Wave CT 2 Equations

phy-1109
PHY-1109 Physics

I. Forced Vibration and Resonance

Equation / Formula	Context	Meaning of Symbols
Differential Equation: $m\frac{d^{2}y}{d{t}^2}+b\frac{dy}{dt}+ky=F_0\sin pt$	General equation of a damped system driven by an external periodic force.	$m$: Mass. $b$: Damping coefficient/constant. $k$: Spring constant/force constant. $y$: Displacement. $t$: Time. $F_0$: Maximum external force amplitude. $p$: Angular frequency of the external force.
Standardized Differential Equation: $\frac{d^{2}y}{d{t}^2}+2\lambda \frac{dy}{dt}+{\omega }^{2}y=\frac{f_0}{m}\sin pt$	Simplified form used for solution derivation.	$2\lambda =\frac{b}{m}$: 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 ($A$): $A=\frac{f_0}{\sqrt{({\omega }^2-p^2{)}^2+4{\lambda }^2{p}^2}}$	Amplitude of the steady-state forced oscillation.	$A$: Amplitude of forced oscillation. $f_0$: Driving force constant. $\omega$: Natural angular frequency. $p$: Driving angular frequency. $\lambda$: Damping factor.
Phase Angle ($\theta$): $\theta ={\tan }^{-1}\left(\frac{2\lambda p}{{\omega }^2-p^2}\right)$	Phase difference between the external driving force and the resultant displacement.	$\theta$: Phase angle. $\lambda$: Damping factor. $p$: Driving angular frequency. $\omega$: Natural angular frequency.
Resonance Frequency ($P_R$): $P_R=\sqrt{{\omega }^2-2{\lambda }^2}$	Driving angular frequency at which maximum displacement (resonance) occurs.	$P_R$: Resonance angular frequency. $\omega$: Natural angular frequency. $\lambda$: Damping factor.
Maximum Amplitude ($A_{max}$): $A_{max}=\frac{f_0}{2\lambda \sqrt{{\omega }^2-{\lambda }^2}}$	Maximum displacement amplitude achieved at the resonance frequency.	$A_{max}$: Maximum amplitude. $f_0$: Driving force constant. $\lambda$: Damping factor. $\omega$: Natural angular frequency.

II. Progressive Wave Dynamics

Equation / Formula	Meaning of Symbols	Context
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.	Displacement ($y$) of a particle at position $x$ and time $t$.
Progressive Wave Equation (Velocity/Wavelength form): $y=a\sin \frac{2\pi }{\lambda }(vt-x)$	$v$: Wave velocity. $\lambda$: Wavelength.	Expresses displacement using wave velocity ($v$).
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 $f=1/T$).	Relationship between wave properties.
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.	Differential equation describing progressive wave motion.
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.	Relationship between particle velocity, wave velocity, and the slope of the displacement curve.
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.	Maximum acceleration experienced by a particle in the wave medium.
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.	Maximum pressure change in the medium due to the wave.
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).	Total energy per unit volume in the wave.
Intensity ($I$): $I=2{\pi }^2{f}^2\rho a^{2}V$	$I$: Intensity of the wave. $V$: Velocity of the wave. $\rho$: Density.	Energy current or intensity of the wave. (Also: $I\propto a^2$)
Intensity (Pressure relation): $I=\frac{P_{max}^2}{2\rho V}$	$P_{max}$: Maximum pressure amplitude. $\rho$: Density. $V$: Wave velocity.	Intensity expressed in terms of maximum pressure.
Intensity (RMS Pressure relation): $I=\frac{P_{rms}^2}{\rho V}$	$P_{rms}$: RMS value of pressure. $\rho$: Density. $V$: Wave velocity.	Intensity expressed in terms of RMS pressure.

III. Interference and Sound

Equation / Formula	Context	Meaning of Symbols
Doppler Effect: $f^{\prime }=f\times \frac{V\pm V_o}{V\mp V_s}$	Observed frequency due to relative motion between source and observer.	$f^{\prime }$: 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: $f=\frac{1}{2L}\sqrt{\frac{T}{M}}$	Fundamental frequency of a vibrating string (transverse vibration).	$f$: Frequency. $L$: Length of the string. $T$: Tension. $M$: Mass per unit length.
Air Column Frequency (Open Pipe): $f_n=\frac{nV}{2L}$ ($n=1,2,\mathrm{3...}$)	Harmonic frequencies for vibration in an open pipe (longitudinal vibration).	$f_n$: Harmonic frequency. $n$: Harmonic number. $V$: Velocity of sound. $L$: Length of the pipe.
Air Column Frequency (Closed Pipe): $f_n=\frac{nV}{4L}$ ($n=1,3,\mathrm{5...}$)	Harmonic frequencies for vibration in a closed pipe (longitudinal vibration).	$f_n$: Harmonic frequency. $n$: Harmonic number (odd integers). $V$: Velocity of sound. $L$: Length of the pipe.
Tuning Fork Frequency: $f=\frac{1}{2\pi }\sqrt{\frac{K}{m}}$	Frequency of a tuning fork.	$K$: Force constant. $m$: Mass.
Resultant Amplitude (Interference): $A=\sqrt{a^2+b^2+2ab\cos \varphi }$	Amplitude of the resultant wave formed by two interfering waves.	$A$: Resultant amplitude. $a,b$: Amplitudes of the two interfering wave trains. $\varphi$: Phase difference between the two waves.
Time for Maximum Sound (Beats): $t=\frac{k}{n_1-n_2}$	Time instants when maximum sound (loudness) occurs.	$t$: Time. $k$: Integer ($0,1,\mathrm{2...}$). $n_1,n_2$: Frequencies of the two wave trains.
Time for Minimum Sound (Beats): $t=\frac{2k+1}{2(n_1-n_2)}$	Time instants when minimum sound (quietness) occurs.	$k$: Integer ($0,1,2,\mathrm{3...}$). $n_1,n_2$: Frequencies of the two wave trains.
Intensity Level ($L$): $L=k{\log }_{10}\left(\frac{I}{I_0}\right)$	Measurement of the intensity level relative to a reference intensity.	$L$: Intensity Level. $k$: Constant. $I$: Measured intensity. $I_0$: Reference intensity.
Group Velocity (Conceptual): $U=V_P-\lambda \frac{\Delta V}{\Delta \lambda }$	Relation for group velocity (derived from the context of phase velocity).	$U$: Group velocity. $V_P$: Phase velocity. $\lambda$: Wavelength. $\Delta V$: Change in phase velocity. $\Delta \lambda$: Change in wavelength.