Wave CT 2 All Equations

phy-1109
PHY-1109 Physics

I. Forced Vibration and Resonance

Equation / Formula	Meaning of Symbols
Standardized Differential Equation: $\frac{d^{2}y}{d{t}^2}+2\lambda \frac{dy}{dt}+{\omega }^{2}y=\frac{f_0}{m}\sin pt$	$2\lambda =\frac{b}{m}$: Damping factor. $\omega =\sqrt{\frac{k}{m}}$: Natural angular frequency of the oscillator. $f_0$: Amplitude of the accelerating force per unit mass (where $\frac{f_0}{m}$ is the ratio).
Forced Vibration Amplitude ($A$): $A=\frac{f_0}{\sqrt{({\omega }^2-p^2{)}^2+4{\lambda }^2{p}^2}}$	$A$: Amplitude of forced oscillation. $f_0$: Driving force constant. $\omega$: Natural angular frequency. $p$: Driving angular frequency. $\lambda$: Damping factor.
Phase Angle ($\theta$): $\theta ={\tan }^{-1}\left(\frac{2\lambda p}{{\omega }^2-p^2}\right)$	$\theta$: Phase angle. $\lambda$: Damping factor. $p$: Driving angular frequency. $\omega$: Natural angular frequency.
Resonance Frequency ($P_R$): $P_R=\sqrt{{\omega }^2-2{\lambda }^2}$	$P_R$: Resonance angular frequency. $\omega$: Natural angular frequency. $\lambda$: Damping factor.

II. Progressive Wave Dynamics

Equation / Formula	Meaning of Symbols
Progressive Wave Equation (General): $y=a\sin (\omega t-kx)$	$y$: Particle displacement. $a$: Amplitude. $\omega =2\pi f$: Angular frequency. $t$: Time. $k=\frac{2\pi }{\lambda }$: Wave number. $x$: Position.
Progressive Wave Equation (Velocity/Wavelength form): $y=a\sin \frac{2\pi }{\lambda }(vt-x)$	$v$: Wave velocity. $\lambda$: Wavelength.
Wave Velocity (Phase Velocity): $V=\frac{\omega }{k}=f\lambda =\frac{\lambda }{T}$	$V$: Wave velocity (or phase velocity). $f$: Frequency. $\lambda$: Wavelength. $T$: Time Period (where $f=1/T$).
Differential Equation: $\frac{d^{2}y}{d{t}^2}=V^2\frac{d^{2}y}{d{x}^2}$	$\frac{d^{2}y}{d{t}^2}$: Particle acceleration. $\frac{d^{2}y}{d{x}^2}$: Rate of change of the slope (curvature). $V$: Wave velocity.
Particle Velocity: $U=\frac{dy}{dt}=-V\frac{dy}{dx}$	$U$: Particle velocity. $V$: Phase velocity. $\frac{dy}{dx}$: Slope of the displacement curve/strain.
Maximum Particle Acceleration ($A_{max}$): $A_{max}=-\frac{4{\pi }^2{V}^2}{{\lambda }^2}a$	$A_{max}$: Maximum particle acceleration. $V$: Wave velocity. $\lambda$: Wavelength. $a$: Amplitude.
Pressure Amplitude: $P_{max}=\frac{2\pi KaV}{\lambda }$	$P_{max}$: Maximum pressure amplitude. $K$: Bulk Modulus (or Volume Stress/Volume Strain ratio). $a$: Amplitude. $V$: Wave velocity. $\lambda$: Wavelength.
Energy Density ($E$): $E=2{\pi }^2{f}^2{a}^2\rho$	$E$: Energy density. $f$: Frequency. $a$: Amplitude. $\rho$: Density (written as ‘p’ in the source).
Intensity ($I$): $I=2{\pi }^2{f}^2\rho a^{2}V$	$I$: Intensity of the wave. $V$: Velocity of the wave. $\rho$: Density.
Intensity (Pressure relation): $I=\frac{P_{max}^2}{2\rho V}$	$P_{max}$: Maximum pressure amplitude. $\rho$: Density. $V$: Wave velocity.
Intensity (RMS Pressure relation): $I=\frac{P_{rms}^2}{\rho V}$	$P_{rms}$: RMS value of pressure. $\rho$: Density. $V$: Wave velocity.

III. Interference and Sound

Equation / Formula	Meaning of Symbols
Doppler Effect: $f^{\prime }=f\times \frac{V\pm V_o}{V\mp V_s}$	$f^{\prime }$: Observed frequency. $f$: Actual source frequency. $V$: Speed of sound in air. $V_o$: Speed of the observer. $V_s$: Speed of the source.
String Vibration Frequency: $f=\frac{1}{2L}\sqrt{\frac{T}{M}}$	$f$: Frequency. $L$: Length of the string. $T$: Tension. $M$: Mass per unit length.
Resultant Amplitude (Interference):A = \sqrt{a^2 + b^2 + 2ab \cos\phi}$$	$A$: Resultant amplitude. $a,b$: Amplitudes of the two interfering wave trains. $\varphi$: Phase difference between the two waves.
Time for Maximum Sound (Beats): $t=\frac{k}{n_1-n_2}$	$t$: Time. $k$: Integer ($0,1,\mathrm{2...}$). $n_1,n_2$: Frequencies of the two wave trains.
Intensity Level ($L$): $L=k{\log }_{10}\left(\frac{I}{I_0}\right)$	$L$: Intensity Level. $k$: Constant. $I$: Measured intensity. $I_0$: Reference intensity.
Group Velocity (Conceptual): $U=V_P-\lambda \frac{\Delta V}{\Delta \lambda }$	$U$: Group velocity. $V_P$: Phase velocity. $\lambda$: Wavelength. $\Delta V$: Change in phase velocity. $\Delta \lambda$: Change in wavelength.