Short Notes: Waves
14.1 Wave Equation
A wave is a disturbance that travels through a medium (or through space for electromagnetic waves) transferring energy and information without bulk transport of matter. Common wave quantities are amplitude (maximum displacement), wavelength (distance between successive identical points on the wave), frequency (number of oscillations per unit time) and phase.
The general form of a one-dimensional travelling sinusoidal wave travelling in the +x direction is
\(y = A \sin(kx - \omega t + \phi)\)
- A - amplitude.
- k - wave number; \(k = \dfrac{2\pi}{\lambda}\), where \(\lambda\) is wavelength.
- \(\omega\) - angular frequency; \(\omega = 2\pi f = \dfrac{2\pi}{T}\), where f is frequency and T period.
- \(\phi\) - initial phase (phase constant).
- Phase velocity (wave speed): \(v = f\lambda = \dfrac{\omega}{k}\).
The sinusoidal wave function satisfies the one-dimensional wave equation (a second-order linear partial differential equation):
\[\dfrac{\partial^{2} y}{\partial x^{2}} = \dfrac{1}{v^{2}} \dfrac{\partial^{2} y}{\partial t^{2}}\]
Substituting a travelling-wave form into the wave equation shows that the wave travels with speed \(v\). For example if \(y = A \sin(kx - \omega t)\), then \(k\) and \(\omega\) must satisfy \(\omega = vk\).
Example. If \(y = 0.05\sin(5x - 20t)\) (SI units), then \(k = 5\) rad·m⁻¹ and \(\omega = 20\) s⁻¹ so the wave speed is \(v = \dfrac{\omega}{k} = \dfrac{20}{5} = 4\) m·s⁻¹.
14.2 Wave Speeds
| Wave Type | Speed Formula (and meaning of symbols) |
|---|---|
| Transverse wave on a stretched string | \(v = \sqrt{\dfrac{T}{\mu}}\) (T: tension in the string, μ: linear mass density = mass/length) |
| Sound in an ideal gas | \(v = \sqrt{\dfrac{\gamma P}{\rho}} = \sqrt{\dfrac{\gamma R T}{M}}\) (γ: ratio of specific heats, P: pressure, ρ: density, R: universal gas constant, T: absolute temperature, M: molar mass) |
| Longitudinal wave in a solid rod | \(v = \sqrt{\dfrac{Y}{\rho}}\) (Y: Young's modulus, ρ: density) |
| Sound in a liquid | \(v = \sqrt{\dfrac{B}{\rho}}\) (B: bulk modulus, ρ: density) |
Notes:
- Increasing the tension T of a string increases the wave speed; increasing the linear density μ decreases it.
- For sound in air, using ideal-gas relations and approximations, speed depends on temperature: \(v \approx 331 + 0.6\,T\) where T is temperature in °C (valid near standard conditions).
- Mechanical wave speed depends on the medium's elastic property (modulus) and inertia (density).
14.3 Standing Waves
A standing wave is formed by the superposition of two waves of the same frequency and amplitude travelling in opposite directions. The pattern does not travel; it has fixed nodes (points of zero displacement) and antinodes (points of maximum displacement).
| System | Frequency formula / allowed modes |
|---|---|
| String fixed at both ends | \(f_n = \dfrac{n}{2L}\sqrt{\dfrac{T}{\mu}}\), where n = 1, 2, 3,... (fundamental n=1) |
| Open pipe (both ends open) | \(f_n = \dfrac{n v}{2L}\), where n = 1, 2, 3,... |
| Closed pipe (one end closed) | \(f_n = \dfrac{n v}{4L}\), where n = 1, 3, 5,... (only odd harmonics) |
Key relations for standing waves on a string of length L fixed at both ends:
- Allowed wavelengths: \(\lambda_n = \dfrac{2L}{n}\).
- Fundamental (first harmonic): \(\lambda_1 = 2L,\; f_1 = \dfrac{v}{2L}\).
- Overtones/harmonics: f_n = n f_1 for a string or open pipe; for closed pipe only odd n appear.
Practical importance: strings on musical instruments and air columns in wind instruments produce discrete harmonic series determined by boundary conditions.
14.4 Superposition and Beats
Superposition principle: When two or more waves propagate in the same region, the resultant displacement is the algebraic sum of individual displacements provided the medium is linear.
When two sinusoidal waves of nearly equal frequency and same amplitude A₀ superpose, using the trigonometric identity for sum of cosines, we get an amplitude-modulated resultant:
\(y = A_0\cos(\omega_1 t) + A_0\cos(\omega_2 t) = 2A_0\cos\!\bigg(\dfrac{\omega_1 - \omega_2}{2}t\bigg)\cos\!\bigg(\dfrac{\omega_1 + \omega_2}{2}t\bigg)\)
The envelope oscillates at the beat angular frequency \(\Delta\omega/2 = (\omega_1-\omega_2)/2\), producing audible beats when acoustic frequencies are close.
- Beat frequency: \(f_{\text{beat}} = |f_1 - f_2|\).
- Resultant amplitude for two waves of same frequency but phase difference Δφ: \(A = 2A_0\cos\!\bigg(\dfrac{\Delta\phi}{2}\bigg)\).
- Constructive interference: path difference = nλ (n = 0,1,2,...).
- Destructive interference: path difference = (n + ½)λ.
Application: tuning musical instruments (listen for beats), noise-cancelling using destructive interference, interferometry.
14.5 Doppler Effect
The Doppler effect is the change in observed frequency when there is relative motion between the source of waves and the observer. For sound in air (where the medium is the air), the general formula is
\(f' = f\,\dfrac{v \pm v_o}{v \mp v_s}\)
- Here v is the speed of sound in the medium; v_o is the speed of the observer relative to the medium; v_s is the speed of the source relative to the medium.
- Choose the sign in the numerator as + when the observer moves towards the source and - when the observer moves away.
- Choose the sign in the denominator as - when the source moves towards the observer and + when the source moves away.
Common special cases:
- Source moving towards stationary observer: \(f' = \dfrac{f v}{v - v_s}\) (observed frequency increases).
- Source moving away from stationary observer: \(f' = \dfrac{f v}{v + v_s}\) (observed frequency decreases).
- Observer moving towards stationary source: \(f' = f\dfrac{v + v_o}{v}\) (observed frequency increases).
- Observer moving away from stationary source: \(f' = f\dfrac{v - v_o}{v}\) (observed frequency decreases).
Example application: change in pitch of a passing ambulance siren, radar and sonar velocity measurements, astronomy (redshift/blueshift) for electromagnetic waves (formulas differ as medium is absent and relativistic effects may be required).
14.6 Important Wave Relations and Notes
- Intensity (I) is proportional to the square of amplitude: I ∝ A².
- Intensity (definition): I = \dfrac{\text{Power}}{\text{Area}}.
- Sound level in decibels (dB): \(L = 10\log_{10}\!\bigg(\dfrac{I}{I_0}\bigg)\), where reference intensity \(I_0 = 10^{-12}\) W·m⁻² (threshold of hearing).
- Speed of sound in air (approximate): \(v \approx 331 + 0.6\,T\), where T is temperature in °C.
- Transverse waves: particle displacement is perpendicular to the direction of wave propagation (e.g., waves on a string, electromagnetic waves).
- Longitudinal waves: particle displacement is parallel to the direction of wave propagation (e.g., sound waves in fluids).
- Energy transport: waves carry energy; the power transmitted depends on intensity and area through which the wave passes.
These relations form the foundation for solving wave problems in mechanics and acoustics. Familiarity with boundary conditions, normal modes, and sign conventions for Doppler problems is essential for correct application.
FAQs on Short Notes: Waves
| 1. What are the main types of waves in physics? |  |
| 2. How do transverse and longitudinal waves differ? | |
| 3. What is the significance of amplitude in wave motion? | |
| 4. Can you explain the concept of wave frequency and its importance? | |
| 5. What role does wavelength play in wave behaviour? | |
