}p }{\partial x^{2}}+\frac{(c+c_{t})\mathrm{d}t_{e}\partial p }{T_{e}\mathrm{d}x\partial x}]=0$ (39) $\text{ }a_{e}(x)\text{ \sqrt{\gamma po/\rho_{e}}\text{ }$ I=P(x)e $(1-2C \delta_{\mathrm{e}})a_{e}^{2}\frac{\mathrm{d}^{2}p}{\mathrm{d}x^{2}}+[1-2(c+c_{t})\delta_{e}]\frac{a_{\mathrm{e}}^{2}}{t_{\mathrm{e}}}\frac{\mathrm{d}t_{e}}{\mathrm{d}x}\frac{\mathrm{d}p}{\mathrm{d}x}+\omega^{2}p=0$ (310) $P$ $\delta_{e}$ $\delta_{e}(x)=\frac{(1-\mathrm{i})}{r}\sqrt{\frac{\nu_{e}}{2\omega}}=\frac{1}{r}(\frac{\nu_{e}}{\mathrm{i}\omega})^{1/2}$ (311) $( \delta_{e} \ll 1)$ (310) 1 [6]

5 116 $x=0$ $P=0$ ( ) $x=l$ dp/dx=0 [11] $\frac{dp}{\mathrm{d}x}=-\frac{(\gamma-1)}{\sqrt{pr}}\frac{\sqrt{\nu_{e}}}{a_{e}^{2}}(\mathrm{i}\omega)^{3/2}p$ (312) 32 (310) $P$ $F=ZP(\text{ }$ $Z=(1-K\delta_{e})a_{e}/a_{0}$ $K$ $a_{0=}a_{\mathrm{e}}(0))$ (310) $X \frac{\mathrm{d}}{\mathrm{d}_{x}}(x\frac{\mathrm{d}f}{\mathrm{d}_{x}})+mx\frac{\mathrm{d}}{d_{x}}(\frac{a_{e}}{t_{e}}\frac{\mathrm{d}t_{e}}{d_{x}}f)+\mathrm{y}f=0$ (310) $\delta_{\mathrm{e}}$ (313) $X$ $K$ $M$ $\mathrm{y}$ $\Omega_{e}$ 1 $X=(1-C\delta_{e})a_{e}$ $\mathrm{y}=^{\omega^{2}}+c_{t}\delta_{\mathrm{c}}(a_{e}\mathrm{d}\omega_{e}/\mathrm{d}_{x}+\omega_{e}^{2})$ $K=\mathit{2}C_{T}-C$ $M=-1/\mathit{2}$ (310) $\Omega_{e}$ - $\Omega_{e}$ $K$ $\Omega_{e}$ $2a_{0}\lambda/L$ $\lambda$ ( ) (314) $\frac{t_{\mathrm{e}}}{t_{0}}=(1+\lambda\frac{x}{l})^{2}$ $X$ $\xi$ $X\mathrm{d}/dx=a_{0}\mathrm{d}/d\xi$ $\mathrm{y}$ $C_{T}\delta_{e}\Omega_{\mathrm{e}}^{2}$ $\xi$ $C_{T}\delta_{e}\Omega_{e}^{2}$ \mbox{\boldmath $\delta$} -- $F=B^{+} \mathrm{e}^{\mathrm{i}k^{+}\xi}+b^{-}\mathrm{e}^{\mathrm{i}k^{-}\xi}+\mathrm{i}\frac{2c_{t}}{c}\frac{\lambda b}{l}(\frac{b^{+}}{k^{+}}\mathrm{e}^{\mathrm{i}k^{+}\xi}+\frac{b^{-}}{k^{-}}\mathrm{e}^{\mathrm{i}k^{-}\xi})\mathrm{e}^{\lambda}\mathrm{r}^{\xi}$ (315) $B^{\pm}$ $b=c\delta_{e}(0)$ $\sigma$ $\omega L/a_{0}$ $\psi=(\sigma^{2}-\lambda^{2}/4)^{1/2}$ $k^{\pm}l=-\mathrm{i}\lambda/\mathit{2}\pm\psi$ ( ) $\xi$ $x$ $\xi=\int_{\mathit{0}}^{x}\frac{a_{0}}{(1-c\delta_{e})a_{e}}dx=\frac{l}{\lambda}[\log\zeta+b(\zeta-1)]+o(b^{2})$ (316) $\zeta=1+\lambda x/l$ $x=l$ $\xi$ \xi =\xi L (312) df/d\xi $\frac{df}{\mathrm{d}\xi}=\{\frac{\lambda}{l}[1-\frac{2c_{t}}{c}(1+\lambda)b]+(1-\frac{1}{c})\frac{r\sigma^{2}}{l^{2}}b\}f+o(b^{2})$ (317)

6 \frac{e^{\mathrm{i}k^{+}\xi_{l}}+\mathrm{e}^{\mathrm{i}k\xi_{l}}}{e^{\mathrm{i}k^{+}\xi_{l}}-\mathrm{e}^{\mathrm{i}k\xi_{l}}}=)=w$ 117 \xi =0 F=0 B\pm izb $( $W= \frac{\lambda}{\mathit{2}}-\frac{2c_{t}}{c}\lambda b+(1-\frac{1}{c})\frac{r}{l}\sigma^{2}b$ (318) $barrow \mathrm{o}$ 1/4 4 $\sigma$ (318) $e^{\lambda 1L/L}$ (316) $\xi_{l}$ $\log(1+\lambda)+\lambda b=-\mathrm{i}\frac{\lambda}{2\psi}\log(\frac{w+\mathrm{i}\psi}{w-\mathrm{i}\psi})$ 3 (41) 41 b b $\log(1+\lambda)+\lambda b=-\cdot\frac{\lambda}{2\psi}[\log(\frac{1+2\mathrm{i}\psi/\lambda}{1-2\mathrm{i}\psi/\lambda})\pm 2\mathrm{i}n\pi]$ $+ \frac{\lambda}{\lambda^{2}/4+\psi^{2}}[\frac{2c_{t}}{c}\lambda-(1-\frac{1}{c})\frac{r}{l}\sigma^{2}]b+o(b^{2})$ (42) $n$ (b=o) \psi $\log[(1+2\mathrm{i}\psi/\lambda)/(1-2\mathrm{i}\psi/\lambda)]=2\mathrm{i}\tan^{-1}$ (2\psi /\mbox{\boldmath $\lambda$}) b $\cot[\frac{\psi}{\lambda}\log(1+\lambda)\pm n\pi]=\frac{\lambda}{2\psi}$ (43) - $\psi$ $\psi=\mathrm{i}\phi$ $1+ \lambda=(\frac{1+\mathit{2}\phi/\lambda}{1-2\phi/\lambda})^{\lambda/2\phi}$ (44)

7 $\lambda$ 118 (43) (44) $\mathrm{e}^{2}-1$ \mbox{\boldmath $\lambda$} \mbox{\boldmath $\sigma$} \mbox{\boldmath $\lambda$} \mbox{\boldmath $\sigma$} C CT 3 \mbox{\boldmath $\sigma$} $T_{L}/T_{0}[=(1+\lambda)^{2}]$ $\lambdaarrow\infty$ $\sigma^{2}=\lambda+\mathit{2}\log\lambda-3+\mathrm{o}(1)$ b\rightarrow 0 $\lambdaarrow 0$ $\sigmaarrow\pi/2$ (42) b $b\propto 1-\mathrm{i}$ $b^{1}$ $b^{\mathit{0}}$ (42) \psi $\lambda^{2}/4+\psi^{2}=\sigma^{2}=c\lambda$ $c= \frac{2c_{t}}{c+(c-1)r/l}$ (45) $\lambda^{2}/4-\phi^{2}=\sigma^{2}=c\lambda$ (46) c R/L 3 $(C=147 C_{T}=114)$ $(C=182 C_{T}=117)$ (45) (46) R/L=005 (b\rightarrow 0) 42 (41) b l b $b=(1-\mathrm{i})\zeta$ $\text{ })\zeta=\frac{\zeta_{0}}{\sqrt{\sigma}}$ $\zeta_{0}=\frac{c}{r}\sqrt{\frac{\nu_{0}l}{2a_{0}}}$ (47) $W= \frac{1}{2}\lambda[1-(\alpha+\mathrm{i}\beta)]$ (48) $\alpha=-\beta=[\frac{4c_{t}}{c}-2(1-\frac{1}{c})\frac{r}{l}\frac{\sigma^{\mathrm{a}}\prime}{\lambda }]\zeta$ (49) $\log(\frac{w+\mathrm{i}\psi}{w-\mathrm{i}\psi})=\log[==\frac{1\alpha \mathrm{i}(\beta-2\psi/\lambda)}{1\alpha \mathrm{i}(\beta+2\psi/\lambda)}]$ (410)

8 $\zeta_{0}$ $T_{L}/T_{\mathit{0}}$ $[=(1+\lambda)^{2}]$ $\sigma$ \psi =0 \mbox{\boldmath $\sigma$}=\mbox{\boldmath $\lambda$}/2 (a) (b) $(\zeta_{0}arrow 0)$ $\psi$ (41) $\log(1+\lambda)+\lambda\zeta$ $- \frac{\lambda}{2\psi}[-\mathrm{t}\bm{\mathrm{t}}^{-1}(\frac{\beta-2\psi/\lambda}{1-\alpha})+\tan^{-1}(\frac{\beta+2\psi/\lambda}{1-\alpha})\pm 2n\pi]=0$ (411) $- \lambda\zeta+\frac{\lambda}{4\psi}\log[=\frac{(1\alpha)^{2}+(\beta-2\psi/\lambda)^{2}}{(1\alpha)^{2}+(\beta+2\psi/\lambda)^{2}}]=0$ (412) $\psi$ $SS $\log(1+\lambda)+\lambda\zeta+\frac{\lambda}{4\phi}\log[=\frac{(1\alpha-2\phi/\lambda)^{2}+\beta^{2}}{(1\alpha+2\phi/\lambda)^{2}+\beta^{2}}]=0$ (413) $- \lambda\zeta+\frac{\lambda}{2\phi}[-\tan^{-1}(\frac{\beta}{1-\alpha-2\phi/\lambda})+\tan^{-1}(\frac{\beta}{1-\alpha+2\phi/\lambda})\pm 2n\pi]=0$ (414) \mbox{\boldmath $\zeta$}o $\zeta_{0}\text{ }$ 4 $T_{L}/T_{0}\approx 7897(\zeta_{0}\approx 047)$ /T0 $\approx 8213(\zeta_{0}\approx 054)$ $n$ 1 2 $\zeta_{0}$ 5 $R/\sqrt{\nu_{0}/\omega}(=R/\sqrt{\nu_{0}L/a_{0}\sigma})$ $\ovalbox{\tt\small REJECT}(L)/T_{\mathrm{e}}(\mathrm{O})$ $[=(1+\lambda)^{2}]$

9 120 5 $R/\sqrt{\nu_{0}}/\omega(=R/\sqrt{\nu_{0}L}/a_{0}\sigma)$ (a) $T_{L}/T_{\mathit{0}}[=(1+\lambda)^{2}]$ $(b)$ \mbox{\boldmath $\sigma$} 6 \mbox{\boldmath $\sigma$} \mbox{\boldmath $\sigma$} $R/(\nu_{0}/\omega_{\mathrm{r}})$ $\omega_{f}$ $\omega$ ( ) 5 $[710]$ 5 6 \mbox{\boldmath $\sigma$} $R/(\nu_{0}/\omega_{f})$

tube in 121 8 [9] ( b2 ) [1] RAYLEIGH LORD 1945 The theory of sound Vol2 Dover publications 230-231 [2] 2005 24 381-393 [3] 1985 4 110-120 [4] SUGIMOTO N &TSUJIMOTO K 2002 Amplification of energy

10 tube in [9] ( b2 ) [1] RAYLEIGH LORD 1945 The theory of sound Vol2 Dover publications [2] [3] [4] SUGIMOTO N &TSUJIMOTO K 2002 Amplification of energy flux of nonlinear acoustic waves in a gas-filled tube under an axial temperature gradient J Fluid Mech [5] TACONIS K W BEENAKKER J J M NIER A O C &ALDRICH L T 1949 $\mathrm{h}\mathrm{e}^{3}$ $\mathrm{h}\mathrm{e}^{4}$ Measurements concerning the vapour-liquid equilibrium of solutions of below $219^{\mathrm{o}}\mathrm{K}$ Physica [6] ROTT N 1969 Damped and thermally driven acoustic oscillations in wide and narrow tubes Z Angew Math Phys [7] Rorr N 1973 Thermally driven acoustic oscillations Part II stability limit for helium Z Angew Math Phys [8] ROTT N 1980 Thermoacoustics $Adv$ Appl Mech [9] ROTT N 2001 Linear thermoacoustics Proc 1st Intl Workshop on Thermoacoustics Technische Universiteit Eindhoven and the Acoustical Society of America 1-63 [10] YAZAKI T TOMINAGA A &NARAHARA Y 1980 Experiments on thermally driven acoustic oscillations of gaseous helium J Low Temp Phys [11] SUGIMOTO N &YOSHIDA M Marginal instability for thermoacoustic osciltioo of gas in $\mathrm{a}$ $J$ Fluid Mech

110 $\ovalbox{\tt\small REJECT}^{\mathrm{i}}1W^{\mathrm{p}}\mathrm{n}$ 2 DDS 2 $(\mathrm{i}\mathrm{y}\mu \mathrm{i})$ $(\mathrm{m}\mathrm{i})$ 2

$110 $\ovalbox{\tt\small REJECT}^{\mathrm{i}}1W^{\mathrm{p}}\mathrm{n}$ 2 DDS 2 $(\mathrm{i}\mathrm{y}\mu \mathrm{i})$ $(\mathrm{m}\mathrm{i})$ 2$ 1539 2007 109-119 109 DDS (Drug Deltvery System) (Osamu Sano) $\mathrm{r}^{\mathrm{a}_{w^{1}}}$ $\mathrm{i}\mathrm{h}$ 1* ] $\dot{n}$ $\mathrm{a}g\mathrm{i}$ Td (Yisaku Nag$) JST CREST 1 ( ) DDS ($\mathrm{m}_{\mathrm{u}\mathrm{g}}\propto