Wave Properties and Propagation (4D) - MCAT Chemical and Physical Foundations of Biological Systems

$k=\frac{2\pi}{\lambda}$. Wavenumber is the spatial analog of frequency, equaling $2\pi$ radians per wavelength.

Transverse wave. Transverse waves feature particle oscillations orthogonal to the propagation direction, exemplified by light waves.

Longitudinal wave. In longitudinal waves, particle displacement aligns with the wave's travel direction, as seen in sound waves.

Maximum displacement from equilibrium. Amplitude represents the peak deviation from the wave's equilibrium position, determining the wave's energy content.

$\text{m},\text{s}^{-1}$. Wave speed's SI unit derives from distance per time, consistent with wavelength in meters and frequency in hertz.

$\omega=2\pi f$. Angular frequency converts linear frequency to radians per second, multiplying by $2\pi$ for rotational equivalence.

$y(x,t)=A\sin(kx+\omega t+\phi)$. The positive time term in the argument denotes wave travel in the negative x-direction.

Intensity increases by a factor of $4$. Given $I\propto A^2$, doubling $A$ quadruples $I$ due to the squared term.

Time for one cycle; unit is seconds. Period is the duration required for one full oscillation or cycle of the wave to occur.

Cycles per second; unit is Hz ($\text{s}^{-1}$). Frequency quantifies how many complete wave cycles pass a point per unit time, derived from the reciprocal of the period.

Distance between identical phase points (e.g., crest to crest). Wavelength measures the spatial period of a wave, representing the shortest distance over which the wave pattern repeats itself.

$\lambda$ is halved. Since $v=\lambda f$ and $v$ is fixed, doubling $f$ requires halving $\lambda$ to maintain equality.

$v=\lambda f$. Wave speed equals the product of wavelength and frequency, indicating how far the wave travels per cycle.

$f=\frac{1}{T}$. Frequency and period are inversely related, as frequency counts cycles per second while period measures seconds per cycle.

$y(x,t)=A\sin(kx-\omega t+\phi)$. The form uses a negative time term to indicate propagation in the positive x-direction for sinusoidal waves.

Net displacement equals the sum of individual displacements. Superposition arises from the linearity of wave equations, allowing waves to overlap without interaction.

Phase difference $\Delta\phi=2\pi m$ (integer $m$). Constructive interference occurs when waves are in phase, maximizing amplitude through aligned crests and troughs.

$6.0,\text{m},\text{s}^{-1}$. Wave speed is calculated using $v=\lambda f$, yielding $6.0,\text{m},\text{s}^{-1}$ from given values.

$0.020,\text{s}$. Period is the inverse of frequency, so $T=1/f$ gives $0.020,\text{s}$ at $50,\text{Hz}$.

$400,\text{Hz}$. Frequency is found via $f=v/\lambda$, resulting in $400,\text{Hz}$ for the provided speed and wavelength.

$I\propto A^2$. For waves carrying energy proportional to amplitude squared, intensity follows this quadratic relationship.

$I=\frac{P}{A}$. Intensity measures power flux per unit area, quantifying energy transport by the wave.

$\Delta L=\left(m+\frac{1}{2}\right)\lambda$. Destructive interference results from path differences that shift phases by half-wavelength multiples.

$\Delta L=m\lambda$. Constructive interference requires path differences that are integer multiples of the wavelength for phase alignment.