PHYS212 : Homework


Homework Assignment 6 (due 3/23)

Acoustic Waves

  1. We found expressions for the phase (and group) velocity

    \begin{displaymath}
v_p = \sqrt{\left. \frac{\partial P}{\partial \rho} \right\vert _{\rho_o}}
\end{displaymath}

    for displacement waves and

    \begin{displaymath}
v_p = \sqrt{\frac{B}{\rho_o}}
\end{displaymath}

    for pressure waves. These equations must describe the same velocity. Use the definition of the bulk modulus $B$

    \begin{displaymath}
\frac{1}{B} = -\frac{1}{V} \frac{dV}{dP}
\end{displaymath}

    or

    \begin{displaymath}
B = -V \frac{dP}{dV}
\end{displaymath}

    to show that they do.

  2. The geometries of your voicebox, throat, and mouth are complex, but they are independent of the kind of gas running through them. Hence, the wavelengths you produce when you vocalize a particular way are also independent of the medium. Use this assumption to explain the shift to higher pitch which occurs when a person who has taken a full breath from a helium balloon speaks or sings. By what factor is the frequency increased? At STP, $\rho_{air} = 1.29$ kg/m$^3$ and $\rho_{\mathrm{He}} = 0.164$ kg/m$^3$. You may assume air is a diatomic. Helium is monatomic.

  3. Consider a 1.0 m long aluminum bar struck by a hammer on one end ( $\rho_{\mathrm{Al}} = 2.7\mathrm{e}{3}$ kg/m$^3$, $B = 7.4\mathrm{e}{10}$ N/m$^2$). For the following calculations assume only the fundamental longitudinal mode of the bar is excited by the hammer, and use the STP values $\rho_{air} = 1.29$ kg/m$^3$ and $B = 1.41\mathrm{e}5$ N/m$^2$. You may also assume that the sound waves in question are acoustic plane waves.

    (a) If the sound level is attenuated by -40 dB between the bar and your eardrum, and the sound level is 60 dB at your eardrum, what are the amplitudes $\delta_{x0}$ of the sound waves at the surface of the bar and at your eardrum?

    (b) What are the gauge pressure amplitudes corresponding to the displacement amplitudes you found in part (a)?

Lab 3: Fourier Analysis


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