12.1 Waves, Speed of Sound, Standing Waves & Doppler Effect

Waves, Speed of Sound, Standing Waves & Doppler Effect

In the Intermediate Physics Part I syllabus and the University of Engineering and Technology (UET) ECAT entry test, wave mechanics, acoustics, and the Doppler effect are heavily weighted areas. Engineering entrance questions test speed-of-sound corrections, boundary-dependent standing wave harmonics, beat frequencies, and multi-case Doppler shift calculations without calculators. Mastering the fundamental derivations and physical mechanisms behind these concepts provides immediate analytical advantages.

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1. Nature of Waves and Wave Kinematics

A wave is a physical disturbance that travels through space or an elastic medium, transporting energy and momentum from one location to another without transporting matter. Waves are broadly classified into two categories:

1. Mechanical Waves: Require an elastic, inertial material medium for propagation (e.g., sound waves, water ripples, seismic waves, vibrations in stretched strings).
2. Electromagnetic Waves: Periodic oscillations of orthogonal electric and magnetic fields that propagate through vacuum at the speed of light $c = 3.0 \times 10^8\text{ m/s}$ as well as through transparent material media (e.g., light, radio waves, X-rays).

Mechanical Waves: Transverse vs. Longitudinal

Mechanical waves depend entirely on the restoring force (elasticity) and inertial properties (density) of the transmitting medium:

• Transverse Waves: The particles of the medium oscillate perpendicular to the direction of wave propagation. Transverse mechanical waves propagate as alternating crests (points of maximum positive displacement) and troughs (points of maximum negative displacement). Because transverse waves require shear elasticity to sustain lateral displacements, they can propagate only through solid media and across liquid surfaces (under surface tension and gravity), but cannot travel through the interior of liquids or gases.
• Longitudinal Waves: The particles of the medium oscillate parallel to the direction of wave propagation. Longitudinal waves propagate as alternating regions of compressions (high density, high pressure) and rarefactions (low density, low pressure). Because all states of matter (solids, liquids, and gases) possess volume elasticity (bulk modulus), longitudinal waves travel through all three phases. Sound in air is strictly longitudinal.

The Fundamental Wave Speed Equation

For any periodic wave of frequency $f$, period $T = \frac{1}{f}$, and wavelength $\lambda$ (the spatial distance between two successive points in identical phase, such as two consecutive crests or compressions):

$v = \frac{\text{Distance}}{\text{Time}} = \frac{\lambda}{T} = f\lambda$

In angular notation, with angular frequency $\omega = 2\pi f$ and wave number (propagation constant) $k = \frac{2\pi}{\lambda}$:

$v = \frac{\omega}{k}$

ECAT Conceptual Rule: The frequency $f$ of a wave is determined exclusively by the source and remains strictly invariant when the wave crosses the boundary between two different media. By contrast, the wave speed $v$ is determined strictly by the physical properties of the medium. Consequently, when a wave enters a medium where its speed changes, its wavelength $\lambda = \frac{v}{f}$ changes proportionally.

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2. Speed of Sound in Gases: Newton vs. Laplace

The speed of a longitudinal mechanical wave through an elastic fluid of bulk modulus $E$ and density $\rho$ is given by the hydrodynamic relation:

$v = \sqrt{\frac{E}{\rho}}$

Newton's Formula (Isothermal Assumption)

Sir Isaac Newton assumed that when sound travels through air, the compressions and rarefactions occur so slowly that heat generated during compression escapes to the surroundings, and heat lost during rarefaction is absorbed from the surroundings. Consequently, Newton assumed the acoustic process is isothermal ($T = \text{constant}$).

Applying Boyle's law for an ideal gas at constant temperature:

$P V = \text{constant}$

Differentiating both sides with respect to volume:

$P(dV) + V(dP) = 0 \implies P = -\frac{dP}{dV / V} = E_T$

Thus, the isothermal bulk modulus $E_T$ equals the ambient equilibrium pressure $P$. Substituting $E_T = P$ into the wave speed formula yields Newton's formula for the speed of sound:

$v = \sqrt{\frac{P}{\rho}}$

At standard temperature and pressure (STP: $P = 1.013 \times 10^5\text{ N/m}^2$, $\rho = 1.293\text{ kg/m}^3$ for dry air):

$v_{\text{Newton}} = \sqrt{\frac{1.013 \times 10^5}{1.293}} \approx 280\text{ m/s}$

The experimentally observed speed of sound in air at $0^\circ\text{C}$ is approximately $332\text{ m/s}$. Newton's theoretical value is about $16\%$ lower than the true physical value. This discrepancy proved that Newton's isothermal assumption was incorrect.

Laplace's Correction (Adiabatic Assumption)

In 1816, the French mathematician Pierre-Simon Laplace identified the flaw in Newton's reasoning. Sound waves in air oscillate at audible frequencies ($20\text{ Hz}$ to $20{,}000\text{ Hz}$). At these rapid oscillation rates:

1. Compressions and rarefactions occur extremely fast.
2. Air is a poor thermal conductor.

Therefore, heat has no time to flow between compressions and rarefactions. The propagation of sound is an adiabatic process ($Q = 0$), not an isothermal one.

For an adiabatic process in an ideal gas:

$P V^\gamma = \text{constant}$

where $\gamma = \frac{C_p}{C_v}$ is the ratio of molar heat capacities (the adiabatic index). Differentiating both sides:

$P(\gamma V^{\gamma-1} dV) + V^\gamma dP = 0$

Dividing through by $V^{\gamma-1}$:

$\gamma P dV + V dP = 0 \implies \gamma P = -\frac{dP}{dV / V} = E_S$

Thus, the adiabatic bulk modulus $E_S$ is $\gamma$ times the pressure: $E_S = \gamma P$. Substituting this into the hydrodynamic formula yields Laplace's corrected formula for the speed of sound:

$v = \sqrt{\frac{\gamma P}{\rho}}$

Air consists primarily of diatomic molecules ($\text{N}_2$ and $\text{O}_2$), for which $\gamma = 1.40$ (or $\frac{7}{5}$). Substituting STP values:

$v_{\text{Laplace}} = \sqrt{\frac{1.4 \times 1.013 \times 10^5}{1.293}} = \sqrt{1.4} \times 280\text{ m/s} \approx 1.183 \times 280\text{ m/s} \approx 332\text{ m/s}$

Laplace's corrected theoretical value matches experimental measurements within experimental uncertainty.

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3. Physical Factors Affecting the Speed of Sound in Air

1. Effect of Pressure

Using the ideal gas equation $P M = \rho R T \implies \frac{P}{\rho} = \frac{R T}{M}$, we express Laplace's formula as:

$v = \sqrt{\frac{\gamma R T}{M}}$

If the pressure of a gas changes while its temperature $T$ remains constant, the density $\rho$ changes in direct proportion (Boyle's law: $\frac{P}{\rho} = \text{constant}$). Consequently, the speed of sound is completely independent of pressure changes at constant temperature.

2. Effect of Temperature

Because $v = \sqrt{\frac{\gamma R T}{M}}$, the speed of sound is directly proportional to the square root of the absolute thermodynamic temperature $T$ (in Kelvin):

$v \propto \sqrt{T} \implies \frac{v_1}{v_2} = \sqrt{\frac{T_1}{T_2}}$

To find the speed of sound $v_t$ at a temperature $t$ in Celsius ($T = T_0 + t$, where $T_0 = 273.15\text{ K}$):

$v_t = v_0\sqrt{\frac{T_0 + t}{T_0}} = v_0\left(1 + \frac{t}{273.15}\right)^{1/2}$

Applying the binomial expansion for $t \ll 273.15$:

$v_t \approx v_0\left(1 + \frac{1}{2} \cdot \frac{t}{273.15}\right) = v_0 + \left(\frac{v_0}{546.3}\right)t$

Taking $v_0 = 332\text{ m/s}$:

$\frac{332}{546.3} \approx 0.61\text{ m/s per }^\circ\text{C}$

$v_t = v_0 + 0.61 t$

ECAT Speed Rule: For every $1^\circ\text{C}$ rise in temperature, the speed of sound in air increases by approximately $0.61\text{ m/s}$ (about $61\text{ cm/s}$); the linear rule is an approximation that works near ordinary temperatures.

3. Effect of Moisture (Humidity)

The molar mass of water vapor ($M_{\text{H}_2\text{O}} = 18\text{ g/mol}$) is significantly less than the effective molar mass of dry air ($M_{\text{air}} \approx 29\text{ g/mol}$). Consequently, adding moisture decreases the effective density $\rho$ of humid air relative to dry air at identical temperature and pressure. Because $v = \sqrt{\frac{\gamma P}{\rho}}$, sound travels faster in humid air than in dry air.

4. Effect of Wind

If wind blows with velocity $w$ at an angle $\theta$ relative to the sound propagation vector, the effective speed of sound observed on the ground is the vector sum:

$v_{\text{eff}} = v + w\cos\theta$

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4. Principle of Superposition & Standing (Stationary) Waves

The Superposition Principle

When two or more waves traverse the same elastic medium simultaneously, the resultant displacement of any particle of the medium at any instant is the vector sum of the individual displacements caused by each wave:

$y(x, t) = y_1(x, t) + y_2(x, t) + \dots + y_n(x, t)$

Depending on the frequencies, directions, and phases of the interfering waves, superposition manifests as:

1. Interference: Superposition of two waves of identical frequency propagating in the same direction.
2. Beats: Superposition of two waves of slightly different frequencies propagating in the same direction.
3. Stationary (Standing) Waves: Superposition of two waves of identical amplitude, frequency, and wavelength propagating in opposite directions.

Mathematical Formulation of Standing Waves

Consider two identical sinusoidal waves travelling along a stretched string in opposite directions:

$y_1(x, t) = A\sin(kx - \omega t), \quad y_2(x, t) = A\sin(kx + \omega t)$

Applying the trigonometric identity $\sin(A - B) + \sin(A + B) = 2\sin A\cos B$:

$y(x, t) = y_1 + y_2 = 2A\sin(kx)\cos(\omega t)$

This expression describes a stationary wave:

• The spatial term $A'(x) = 2A\sin(kx)$ represents the position-dependent amplitude of oscillation.
• The temporal term $\cos(\omega t)$ shows that all particles oscillate in simple harmonic motion with frequency $\omega$, but with different amplitudes.
• There is no travelling phase factor of the form $(kx \pm \omega t)$; hence, wave energy is trapped between boundaries and does not propagate along the medium.

Nodes and Antinodes

1. Nodes ($N$): Points where the amplitude is permanently zero: $A'(x) = 0$.

$\sin(kx) = 0 \implies kx = n\pi \implies \frac{2\pi}{\lambda}x = n\pi \implies x = n\frac{\lambda}{2} \quad (n = 0, 1, 2, 3, \dots)$

• Position of nodes: $x = 0, \frac{\lambda}{2}, \lambda, \frac{3\lambda}{2}, \dots$
• Distance between two consecutive nodes: $d_{N-N} = \frac{\lambda}{2}$.
• At nodes, displacement is zero, but strain and pressure variations are maximum.

1. Antinodes ($A$): Points where the amplitude reaches maximum value: $A'(x) = \pm 2A$.

$\sin(kx) = \pm 1 \implies kx = \left(n + \frac{1}{2}\right)\pi \implies \frac{2\pi}{\lambda}x = (2n+1)\frac{\pi}{2} \implies x = (2n+1)\frac{\lambda}{4}$

• Position of antinodes: $x = \frac{\lambda}{4}, \frac{3\lambda}{4}, \frac{5\lambda}{4}, \dots$
• Distance between two consecutive antinodes: $d_{A-A} = \frac{\lambda}{2}$.
• Distance between an adjacent node and antinode: $d_{N-A} = \frac{\lambda}{4}$.

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5. Resonant Standing Waves in Stretched Strings & Organ Pipes

1. Stretched String Fixed at Both Ends

A string of length $L$, tension $T$, and linear mass density (mass per unit length) $m = \frac{M}{L}$ supports transverse waves with speed:

$v = \sqrt{\frac{T}{m}}$

Because both ends are rigidly clamped, displacement nodes must occur at $x = 0$ and $x = L$. This boundary condition requires that the length $L$ equals an integer number of half-wavelengths:

$L = n\frac{\lambda_n}{2} \implies \lambda_n = \frac{2L}{n} \quad (n = 1, 2, 3, \dots)$

The resonant natural frequencies (harmonics) are:

$f_n = \frac{v}{\lambda_n} = n\frac{v}{2L} = \frac{n}{2L}\sqrt{\frac{T}{m}} = n f_1$

• Fundamental Frequency (First Harmonic, $n = 1$): $f_1 = \frac{v}{2L} = \frac{1}{2L}\sqrt{\frac{T}{m}}$, with 2 nodes and 1 antinode.
• Second Harmonic (First Overtone, $n = 2$): $f_2 = 2f_1$, with 3 nodes and 2 antinodes.
• Third Harmonic (Second Overtone, $n = 3$): $f_3 = 3f_1$, with 4 nodes and 3 antinodes.
• Harmonic Series: Stretched strings produce all integer harmonics ($f_1 : f_2 : f_3 : \dots = 1 : 2 : 3 : \dots$).

2. Standing Waves in Organ Pipes

In an acoustic pipe, air molecules at a closed end cannot oscillate longitudinally, creating a displacement node (pressure antinode). At an open end, air molecules oscillate freely into the open atmosphere, creating a displacement antinode (pressure node).

A. Open Organ Pipe (Both Ends Open)

Both ends must be displacement antinodes. The resonant condition requires:

$L = n\frac{\lambda_n}{2} \implies \lambda_n = \frac{2L}{n} \implies f_n = n\frac{v}{2L} = n f_1 \quad (n = 1, 2, 3, \dots)$

• Fundamental mode ($n = 1$): $L = \frac{\lambda_1}{2} \implies f_{1,\text{open}} = \frac{v}{2L}$.
• Open organ pipes produce all integer harmonics ($1 : 2 : 3 : 4 : \dots$).

B. Closed Organ Pipe (One End Closed, One End Open)

The closed end is a displacement node; the open end is an antinode. The length must accommodate an odd number of quarter-wavelengths:

$L = (2n - 1)\frac{\lambda_n}{4} \implies \lambda_n = \frac{4L}{2n - 1} \implies f_n = (2n - 1)\frac{v}{4L} = (2n - 1)f_1 \quad (n = 1, 2, 3, \dots)$

• Fundamental mode ($n = 1$): $L = \frac{\lambda_1}{4} \implies f_{1,\text{closed}} = \frac{v}{4L}$.
• Third harmonic (First overtone, $n = 2$): $f_2 = 3f_{1,\text{closed}}$.
• Fifth harmonic (Second overtone, $n = 3$): $f_3 = 5f_{1,\text{closed}}$.
• Closed organ pipes produce only odd harmonics ($1 : 3 : 5 : 7 : \dots$).

ECAT Key Relationship: For pipes of identical physical length $L$: $f_{1,\text{closed}} = \frac{1}{2} f_{1,\text{open}}$ An open pipe has twice the fundamental frequency of a closed pipe of the same length and is acoustically richer in tonal quality because all harmonics are present.

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6. Acoustic Beats

When two sound waves of slightly different frequencies $f_1$ and $f_2$ (typically $|f_1 - f_2| \le 10\text{ Hz}$) travel along the same path in the same direction, their superposition produces periodic waxing and waning of sound intensity known as beats.

Let the displacements be $y_1 = A\sin(2\pi f_1 t)$ and $y_2 = A\sin(2\pi f_2 t)$. By trigonometric identity:

$y = y_1 + y_2 = \left[2A\cos\left(2\pi\frac{f_1 - f_2}{2}t\right)\right] \sin\left(2\pi\frac{f_1 + f_2}{2}t\right)$

The modulated amplitude is $A_{\text{mod}} = 2A\cos\left(2\pi\frac{f_1 - f_2}{2}t\right)$. Intensity is proportional to $A_{\text{mod}}^2$, reaching maximum when $\cos = \pm 1$:

$\text{Frequency of intensity maxima (Beat Frequency):} \quad f_{\text{beat}} = |f_1 - f_2|$

Practical Applications of Beats

1. Tuning Musical Instruments: A musician sounds an instrument alongside a standard tuning fork. The tension is adjusted until the beat frequency decreases to zero, indicating perfect unison ($f_1 = f_2$).
2. Detection of Toxic Mine Gases: Two identical resonant pipes, one filled with pure air and the other exposed to mine air, sound simultaneously. If methane or coal gas is present, the speed of sound inside changes, producing audible beats that warn miners.

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7. The Doppler Effect

The Doppler effect is the apparent change in the observed frequency (or pitch) of a wave when there is relative motion between the source of waves and the observer.

Let $v$ be the speed of sound in the stationary medium, $f$ the true source frequency, and $\lambda = \frac{v}{f}$ the stationary wavelength.

Case 1: Observer Moving, Source Stationary ($u_s = 0$)

When the source is stationary, the physical wavelength $\lambda$ in the medium remains unchanged. The observer moves relative to the wavefronts, altering the relative speed of wave interception:

• Observer moves toward stationary source with speed $u_o$: Relative speed of sound is $v' = v + u_o$.

$f' = \frac{v'}{\lambda} = \frac{v + u_o}{v / f} = f\left(\frac{v + u_o}{v}\right) > f$

• Observer moves away from stationary source with speed $u_o$: Relative speed of sound is $v' = v - u_o$.

$f' = \frac{v'}{\lambda} = \frac{v - u_o}{v / f} = f\left(\frac{v - u_o}{v}\right) < f$

Case 2: Source Moving, Observer Stationary ($u_o = 0$)

When the source moves, it compresses or stretches the physical wavelength in the medium before the waves reach the observer:

• Source moves toward stationary observer with speed $u_s$: Wavefronts are compressed in front of the source. During one period $T = \frac{1}{f}$, the source advances by distance $u_s T$. The modified wavelength is $\lambda' = \lambda - u_s T = \frac{v - u_s}{f}$.

$f' = \frac{v}{\lambda'} = f\left(\frac{v}{v - u_s}\right) > f$

• Source moves away from stationary observer with speed $u_s$: Wavefronts are stretched behind the source: $\lambda' = \lambda + u_s T = \frac{v + u_s}{f}$.

$f' = \frac{v}{\lambda'} = f\left(\frac{v}{v + u_s}\right) < f$

Generalized Doppler Formula

Combining both motions into a single universal equation:

$f' = f\left(\frac{v \pm u_o}{v \mp u_s}\right)$

• Upper signs ($+u_o, -u_s$): Used for approaching motion (frequency increases).
• Lower signs ($-u_o, +u_s$): Used for receding motion (frequency decreases).

Asymmetry of the Doppler Effect in Sound

Because sound requires a material medium, motion of the source is physically distinct from motion of the observer:

• If observer moves toward stationary source at $u = 0.5v$: $f' = f(1 + 0.5) = 1.5 f$.
• If source moves toward stationary observer at $u = 0.5v$: $f' = f\left(\frac{1}{1 - 0.5}\right) = 2.0 f$.

Thus, the frequency shift is greater when the source moves than when the observer moves at the same speed.

Astronomical Doppler Effect: Redshift and Blueshift

For light, electromagnetic waves do not require a material medium. For velocities $u \ll c$:

$\frac{\Delta f}{f} = \frac{|\Delta \lambda|}{\lambda} \approx \frac{u}{c}$

1. Redshift (Receding Celestial Bodies): If a distant star or galaxy recedes from Earth, the observed wavelength increases toward the longer-wavelength red end of the optical spectrum. Edwin Hubble used this redshift to discover cosmic expansion (Hubble's Law: $v = H_0 d$).
2. Blueshift (Approaching Celestial Bodies): If a celestial object moves toward Earth, the observed wavelength shifts toward shorter blue wavelengths ($f' > f$).

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8. Standing Waves and Doppler Formulas Summary Table