Waves, Sound & Doppler Effect – Comprehensive Study Notes

Definition and Classification of Waves

A wave is a disturbance that propagates through space carrying energy and momentum while the bulk mass of the medium remains, on the average, at rest.

Waves are categorised along several independent lines:

• Necessity of a material medium
– Mechanical waves (require a medium): sound, water-surface ripples, waves on a string, pulses on a spring.
– Non-mechanical or electromagnetic (EM) waves (do not require a medium): visible light, radio, microwaves, γ-rays.

• Dimensionality of the disturbance
– 1-D (strings, slinky), 2-D (water surface), 3-D (sound in air).

• Nature of particle vibration compared with direction of wave propagation
– Transverse wave: particle displacement ⟂ direction of energy flow, e.g.
waves on a stretched string, all EM waves.
– Longitudinal wave: particle displacement ∥ direction of energy flow, e.g.
sound in air, compressions in a spring.

• Mode of progression
– Progressive (travelling) waves: energy is transported continuously through the medium.
– Standing (stationary) waves: formed by the superposition of two identical counter-propagating progressive waves; energy is not transported beyond a local region.

Fundamental Wave Parameters

For a sinusoidal (harmonic) wave:

• Amplitude (A) – the maximum displacement of a particle measured from its equilibrium position; numerically the height of a crest or the depth of a trough.

• Wavelength (λ) – the shortest distance between two consecutive points in identical phase (e.g. crest-to-crest or trough-to-trough).

• Frequency (f) – number of complete waves produced per second. Unit: hertz (Hz).

• Wave number (ν) – number of wavelengths accommodated per unit length; $\nu = \frac{1}{\lambda}$. Unit: $\text{m}^{-1}$.

• Angular wave number (propagation constant, k) – number of radians of phase variation per metre; $k = \frac{2\pi}{\lambda}$.

• Angular frequency (ω) – number of radians of phase completed per second; $\omega = 2\pi f$.

• Phase/path relation – if two points are separated by a distance $\Delta x$, their phase difference is

\phi = \frac{2\pi}{\lambda}\,\Delta x = k\,\Delta x.

• Wave (phase) velocity – the speed with which a point of constant phase (e.g. a crest) travels:

v = f\lambda = \frac{\omega}{k}.

Mathematical Representation of Progressive Waves

Take the positive x-direction as the preferred direction.

• Wave travelling to the right ( + x ): $y(x,t) = A\,\sin(\omega t - kx)$.
• Wave travelling to the left ( – x ): $y(x,t) = A\,\sin(\omega t + kx)$.

Alternative but equivalent forms encountered:

• $y = A\,\sin\bigl[2\pi\bigl(f t \mp \tfrac{x}{\lambda}\bigr)\bigr]$
• $y = A\,\sin\bigl[\omega\bigl(t \mp \tfrac{x}{v}\bigr)\bigr]$
• Generic functional notation: $y = f(ax \pm bt)$. Same sign for a and b ⇒ motion toward –x; opposite signs ⇒ motion toward +x.

Example (worked in lecture): For $y = (5\,\text{cm})\,\sin(3\pi t - 3x)$,
Amplitude = 0.05 m, $\omega = 3\pi\,\text{rad s}^{-1}$, $k = 3\,\text{m}^{-1}$, $\lambda = \tfrac{2\pi}{k}=\tfrac{2\pi}{3}\,\text{m}$, $v = \tfrac{\omega}{k}= \tfrac{3\pi}{3}=\pi\,\text{m s}^{-1} \approx 3.14\,\text{m s}^{-1}$.

The Linear Wave Equation

Starting from $y = A\,\sin(\omega t - kx)$ and differentiating twice,

\frac{\partial^2 y}{\partial t^2} = -\omega^{2}y, \qquad
\frac{\partial^2 y}{\partial x^{2}} = -k^{2}y.

Eliminating y gives the one-dimensional wave equation

\boxed{\frac{\partial^{2}y}{\partial t^{2}} = v^{2}\,\frac{\partial^{2}y}{\partial x^{2}}}, \qquad v = \frac{\omega}{k}.

Speed of Waves in Different Media

Transverse Waves on a Stretched String

A short element of string (mass dm, length dx) under tension T undergoes centripetal acceleration when a wave passes. For small slopes $\sin\theta \approx \theta$ and one obtains

v = \sqrt{\frac{T}{\mu}}, \qquad \mu = \frac{\text{mass}}{\text{length}}.

Sound in Solids

Using Young’s modulus (Y) for longitudinal strain,

v_{\text{solids}} = \sqrt{\frac{Y}{\rho}}.

Sound in Fluids

Using bulk modulus (B),

v{\text{fluids}} = \sqrt{\frac{B}{\rho}}. For gases, Laplace’s adiabatic correction with $P\,V^{\gamma}=\text{const}$ leads to v{\text{gas}} = \sqrt{\frac{\gamma P}{\rho}} = \sqrt{\frac{\gamma R T}{M}}.

Consequences:
– At fixed mass, $v \propto \sqrt{T}$ (T in kelvin).
– At fixed temperature, $v \propto \tfrac{1}{\sqrt{M}}$ (lighter gases carry sound faster).

Ordering: v{\text{solids}} > v{\text{liquids}} > v_{\text{gases}}.

Reflection of Waves at a Boundary

• Fixed (rigid) end: the reflected wave undergoes a phase inversion (π rad).
Incident $A\sin(\omega t - kx) \to$ Reflected $-A\sin(\omega t + kx).$

• Free (loosely attached) end: reflection without inversion.
Reflected $A\sin(\omega t + kx).$

Phase differences therefore: fixed end → $\pi$, free end → $0.$

Principle of Superposition

When two or more waves overlap in the same medium, the instantaneous displacement of any particle equals the algebraic (vector) sum of the displacements that each wave would produce separately:

y{\text{total}} = y1 + y_2 + \dots

From superposition flow three important phenomena: interference, beats, and standing waves.

Standing (Stationary) Waves

Generated by two progressive sinusoidal waves with identical A, f, λ but travelling in opposite directions.

Take
$y<em>1 = A\sin(\omega t - kx), \quad y</em>2 = A\sin(\omega t + kx).$
Then

y = y1 + y2 = 2A\sin kx\,\cos \omega t.

Interpretation:
– The spatial factor $\sin kx$ dictates the amplitude distribution (nodes & antinodes).
– The temporal factor $\cos \omega t$ shows all points oscillate in phase or in antiphase; energy is exchanged but not transported.

Node condition: $\sin kx = 0 \Rightarrow kx = n\pi$ ⇒ $x_n = n\,\frac{\lambda}{2}.$

Antinode condition: $\sin kx = \pm1 \Rightarrow kx = (n + \tfrac12)\pi$ ⇒ $x_{an} = (n+\tfrac12)\frac{\lambda}{2}.$

Spacing: node-to-adjacent-node or antinode-to-adjacent-antinode = $\frac{\lambda}{2}$; node-to-antinode = $\frac{\lambda}{4}.$

Standing Waves on a String (Both Ends Fixed)

Let the string length be l, speed of transverse waves v.

Fundamental (1st harmonic): $\lambda<em>1 = 2l, \; f</em>1 = \frac{v}{2l}.$

nth harmonic (n loops):

\lambdan = \frac{2l}{n}, \quad fn = n\,f_1 = n\frac{v}{2l}, \; n = 1,2,3,\dots

Empirical laws (keeping other variables constant):
(1) Law of length: $f \propto \tfrac{1}{l}$.
(2) Law of tension: $f \propto \sqrt{T}$.
(3) Law of mass/diameter: $f \propto \tfrac{1}{\sqrt{\mu}}.$

Air Columns

Speed of sound = v, effective length = l.

Closed Pipe (one end closed)

Only odd harmonics exist.

\lambda1 = 4l, \quad f1 = \frac{v}{4l}.


fn = (2n-1)\,f1, \; n = 1,2,3,\dots

Open Pipe (both ends open)

All harmonics present.

\lambda1 = 2l, \quad f1 = \frac{v}{2l}, \quad fn = n\,f1.

Doppler Effect

Apparent frequency $f'$ heard by an observer when there is relative motion between source and observer.

Let
v = speed of sound,
vs = speed of source (positive if moving toward the observer), vo = speed of observer (positive if moving toward the source).

General formula

\boxed{\displaystyle f' = \frac{v \pm vo}{v \mp vs}\,f}.

Signs: use upper ( + ) when the observer moves toward the source / the source moves toward the observer; use lower ( – ) for motion away.

Special cases (stationary counterpart at rest):
• Source toward stationary observer: $f' = \frac{v}{v - v<em>s}\,f.$ • Source away: $f' = \frac{v}{v + v</em>s}\,f.$
• Observer toward stationary source: $f' = \frac{v+v<em>o}{v}\,f.$ • Observer away: $f' = \frac{v - v</em>o}{v}\,f.$

Example (train whistle, v = 340 m s⁻¹, v_s = 10 m s⁻¹, f = 400 Hz):
Approaching: $f' = \frac{340}{340-10}\,400 \approx 412\,\text{Hz}$.
Receding: $f' = \frac{340}{340+10}\,400 \approx 389\,\text{Hz}.$

Beats

When two waves of nearby frequencies $f<em>1, f</em>2$ superpose, the combined displacement at a fixed position (x = 0) is

y = 2A\cos\bigl[2\pi(f1 - f2)\tfrac{t}{2}\bigr]\,\sin\bigl[2\pi(f1+f2)\tfrac{t}{2}\bigr].

The envelope (cosine term) modulates the amplitude, leading to periodic waxing and waning of loudness. The beat frequency is

f{\text{beat}} = |f1 - f_2|.

Practical hints with tuning forks:
– Filing (removing mass) raises the fork’s frequency ((f \uparrow)).
– Loading (adding wax) lowers the frequency ((f \downarrow)).
– Increasing string tension increases frequency ((f \propto \sqrt{T})).

Illustrative Problems Mentioned in Lecture

1. Two closed organ pipes of lengths 0.20 m and 0.205 m:
   Fundamental frequencies $f<em>1 = \tfrac{v}{4l</em>1} = 412.5\,\text{Hz}, \; f_2 = 402.4\,\text{Hz}.$

2. Open pipe of length 0.30 m resonates with a 1.1 kHz source:
   Allowed open-pipe frequencies $f_n = n\tfrac{v}{2l} = n(550\,\text{Hz})$.
   Hence n = 2 ⇒ second harmonic (1st overtone) matches 1.1 kHz.
   If one end is closed the permitted closed-pipe frequencies are odd multiples 275 Hz, 825 Hz, 1375 Hz … none equals 1.1 kHz; resonance not observed.

3. Guitar string 90 cm, fundamental 124 Hz. Required length for 186 Hz fundamental:
   $f<em>1l</em>1 = f<em>2l</em>2 \Rightarrow l_2 = \frac{124\times90}{186} \approx 60\,\text{cm}.$

4. Two tuning forks give 5 beats s⁻¹; one fork is 512 Hz. Filing the other increases the beat rate implying its original frequency was lower. So unknown fork has 507 Hz (if decrease) or 517 Hz (if increase); beat increase after filing ⇒ initial < 512 Hz ⇒ 507 Hz.

5. Source & observer each move toward one another at 50 m s⁻¹; apparent frequency 434 Hz; v = 332 m s⁻¹.
   $434 = \frac{332+50}{332-50}f \Rightarrow f \approx 320\,\text{Hz}.$

These problems illustrate practical application of formulae for strings, organ pipes, and Doppler shift.

Ethical, Practical and Historical Notes

• Acoustic design (concert halls, organ building) relies on precise knowledge of standing waves and harmonics.
• Doppler radar, velocity measurement in traffic control and astrophysics (red/blue shift) are direct technological extensions of the Doppler principle.
• Safety sirens exploit Doppler shifts to warn pedestrians of approaching vehicles.
• The wave equation is foundational in fields ranging from quantum mechanics to electromagnetic theory, emphasising the unity of wave phenomena across physics.