Wave Motion — Revision Notes

⚡ 30-Second Revision

Wave Equation: — $v = lambda

u$

Period: — $T = 1/

u$

Angular Frequency: — $omega = 2pi

u = 2pi/T$

Wave Number: — $k = 2pi/lambda$
General Wave Function: — $y(x, t) = A sin(kx mp omega t + phi)$
Speed of Transverse Wave on String: — $v = sqrt{T/mu}$ (where $T$ = tension, $mu$ = linear mass density)
Speed of Sound in Gas: — $v = sqrt{gamma P/\rho} = sqrt{gamma RT/M}$ (where $gamma$ = adiabatic index, $P$ = pressure, $ho$ = density, $R$ = gas constant, $T$ = absolute temperature, $M$ = molar mass)
Reflection: — Fixed end: phase change of $pi$ (crest reflects as trough). Free end: no phase change.
Superposition Principle: — Resultant displacement is vector sum of individual displacements.

2-Minute Revision

Wave motion is the propagation of a disturbance that transfers energy and momentum without net transfer of matter. Key parameters are amplitude ($A$), wavelength ($lambda$), frequency ($u$), period ($T$), and wave speed ($v$).

They are linked by $v = lambda u$ and $T = 1/ u$. Waves are classified as transverse (particles oscillate perpendicular to propagation, e.g., light, string waves) or longitudinal (particles oscillate parallel, e.

g., sound). Mechanical waves require a medium (sound), while electromagnetic waves do not (light). The speed of a transverse wave on a string depends on tension ($T$) and linear mass density ($mu$) as $v = sqrt{T/mu}$.

The speed of sound in a gas depends on temperature ($T$) as $v propto sqrt{T}$. The principle of superposition states that overlapping waves add their displacements, leading to interference (constructive or destructive) and standing waves.

Remember phase changes upon reflection: $pi$ for a fixed boundary, zero for a free boundary. Focus on applying these formulas and understanding the conceptual distinctions.

5-Minute Revision

Wave motion is the backbone of many physical phenomena, describing how energy propagates through space or a medium without transporting matter. The fundamental wave equation, $v = lambda u$, connects wave speed ($v$), wavelength ($lambda$), and frequency ($u$).

Remember that frequency is the inverse of the period ($T = 1/ u$). Waves are broadly categorized: mechanical waves (like sound or water waves) require a material medium, while electromagnetic waves (like light or radio waves) do not and can travel through a vacuum.

Further, waves are classified by particle oscillation direction: transverse waves (oscillations perpendicular to propagation, e.g., waves on a string, light) and longitudinal waves (oscillations parallel to propagation, e.

g., sound). The speed of a transverse wave on a string is given by $v = sqrt{T/mu}$, where $T$ is tension and $mu$ is linear mass density. For sound in an ideal gas, $v = sqrt{gamma RT/M}$, highlighting its dependence on the square root of absolute temperature.

The principle of superposition is crucial: when waves overlap, their displacements add vectorially. This leads to interference (constructive if in phase, destructive if out of phase by $pi$) and the formation of standing waves, which have fixed nodes (zero displacement) and antinodes (maximum displacement).

Reflection at a fixed boundary causes a $pi$ phase change, while at a free boundary, there's no phase change. Practice problems involving these formulas and conceptual distinctions, paying attention to units and temperature conversions.

Prelims Revision Notes

1
Definition: — Wave is a disturbance that propagates, transferring energy and momentum, but not matter. Particles oscillate about mean positions.
2
Types of Waves:

* Mechanical Waves: Require medium (e.g., sound, water waves). Cannot travel in vacuum. * Electromagnetic Waves: Do not require medium (e.g., light, radio waves). Travel in vacuum. * Transverse Waves: Particle oscillation $perp$ to wave propagation (e.g., string waves, EM waves). Characterized by crests and troughs. * Longitudinal Waves: Particle oscillation $parallel$ to wave propagation (e.g., sound waves). Characterized by compressions and rarefactions.

1
Wave Parameters & Formulas:

* Amplitude ($A$): Max displacement. * Wavelength ($lambda$): Distance for one complete cycle. * Frequency ($u$ or $f$): Cycles per second. $1/\text{T}$. * Period ($T$): Time for one cycle. $1/ u$. * Wave Speed ($v$): $v = lambda u$. * Angular Frequency ($omega$): $omega = 2pi u = 2pi/T$. * Wave Number ($k$): $k = 2pi/lambda$. * General Wave Equation: $y(x, t) = A sin(kx mp omega t + phi)$.

1
Speed of Waves:

* Transverse wave on a stretched string: $v = sqrt{T/mu}$, where $T$ is tension, $mu$ is linear mass density (mass/length). * Sound in a gas: $v = sqrt{gamma P/\rho} = sqrt{gamma RT/M}$. $v propto sqrt{T}$ (absolute temperature). $v propto 1/sqrt{M}$. * Sound in a liquid: $v = sqrt{B/\rho}$ (B = Bulk modulus). * Sound in a solid rod: $v = sqrt{Y/\rho}$ (Y = Young's modulus).

1
Wave Phenomena:

* Principle of Superposition: $y_{resultant} = y_1 + y_2 + dots$ * Interference: Constructive (in phase, $Deltaphi = 2npi$) or Destructive (out of phase, $Deltaphi = (2n+1)pi$). * Reflection: Fixed end: $pi$ phase change. Free end: no phase change. * Standing Waves: Formed by superposition of two identical waves traveling in opposite directions. Have fixed nodes (zero displacement) and antinodes (max displacement). No net energy transfer.

Vyyuha Quick Recall

To remember the factors for wave speed on a string: Tension Makes Vibrations Speedy. (T for Tension, M for Mass density, V for Velocity, S for Square root relationship: $v = sqrt{T/mu}$)