Waves

Any function of the form \(y = f(x - vt)\) is a wave moving toward \(+x\) with speed \(v\); \(f(x + vt)\) moves toward \(-x\). The shape stays the same and slides along. For example, \(y = \dfrac{1}{1+(x-2t)^2}\) is a pulse moving toward \(+x\) at 2 m/s.

For the harmonic wave \(y = A\sin(kx - \omega t + \phi)\):

• the coefficient of \(x\) is \(k = 2\pi/\lambda\); the coefficient of \(t\) is \(\omega = 2\pi f\);
• the wave speed is \(v = \omega/k\) (ratio of the coefficients, with signs dropped);
• opposite signs of the \(x\) and \(t\) terms mean motion toward \(+x\); the same sign means \(-x\).

If the equation is written as \(y = A\sin2\pi(x/\lambda - t/T)\), read \(\lambda\) and \(T\) directly.

Each particle oscillates up and down; the wave pattern moves sideways. Particle velocity is \(v_p = \partial y/\partial t = -A\omega\cos(kx - \omega t)\), with maximum \(A\omega\). The wave speed is \(v = \omega/k\), fixed by the medium.

• A useful link: \(v_p = -v\,\dfrac{\partial y}{\partial x}\). Where the snapshot slopes upward (in the direction of travel), the particle is moving down, and vice versa.
• \(v_{p,\max}/v = Ak = 2\pi A/\lambda\). The two are equal only if \(\lambda = 2\pi A\).
• Particle acceleration is \(-\omega^2y\): each particle does SHM.

Two points a distance \(\Delta x\) apart along the wave differ in phase by \(\Delta\phi = \dfrac{2\pi}{\lambda}\Delta x\). One point at two times \(\Delta t\) apart differs by \(\Delta\phi = \dfrac{2\pi}{T}\Delta t\).

Points \(\lambda\) apart are in phase. Points \(\lambda/2\) apart are in opposite phase. Points \(\lambda/4\) apart are a quarter cycle apart.

Dimensional check. Tension has units N = kg m s⁻²; linear density has kg m⁻¹. Their ratio has units m² s⁻², so \(\sqrt{T/\mu}\) is a speed. No other combination of \(T\) and \(\mu\) gives a speed.

Physical argument. Ride along with a pulse. A small piece of string of length \(\Delta l\) at the top of the pulse moves on a circular arc of radius \(R\) at speed \(v\). The two tension forces, each at a small angle, give a net inward force \(T\Delta l/R\). This must equal \(\mu\Delta l\,v^2/R\). So \(v^2 = T/\mu\).

For a wire of density \(\rho\) and cross-section \(A\): \(\mu = \rho A\), so \(v = \sqrt{T/(\rho A)} = \sqrt{\text{stress}/\rho}\).

A uniform rope of length \(L\) hangs from the ceiling. At height \(x\) above its lower end, it supports the rope below it: \(T = \mu gx\). So \(v = \sqrt{gx}\), zero at the bottom and \(\sqrt{gL}\) at the top.

The time for a pulse to travel the full length is \(t = \int_0^L dx/\sqrt{gx} = 2\sqrt{L/g}\). It does not depend on the rope’s mass.

If a block hangs from a string and is then dipped in a liquid, buoyancy lowers the tension, which lowers the wave speed and the string’s frequencies.

Wave speed always has the form \(\sqrt{\text{elastic property}/\text{inertial property}}\).

• Longitudinal waves in a thin rod: \(v = \sqrt{Y/\rho}\).
• Sound in a liquid or gas: \(v = \sqrt{B/\rho}\), with \(B\) the bulk modulus.
• Solids are far stiffer than gases, so sound typically travels fastest in solids (steel about 5000 m/s), then liquids (water about 1500 m/s), then gases (air about 340 m/s).
• Transverse waves need a shear stiffness, so they cannot travel through the bulk of a liquid or gas.

The details for gases (Newton’s formula and Laplace’s correction) are in the next section.

Newton treated the compressions in a sound wave as isothermal. For an isothermal ideal gas, \(pV = \text{constant}\), so \(B = -V\,dp/dV = p\). That gives \(v = \sqrt{p/\rho}\approx280\) m/s at STP. The measured value is about 332 m/s, roughly 18% higher.

Laplace pointed out that compressions and rarefactions alternate very fast, and air is a poor conductor of heat. A compressed region warms up and cannot lose that heat before it expands again. The process is adiabatic: \(pV^\gamma = \text{constant}\), so \(B = \gamma p\). The speed rises by \(\sqrt\gamma = \sqrt{1.4}\approx1.18\), giving about 331 m/s.

• Temperature: \(v\propto\sqrt T\) (absolute). Near room temperature, \(v_t\approx v_0 + 0.61\,t\) m/s, with \(t\) in °C and \(v_0\approx332\) m/s.
• Pressure: no effect at constant temperature, because \(p/\rho = RT/M\).
• Humidity: moist air is less dense (water vapour, \(M = 18\) g/mol, replaces heavier N₂ and O₂), so sound travels slightly faster on a humid day.
• Nature of the gas: \(v\propto\sqrt{\gamma/M}\) at the same temperature. Sound in hydrogen is 4 times as fast as in oxygen (same \(\gamma\), \(M\) ratio 1 : 16).
• Wind: adds its component along the direction of travel.
• Frequency, wavelength and amplitude: no effect. All audible frequencies travel together, which is why music from far away still sounds in tune.

Kinetic theory gives \(v_{\rm rms} = \sqrt{3RT/M}\). Dividing, \(v_{\rm sound}/v_{\rm rms} = \sqrt{\gamma/3}\), about 0.68 for air. Sound cannot outrun the molecules that carry it, and both scale as \(\sqrt{T/M}\).

1. A particle of mass \(\Delta m\) in SHM has total energy \(\tfrac12\Delta m\,\omega^2A^2\).
2. Energy per unit volume (energy density): \(u = \tfrac12\rho\omega^2A^2\).
3. In time \(\Delta t\), the energy in a column of length \(v\Delta t\) and area \(S\) crosses the area \(S\): energy \(= u\,Sv\Delta t\).
4. Intensity \(= \dfrac{\text{energy}}{S\Delta t} = uv = \tfrac12\rho v\omega^2A^2\).

At fixed medium, \(I\propto A^2\omega^2\), or \(I\propto A^2f^2\).

• Point source (spherical waves): power \(P\) spreads over \(4\pi r^2\). \(I = P/4\pi r^2\propto1/r^2\), and amplitude \(A\propto1/r\).
• Line source (cylindrical waves, such as a long busy road): \(I\propto1/r\) and \(A\propto1/\sqrt r\).
• Plane waves (far from the source, ideal): \(I\) and \(A\) stay constant.

Absorption in the medium makes real sound fade faster than these ideal laws.

The ear responds to an enormous range of intensities, from about \(10^{-12}\) W/m² (threshold of hearing) to about 1 W/m² (threshold of pain). Its sense of loudness grows roughly with the logarithm of intensity. So we use the sound level

\(\beta = 10\log_{10}(I/I_0)\) dB, with \(I_0 = 10^{-12}\) W/m².

• 10 times the intensity adds 10 dB; 100 times adds 20 dB; doubling adds about 3 dB.
• Typical levels: whisper about 20–30 dB, conversation about 60 dB, busy traffic about 80 dB, pain about 120 dB.
• Loudness is the listener’s sensation and depends mainly on intensity (and also on frequency).
• Pitch depends on frequency. A female voice usually has a higher pitch than a male voice.
• Quality (timbre) depends on which overtones are present and how strong they are. It lets you tell a violin from a flute playing the same note.
• Humans hear about 20 Hz to 20 kHz. Below is infrasound; above is ultrasound.

Two waves of the same frequency meeting with phase difference \(\delta\) give \(A_R^2 = A_1^2 + A_2^2 + 2A_1A_2\cos\delta\). Since \(I\propto A^2\):

\(I_R = I_1 + I_2 + 2\sqrt{I_1I_2}\cos\delta\).

• Maximum (\(\delta = 0, 2\pi, \ldots\)): \(A_{\max} = A_1 + A_2\), \(I_{\max} = (\sqrt{I_1}+\sqrt{I_2})^2\).
• Minimum (\(\delta = \pi, 3\pi, \ldots\)): \(A_{\min} = |A_1 - A_2|\), \(I_{\min} = (\sqrt{I_1}-\sqrt{I_2})^2\).
• \(\dfrac{I_{\max}}{I_{\min}} = \left(\dfrac{A_1+A_2}{A_1-A_2}\right)^2\).
• For equal intensities \(I_0\): \(I_{\max} = 4I_0\), \(I_{\min} = 0\), and the average is \(2I_0\). Energy is redistributed, not created.

Two sources vibrating in phase send waves to a point P. If the path difference is \(\Delta x\), the phase difference is \(\delta = 2\pi\Delta x/\lambda\).

• Constructive at P: \(\Delta x = n\lambda\) (\(n = 0, 1, 2, \ldots\)).
• Destructive at P: \(\Delta x = (2n+1)\lambda/2\).

If the sources themselves are in opposite phase, swap the two conditions.

A steady pattern of loud and quiet spots needs a constant phase difference, so the sources must have the same frequency and a fixed phase relation (coherent). Two independent sources change their phase randomly; the \(\cos\delta\) term averages to zero, and intensities simply add: \(I = I_1 + I_2\).

Add two equal waves travelling in opposite directions:

\(y = A\sin(kx - \omega t) + A\sin(kx + \omega t) = 2A\sin kx\cos\omega t\).

• Every point oscillates at the same \(\omega\), but with its own amplitude \(2A|\sin kx|\).
• Nodes (amplitude zero): \(\sin kx = 0\), \(x = 0, \lambda/2, \lambda, \ldots\)
• Antinodes (amplitude \(2A\)): \(x = \lambda/4, 3\lambda/4, \ldots\)
• All points between two neighbouring nodes move in phase. Points on opposite sides of a node are in antiphase.
• No energy flows past a node, so the wave does not transport energy; energy sloshes between kinetic and potential forms within each loop.

If the waves are \(A\sin(kx - \omega t)\) and \(-A\sin(kx+\omega t)\) instead, the result is \(-2A\cos kx\sin\omega t\), with an antinode at \(x = 0\). The origin’s nature depends on the boundary.

• Fixed end or denser medium: the reflected wave is inverted (phase change \(\pi\)). A crest returns as a trough. The end is a node.
• Free end or rarer medium: no phase change. A crest returns as a crest. The end is an antinode.
• Frequency never changes on reflection or transmission. Wavelength changes only in the transmitted wave if the speed changes.

In a sound standing wave, a displacement node is a pressure antinode, and vice versa. Air molecules on both sides of a displacement node move toward it and away from it together, so the pressure there swings the most. At the closed end of a pipe, the air cannot move (displacement node), but the pressure varies most. At an open end, the pressure stays close to atmospheric (pressure node), while the air moves most.

Both ends must be nodes, so a whole number of half-wavelengths fits: \(L = n\lambda/2\).

• \(\lambda_n = 2L/n\), \(f_n = \dfrac{n}{2L}\sqrt{\dfrac{T}{\mu}}\), with \(n = 1, 2, 3, \ldots\). All harmonics are present.
• The nth harmonic is the \((n-1)\)th overtone. It has \(n\) loops, \(n+1\) nodes and \(n\) antinodes.
• Laws of a sonometer: \(f\propto1/L\) (fixed \(T, \mu\)), \(f\propto\sqrt T\) (fixed \(L, \mu\)), \(f\propto1/\sqrt\mu\) (fixed \(L, T\)).
• Consecutive harmonics differ by \(f_1\). If two consecutive resonances are known, their difference gives the fundamental.

Closed at one end (node at the closed end, antinode at the open end): \(L = (2n-1)\lambda/4\), so \(f = (2n-1)\dfrac{v}{4L}\). Only odd harmonics: \(f_1, 3f_1, 5f_1, \ldots\) Consecutive resonances differ by \(2f_1\).

Open at both ends (antinodes at both ends): \(L = n\lambda/2\), so \(f = n\dfrac{v}{2L}\). All harmonics.

For the same length, the open pipe’s fundamental is twice the closed pipe’s. If an open pipe is closed at one end, its fundamental halves. An open pipe sounds richer because it has all harmonics.

The antinode at an open end forms slightly outside the pipe, at a distance \(e\approx0.6r\) (\(r\) is the pipe’s inner radius). So the effective length is \(L + e\) for a closed pipe and \(L + 2e\) for an open pipe.

In a resonance-tube experiment with a tuning fork, the first two resonating lengths satisfy \(l_1 + e = \lambda/4\) and \(l_2 + e = 3\lambda/4\). Subtracting removes \(e\):

\(\lambda = 2(l_2 - l_1)\), \(v = 2f(l_2 - l_1)\), and \(e = \dfrac{l_2 - 3l_1}{2}\).

At one place, two sound waves of equal amplitude give

\(y = A\sin2\pi f_1t + A\sin2\pi f_2t = \left[2A\cos\pi(f_1 - f_2)t\right]\sin\pi(f_1 + f_2)t\).

• The fast factor \(\sin\pi(f_1+f_2)t\) oscillates at the average frequency \((f_1+f_2)/2\). That is the pitch you hear.
• The slow factor \(2A\cos\pi(f_1-f_2)t\) is the varying amplitude. Loudness depends on its size, not its sign.
• \(|\cos\pi\Delta f\,t|\) peaks every \(1/\Delta f\) seconds. So there are \(|f_1 - f_2|\) loud moments (beats) per second.

The ear holds a sound impression for about 0.1 s. If the loud–soft cycles come faster than about 10 per second, they blur together. So beats are clearly heard only for small frequency differences, up to roughly 10 Hz. Larger differences are heard as a rough or a combined tone.

Suppose a known fork of frequency \(f_0\) gives \(b\) beats per second with an unknown fork X. Then \(f_X = f_0\pm b\). To decide the sign, change X slightly:

• Waxing (loading the prongs) lowers \(f_X\). Filing the prongs raises \(f_X\).
• If lowering \(f_X\) makes the beats fewer, \(f_X\) was above \(f_0\): \(f_X = f_0 + b\).
• If lowering \(f_X\) makes the beats more, \(f_X\) was below: \(f_X = f_0 - b\).
• Reverse the reasoning for filing. For a string, more tension raises the frequency.

If the beats stay the same after the change, the frequency crossed over \(f_0\) (for example, from \(f_0 + b\) to \(f_0 - b\)).

Moving source. A source of frequency \(f\) moves toward the observer at \(v_s\). In one period \(1/f\), a wavefront travels \(v/f\) but the source follows it by \(v_s/f\). So the wavelength ahead shrinks to \(\lambda' = (v - v_s)/f\). The observer receives \(f^{\prime} = v/\lambda' = f\dfrac{v}{v - v_s}\).

Moving observer. The wavelength is unchanged, \(\lambda = v/f\), but an observer moving toward the source at \(v_o\) meets the waves at relative speed \(v + v_o\). So \(f^{\prime} = (v + v_o)/\lambda = f\dfrac{v + v_o}{v}\).

Combining: \(f^{\prime} = f\dfrac{v \pm v_o}{v \mp v_s}\). Sign rule: use the upper signs for motion toward the other party (each raises \(f'\)), the lower signs for motion away.

• Same velocity. Source and observer moving together in still air: \(f^{\prime} = f\) (the factors cancel).
• Wind blowing at \(w\) from source to observer: replace \(v\) by \(v + w\) (or \(v - w\) if it blows the other way). If nothing moves except the air, there is no shift.
• Motion at an angle. Use only the component of velocity along the line joining source and observer. At the moment of closest approach (motion perpendicular to the line), there is no shift.
• Reflection from a moving object. Treat it in two steps: the object first acts as a moving observer, then as a moving source. A car approaching a wall at \(u\) hears its own echo at \(f\dfrac{v+u}{v-u}\).
• Source passing by. The pitch drops suddenly from \(f\dfrac{v}{v - v_s}\) to \(f\dfrac{v}{v + v_s}\).

A source approaching at 100 m/s (with \(v = 340\) m/s) gives \(f^{\prime} = f\times340/240 = 1.42f\). An observer approaching at 100 m/s gives \(f^{\prime} = f\times440/340 = 1.29f\). The results differ because sound moves relative to the air, so “who is moving” relative to the air matters. Only the relative velocity matters for light, which needs no medium.