固体力学 Solid mechanics 2 ME10010

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string by trying in the form
$$w(x, t)=f_{1}(x) \cos (\omega t)+f_{2}(x) \sin (\omega t),$$
which is often written in the compact form
$$w(x, t)=\operatorname{Re}\left[f(x) \mathrm{e}^{-\mathrm{i} \omega t}\right]$$

where $f(x)=f_{1}(x)+i f_{2}(x)$. Substitution reveals that $f(x)$ must satisfy the ordinary differential equation
$$\frac{\mathrm{d}^{2} f}{\mathrm{~d} x^{2}}+\frac{\varrho \omega^{2}}{T} f=0$$
whose general solution is
$$f(x)=A \cos (k x)+B \sin (k x), \quad \text { where } \quad k=\omega \sqrt{\frac{\varrho}{T}} .$$

ME10010 COURSE NOTES ：

Hence the Navier reduces to
$$\left(\rho \omega^{2}-\mu|k|^{2}\right)(B \times k)+\left(\rho \omega^{2}-(\lambda+2 \mu)|k|^{2}\right) A k=0$$
which we can only satisfy for nonzero $k$ if either
$$\boldsymbol{B}=0 \quad \text { and } \quad \rho v^{2}=(\lambda+2 \mu)|k|^{2}$$
or
$$A=0 \quad \text { and } \quad \rho \omega^{2}=\mu|k|^{2} .$$
The vectorial nature of the Navier equation has thus led to the existence of two dispersion relations, corresponding to two distinct types of waves.