Key Points

A wave is a disturbance that propagates through a medium or vacuum, transporting energy and information without the physical transfer of matter as a whole.

A wave is a disturbance that transfers energy and momentum from one point to another without the actual physical transfer of matter as a whole.

In transverse waves, the particles of the medium oscillate perpendicular to the direction of wave propagation. Examples include waves on a string and electromagnetic waves.

In transverse waves, particles of the medium oscillate perpendicular to the direction of wave propagation. In longitudinal waves, particles oscillate parallel to the direction of wave propagation.

The displacement of a sinusoidal wave traveling in the positive x-direction is given by $y(x, t) = a \sin(kx - \omega t + \phi)$, where 'a' is amplitude, 'k' is the angular wave number, '$\omega$' is the angular frequency, and '$\phi$' is the phase constant.

In longitudinal waves, the particles of the medium oscillate parallel to the direction of wave propagation. Sound waves are a primary example, consisting of compressions and rarefactions.

The equation for a sinusoidal wave travelling in the positive x-direction is $y(x, t) = a \sin(kx - \omega t + \phi)$. Here, 'a' is amplitude, 'k' is the angular wave number, '$\omega$' is the angular frequency, and '$\phi$' is the initial phase.

Wavelength ($\lambda$) is the minimum distance between two points in the same phase. The angular wave number (k) is related to wavelength by $k = \frac{2\pi}{\lambda}$.

Period (T) is the time for one complete oscillation. Frequency ($\nu$) is $1/T$. Angular frequency ($\omega$) is related by $\omega = \frac{2\pi}{T} = 2\pi\nu$.

Wavelength ($\lambda$) is the minimum distance between two points in the same phase. It is related to the angular wave number (k) by the formula $k = \frac{2\pi}{\lambda}$.

The speed of a progressive wave is related to its wavelength and frequency by the formula $v = \nu\lambda$. It can also be expressed as $v = \frac{\omega}{k}$.

The period (T) is the time for one complete oscillation. Frequency ($\nu$) is $1/T$. Angular frequency ($\omega$) is related by $\omega = \frac{2\pi}{T} = 2\pi\nu$.

The speed of a progressive wave (v) is related to its wavelength and frequency by the formula $v = \nu\lambda$. It can also be expressed as $v = \frac{\omega}{k}$.

The speed of a transverse wave on a stretched string depends on the tension (T) and the linear mass density ($\mu$) of the string, given by $v = \sqrt{\frac{T}{\mu}}$.

The speed of sound in a fluid is given by $v = \sqrt{\frac{B}{\rho}}$, where B is the bulk modulus and $\rho$ is the density. In a solid rod, it is $v = \sqrt{\frac{Y}{\rho}}$, where Y is Young's modulus.

The speed of a transverse wave on a stretched string depends on the tension (T) in the string and its linear mass density ($\mu$). The formula is $v = \sqrt{\frac{T}{\mu}}$.

The speed of sound in an ideal gas is given by $v = \sqrt{\frac{\gamma P}{\rho}}$, where $\gamma$ is the ratio of specific heats, P is the pressure, and $\rho$ is the density.

The speed of a longitudinal wave in a medium depends on its elastic modulus and density ($\rho$). In a fluid, $v = \sqrt{\frac{B}{\rho}}$ where B is the bulk modulus. In a solid rod, $v = \sqrt{\frac{Y}{\rho}}$ where Y is Young's modulus.

When two or more waves overlap, the resultant displacement at any point and at any instant is the algebraic sum of the displacements due to individual waves: $y(x, t) = y_1(x, t) + y_2(x, t)$.

The speed of sound in a gas is given by $v = \sqrt{\frac{\gamma P}{\rho}}$, where $\gamma$ is the ratio of specific heats, P is the pressure, and $\rho$ is the density.

Superposition of two waves of the same frequency results in interference. It is constructive when waves are in phase (phase difference $2n\pi$) and destructive when out of phase (phase difference $(2n+1)\pi$).

When two or more waves overlap in a medium, the resultant displacement at any point is the algebraic sum of the displacements due to individual waves: $y(x, t) = y_1(x, t) + y_2(x, t)$.

Superposition of two waves with the same frequency can lead to interference. Constructive interference occurs when waves are in phase, and destructive interference occurs when they are out of phase.

When a wave reflects from a rigid boundary, it undergoes a phase change of $\pi$ radians ($180^\circ$). When reflecting from an open or free boundary, there is no phase change.

When a wave reflects from a rigid boundary, it undergoes a phase change of $\pi$ radians ($180^\circ$). When reflecting from an open or free boundary, there is no phase change.

The superposition of two identical waves traveling in opposite directions produces standing waves, described by $y(x, t) = (2a \sin kx) \cos \omega t$. Points of zero amplitude are nodes, and points of maximum amplitude are antinodes.

Standing waves are formed by the superposition of two identical waves travelling in opposite directions. They have fixed points of zero displacement called nodes and maximum displacement called antinodes.

For a string of length L fixed at both ends, the allowed frequencies (harmonics) are given by $\nu_n = \frac{nv}{2L}$, where n = 1, 2, 3, ... The n=1 case is the fundamental frequency.

In a standing wave, nodes are points that remain stationary, while antinodes are points that oscillate with maximum amplitude. The distance between two consecutive nodes or antinodes is $\frac{\lambda}{2}$.

For an air column in a pipe of length L open at both ends, all harmonics are present. The frequencies are given by $\nu_n = \frac{nv}{2L}$, where n = 1, 2, 3, ...

For an air column in a pipe of length L closed at one end, only odd harmonics are present. The frequencies are given by $\nu_n = (2n+1)\frac{v}{4L}$, where n = 0, 1, 2, ...

For a string of length L fixed at both ends, the allowed frequencies of vibration (harmonics) are given by $\nu_n = \frac{nv}{2L}$ for $n = 1, 2, 3, ...$. The n=1 frequency is the fundamental frequency.

Beats are the periodic variation in the intensity of sound resulting from the superposition of two sound waves having slightly different frequencies.

For a pipe of length L open at both ends, harmonics are $\nu_n = \frac{nv}{2L}$ for $n = 1, 2, 3, ...$. For a pipe closed at one end, only odd harmonics are present: $\nu = (n + \frac{1}{2}) \frac{v}{2L}$ for $n = 0, 1, 2, ...$.

The beat frequency is the number of intensity maxima per second and is equal to the difference in the frequencies of the two superposing waves: $\nu_{\text{beat}} = |\nu_1 - \nu_2|$.

Beats are periodic variations in the intensity of sound produced by the superposition of two waves with slightly different frequencies. The beat frequency is the difference between the two source frequencies: $\nu_{\text{beat}} = |\nu_1 - \nu_2|$.