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Physics 232. Heat, Sound, and Light


Equations for Exam II - oscillations and waves

Constants, such as values for $B$, bulk moduli, will be provided as needed.


15.1 Force on a mass on a spring
\begin{displaymath}F = -kx \end{displaymath}

15.6 Equation of motion for mass on a spring
\begin{displaymath}x(t) = A\cos(\omega t + \phi) \end{displaymath}

15.9 Angular frequency for mass on a spring
\begin{displaymath}\omega = \sqrt{\frac{k}{m}} \end{displaymath}

15.10-14 Relations between frequencies and periods. 15.15 Velocity of mass on a spring
\begin{displaymath}v(t) = -\omega A \sin (\omega t + \phi) \end{displaymath}

15.16 Acceleration of mass on a spring
\begin{displaymath}a(t) = -\omega^2 A \cos (\omega t + \phi) \end{displaymath}

15.21 Total energy of a simple harmonic oscillator
\begin{displaymath}E_{tot}=K+U=U_{max}=\frac{1}{2}kA^2 \end{displaymath}

15.26 Period of a simple pendulum
\begin{displaymath}T=2\pi \sqrt{\frac{L}{g}} \end{displaymath}

16-1 Equation for travelling wave
\begin{displaymath}y(x,t) = f(x-vt) \end{displaymath}           (To right)

\begin{displaymath}y(x,t) = f(x+vt) \end{displaymath}           (To left)


16.5 Wave function for travelling sinusoidal wave:
\begin{displaymath}y(x,t) = A\sin [(kx-\omega t)] \end{displaymath}


where $k$ and $\omega $ are the spatial and angular frequency or...                                     y(x,t) = A sin [2*pi(x/lambda - t/T)]
where $\lambda$ and $T$ are the spatial and temporal periods.

16.12 ``Golden rule" of waves:
\begin{displaymath}v = \lambda f \end{displaymath}

16.14 Transverse speed of sinusoidal wave on string
\begin{displaymath}v_y = -\omega A \cos (kx - \omega t) \end{displaymath}

16.15 Transverse acceleration of sinusoidal wave on string
\begin{displaymath}a_y = -\omega^2 A \sin (kx - \omega t) \end{displaymath}

16.18 Speed of a wave on a string
\begin{displaymath}v = \sqrt{\frac{T}{\mu } } \end{displaymath}


where $T$ is the tension, and $\mu$ is the linear mass density of string 16.- Kinetic energy stored in 1 $\lambda$ of a wave on a string
\begin{displaymath}K_{\lambda} = \frac{1}{4} \mu \omega^2 A^2 \lambda \end{displaymath}

16.20 Total energy stored in 1 $\lambda$ of a wave on a string
\begin{displaymath}E_{\lambda} = K +U = \frac{1}{2} \mu \omega^2 A^2 \lambda \end{displaymath}

16.21 Power of sinusoidal wave on string (energy per period T)
\begin{displaymath}\wp =\frac{1}{2} \mu \omega^2 A^2 v \end{displaymath}

17.1 Speed of sound in medium with bulk modulus $B=-\Delta P/(
\Delta V/V)$ and density $\rho$
\begin{displaymath}v=\sqrt{ \frac {B}{\rho}} \end{displaymath}

17.1b Speed of sound in air as a function of temperature
\begin{displaymath}v= (331 m/s) \sqrt{ 1 + \frac {T_C}{273^{\circ}C}} \end{displaymath}

17.2 If the displacement of air elements as sound passes is given by
\begin{displaymath}s(x,t) = s_{max} \cos (kx-\omega t) \end{displaymath}

17.3 then the variation in pressure is given by
\begin{displaymath}\Delta P(x,t) = \Delta P_{max} \sin (kx-\omega t) \end{displaymath}

17.4 Maximum pressure amplitude for the above wave
\begin{displaymath}\Delta P_{max} = \rho v \omega s_{max} \end{displaymath}

17.- Power transmitted by a sound wave through area A
\begin{displaymath}\wp = \frac{1}{2} \rho A v (\omega s_{max})^2 \end{displaymath}

17.5 Intensity of a sinusoidal sound wave
\begin{displaymath}I = \frac{\wp}{A}\end{displaymath}

17.- Intensity of a sinusoidal sound wave
\begin{displaymath}I = \frac{1}{2} \rho v (\omega s_{max})^2 \end{displaymath}

17.6 Intensity of a sinusoidal sound wave (in terms of pressure ampl)
\begin{displaymath}I = \frac{\Delta P_{max}^2}{2\rho v} \end{displaymath}

17.7 Inverse square law: intensity as function of distance from source
\begin{displaymath}I = \frac{\wp_{av}}{4\pi r^2} \end{displaymath}

17.8 Sound level in decibels
\begin{displaymath}\beta = 10 \log (\frac{I}{I_0}) \end{displaymath}

17.13 Doppler shifted frequency, $f'$
\begin{displaymath}f' = f(\frac{v+v_O}{v-v_S}) \end{displaymath}


where $f$ is the frequency at the source, $v$ is sound speed, $v_O$ is the observer's speed (negative if away from source), and $v_S$ is the source speed (negative if away from observer).

18.- Superposition principle
\begin{displaymath}y_{result.} = y_1 + y_2 + y_3 + ... \end{displaymath}

18.- Superposition of two sinusoidal waves travelling to the right
\begin{displaymath}y_{result.} = 2A\cos(\frac{\phi}{2})\sin (kx - \omega t + \frac{\phi}{2}) \end{displaymath}


where $A$ is the amplitude of both waves, $\phi$ is the phase shift between the two waves
18.2 Interference conditions for path length difference $\Delta r$ 18.3 Standing wave - sum of two identical waves travelling in opposite directions
\begin{displaymath}y = (2A \sin kx) \cos \omega t \end{displaymath}




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Jason Pinkney 2011-04-21