Waves

1. What this chapter covers

Chapter 13 studied a single oscillator in isolation. A material medium is a vast collection of them, coupled together by elastic forces — displace one, and its neighbours feel it too. That coupling is what makes a wave possible, and everything in this chapter follows from taking Chapter 13's oscillator equations and asking what happens when they are chained together.

2. A wave moves; the medium mostly doesn't

Watch cork pieces floating on a pond after a pebble drop: they bob up and down but never drift outward with the expanding ripple. The disturbance travels; the water, on average, stays put. This is the definition of a wave — a pattern that propagates through a medium without net transport of the medium itself. A wind (air moving as a whole) is not a sound wave (a pressure disturbance propagating through air that stays put).

Picture a chain of springs, each connected to the next. Pull one end and release: the disturbance travels down the chain, but each spring only oscillates about its own equilibrium length — nothing travels bodily from one end to the other, the same way a shove at the front of a coupled train of railway bogies passes backward through the couplings without the whole train lurching forward as one block.

Sound in air works the same way: a compressed region pushes its neighbour, which compresses in turn and leaves the original region rarefied, and the compression-rarefaction pattern walks forward through air that never travels far from where it started.

3. Transverse and longitudinal waves — and why gases can't do both

If the medium's constituents oscillate perpendicular to the direction the wave travels, the wave is transverse (a jerk on a string). If they oscillate along the direction of travel, the wave is longitudinal (sound in a pipe, driven by a piston pushing and pulling).

This distinction is not just descriptive — it decides which media can carry which wave at all. A transverse wave shears each element of the medium sideways, so it needs a medium that can sustain shear stress: solids can, fluids cannot (a fluid just flows instead of springing back).

A longitudinal wave only ever compresses or stretches the medium along its own direction, which needs only a bulk or compressive modulus — present in solids, liquids, and gases. That is the entire reason steel can carry both wave types while air carries only longitudinal ones.

4. The displacement relation: one equation, read two different ways

A sinusoidal travelling wave moving in the positive x-direction is:

$y(x,t)=a\sin (kx-\omega t+\varphi )$

• $a$ — amplitude, the maximum displacement.
• $kx-\omega t+\varphi$ — the phase.
• $\varphi$ — the initial phase angle (phase at $x=0,t=0$); origin and clock start can always be chosen to make $\varphi =0$ without loss of generality.
• $k=2\pi /\lambda$ — angular wave number, radians per metre.
• $\omega =2\pi /T=2\pi \nu$ — angular frequency.

This single function answers two different questions depending on which variable you freeze. Fix time $t=t_0$: the equation becomes a function of $x$ alone, giving the wave's shape in space at that instant. Fix position $x=x_0$: it becomes a function of $t$ alone, showing that every constituent of the medium executes SHM — this is the direct link back to Chapter 13.

A wave written as $a\sin (kx+\omega t+\varphi )$ — with a plus sign between the terms — travels in the negative x-direction instead. The sign between $kx$ and $\omega t$ is the entire direction indicator; nothing else in the equation needs to change.

Worked example. For $y(x,t)=0.005\sin (80.0x-3.0t)$ m: $k=80.0{\text{m}}^{-1}$ gives $\lambda =2\pi /k=7.85$ cm; $\omega =3.0{\text{s}}^{-1}$ gives $T=2\pi /\omega =2.09$ s and $\nu =1/T=0.48$ Hz.

5. Speed of a wave: a property of the medium, not of the wave itself

Track a fixed point on the wave — a crest, say — and ask how fast it moves. The condition "same phase" means $kx-\omega t=\text{constant}$, and differentiating gives the wave speed directly:

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

The genuinely important fact hiding in this simple relation: wave speed is set entirely by the medium's own inertial and elastic properties — tension and mass density for a string, bulk modulus and density for sound — never by the wavelength or frequency of the wave passing through. A source picks the frequency; the medium's speed then fixes the wavelength via $\lambda =v/\nu$, not the other way around.

Speed on a stretched string

Dimensional analysis alone can narrow the formula down but never fix its numerical constant — tension $T$ has dimension $[{\text{MLT}}^{-2}]$, linear mass density $\mu$ has dimension $[{\text{ML}}^{-1}]$, and only $\sqrt{T/\mu }$ combines to the dimension of speed $[{\text{LT}}^{-1}]$. A full derivation from Newton's laws (outside this book's scope) shows the missing constant is exactly 1:

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

Speed of sound: Newton got it wrong, and the fix is worth knowing

For a longitudinal wave, dimensional analysis on bulk modulus $B$ and density $\rho$ gives $v=\sqrt{B/\rho }$. For a solid bar under longitudinal strain, the relevant modulus is Young's modulus instead: $v=\sqrt{Y/\rho }$.

Newton assumed sound propagates through air isothermally, which makes $B=P$ (from the ideal gas law at constant $T$), giving $v=\sqrt{P/\rho }$. Applied to air at STP, this predicts about 280 m/s — roughly 15% below the measured 331 m/s.

Laplace identified the error: pressure changes in a sound wave happen far too fast for heat to flow and keep temperature constant, so the process is adiabatic, not isothermal. For an adiabatic ideal gas, $B_{ad}=\gamma P$ instead of just $P$, giving the corrected formula:

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

With $\gamma =7/5$ for air, this predicts 331.3 m/s — matching experiment. The lesson is as much about scientific method as physics: dimensional analysis got the form right immediately, but only identifying the correct physical assumption (adiabatic, not isothermal) got the number right.

6. Superposition: waves pass through each other unchanged

When two wave pulses meet, the net displacement at every point is simply the algebraic sum of what each pulse would have produced alone — the principle of superposition. After they cross, each pulse continues on exactly as if the other had never been there.

For two waves of equal amplitude $a$, same $\omega$ and $k$, differing only by phase constant $\varphi$:

$y(x,t)=2a\cos \left(\frac{\varphi }{2}\right)\sin \left(kx-\omega t+\frac{\varphi }{2}\right)$

still a travelling wave at the same frequency, but now with amplitude $2a\cos (\varphi /2)$ — a function of the phase difference alone. At $\varphi =0$ (in phase), amplitude is the maximum possible, $2a$: constructive interference. At $\varphi =\pi$ (exactly out of phase), amplitude is zero everywhere, for all time: destructive interference. Nothing about the individual waves' energy is destroyed — it is redistributed to wherever the interference is constructive elsewhere.

7. Reflection: a phase flip at a wall, none at a free end

At a rigid boundary, the medium there is physically forced to stay at zero displacement always — the only way superposition can guarantee that is if the reflected wave is exactly $\pi$ out of phase with the incident one, cancelling it at that point at every instant. Equivalently: the incident pulse pushes on the wall, and by Newton's third law the wall pushes back, generating a reflected pulse inverted relative to the original.

At a free (open) boundary — like a string end tied to a ring that slides freely — nothing constrains the displacement there, so the reflected wave keeps the same phase as the incident one, and the two add constructively at that point, reaching twice the single-pulse amplitude momentarily.

8. Standing waves: when reflection happens at both ends

A wave travelling right, $a\sin (kx-\omega t)$, and its reflection travelling left, $a\sin (kx+\omega t)$, superpose to:

$y(x,t)=[2a\sin (kx)]\cos (\omega t)$

Notice $x$ and $t$ no longer appear in the combination $kx-\omega t$ — they appear separately. This pattern does not travel in either direction; it is a standing (stationary) wave. Every point oscillates with the same angular frequency $\omega$, but the amplitude $2a\sin (kx)$ depends on position: zero at nodes ($\sin kx=0$), maximum at antinodes ($\sin kx=\pm 1$). Adjacent nodes (or adjacent antinodes) are always separated by $\lambda /2$.

Boundary conditions — where the physical setup forces a node or antinode — restrict a standing-wave system to a discrete set of allowed frequencies, unlike a travelling wave which can have any frequency at all.

Worked example. A pipe 30 cm long, open at both ends, needs to resonate with a 1.1 kHz source ($v=330$ m/s). Its harmonics are ${\nu }_n=nv/2L=n(330)/0.6=550n$ Hz, so $n=2$ gives exactly 1100 Hz — the source excites the second harmonic.

Close one end of the same pipe: only odd harmonics survive, at ${\nu }_n=(2n-1)v/4L=(2n-1)(275)$ Hz — 275, 825, 1375 Hz, and so on. Since 1100 Hz never appears in that list, no resonance occurs once the end is closed.

A string or air column rarely vibrates in a single pure mode — real vibration is usually a superposition of several harmonics at once, with the relative strength of each depending on how and where the string is plucked or bowed.

This is why a sitar and a violin playing the identical fundamental note still sound distinguishably different: the note's pitch is set by the fundamental frequency, but its timbre — the mix of overtones riding on top of it — is what your ear actually uses to tell the two instruments apart.

9. Beats: the audible throb of two close frequencies

Superpose two waves of nearly equal (not identical) frequency ${\omega }_1$ and ${\omega }_2$:

$s=a\cos {\omega }_{1}t+a\cos {\omega }_{2}t=\left[2a\cos ({\omega }_{b}t)\right]\cos ({\omega }_{a}t)$

where ${\omega }_a=({\omega }_1+{\omega }_2)/2$ is the fast average frequency you actually hear as pitch, and ${\omega }_b=|{\omega }_1-{\omega }_2|/2$ is a slow envelope modulating the amplitude of that fast oscillation. Because intensity depends on amplitude squared, the loudness waxes and wanes twice per cycle of the envelope, giving a beat frequency of:

${\nu }_{beat}=|{\nu }_1-{\nu }_2|$

Musicians use this directly: two strings are in tune exactly when the beats — that audible throbbing — disappear entirely.

Worked example. Two sitar strings A and B, both playing the same note, produce 5 beats per second. Tightening string B (which raises its frequency) makes the beat frequency drop to 3 beats per second.

Since raising B's frequency reduced the gap, B's frequency must already have been below A's — if it had been above, raising it further would only have widened the gap. With ${\nu }_A=427$ Hz and ${\nu }_A-{\nu }_B=5$, string B's original frequency was $422$ Hz.

Summary

• A wave transports a disturbance and energy through a medium without net transport of the medium itself — the water under a ripple, or the air carrying a sound, mostly stays where it started.
• Transverse waves (perpendicular oscillation) need a shear modulus, so only solids carry them. Longitudinal waves (parallel oscillation) only need a bulk modulus, so solids, liquids, and gases all carry them.
• $y(x,t)=a\sin (kx-\omega t+\varphi )$ describes a wave moving in the $+x$ direction; a $+$ sign between $kx$ and $\omega t$ reverses the direction.
• Wave speed $v=\omega /k=\lambda \nu$ is fixed by the medium's own properties — never by the wave's own frequency or wavelength.
• $v=\sqrt{T/\mu }$ for a string; $v=\sqrt{B/\rho }$ (or $\sqrt{Y/\rho }$ for a solid bar) for longitudinal waves; for a gas, Laplace's adiabatic correction $v=\sqrt{\gamma P/\rho }$ fixes Newton's isothermal formula's 15% error.
• Superposition: two waves overlapping add algebraically; equal-amplitude waves interfere constructively at $\varphi =0$ (amplitude $2a$) and destructively at $\varphi =\pi$ (amplitude $0$).
• Reflection flips phase by $\pi$ at a rigid boundary, and leaves phase unchanged at a free boundary.
• Standing waves form when reflection happens at both ends, restricting the system to a discrete ladder of allowed frequencies — every harmonic for a string or open-open pipe, only odd harmonics for a pipe closed at one end.
• Beats arise from superposing two close-but-different frequencies; the beat frequency is simply their difference, ${\nu }_{beat}=|{\nu }_1-{\nu }_2|$.