Waves

A wave is a way of moving energy (and information) from one place to another without moving matter along with it. Drop a stone in a still pond and the ripples travel outward across the surface, but a leaf floating nearby only bobs up and down in place; it never drifts toward the shore. Every particle of the medium does the disturbing locally, and passes it on to its neighbour, nobody actually makes the trip.

Waves that need a physical medium to exist in, string waves, water waves, sound, are called mechanical waves, and they need two properties from that medium: elasticity, so a displaced particle feels a restoring force, and inertia, so it overshoots and keeps the disturbance moving instead of simply snapping back. Light needs neither and travels through vacuum perfectly well, it is not a mechanical wave at all, and we leave it for a later chapter.

A wave travelling steadily in the $+x$ direction, without changing shape, is written as a function of both position and time:

Here $A$ is the amplitude, the maximum displacement of a particle from equilibrium, and $\phi$ is a phase constant fixed by whatever is happening at $x=0,\,t=0$. The two quantities worth knowing cold are the angular wave number $k$, which packs the wavelength into radians per metre, and the angular frequency $\omega$, which does the same for time:

A full cycle in space (one wavelength $\lambda$) and one cycle in time (one period $T$) are linked by how fast the pattern itself glides along, the wave speed:

A wave of the form $\sin(kx+\omega t)$ (plus sign instead of minus) describes exactly the same kind of disturbance travelling in the $-x$ direction instead. Note also that this wave speed is not the speed of any particle of the medium, it is the speed of the pattern; a single particle at fixed $x$ only oscillates up and down with velocity $\partial y/\partial t = -A\omega\cos(kx-\omega t+\phi)$, a completely different quantity that depends on amplitude and frequency, never on how “fast the wave travels”.

For a wave on a stretched string, the propagation speed is set entirely by two mechanical properties of the string itself: how hard it is pulled, and how heavy it is per unit length. If $T$ is the tension and $\mu = m/L$ the linear mass density (mass per unit length), the full derivation from the string’s equation of motion lands on a remarkably clean result:

A quick dimensional check is reassuring: $[T]=MLT^{-2}$ and $[\mu]=ML^{-1}$, so $[T/\mu] = L^2T^{-2}$, exactly velocity squared. Physically, more tension stiffens the restoring force and speeds the wave up; more mass per length adds inertia and slows it down, precisely as you would guess by tightening or loosening a guitar string.

It is worth being deliberate here, because the chapter uses the letter $v$ for two genuinely different things. The $v=\sqrt{T/\mu}$ above is how fast the wave pattern glides along the string, fixed once and for all by $T$ and $\mu$. It has nothing to do with how fast an individual bit of string is moving up and down at some instant, that transverse particle speed depends on the wave’s amplitude and frequency (§14.2) and changes continuously as the particle oscillates. Mixing up these two speeds is one of the most common slips in this chapter.

What happens when two waves arrive at the same point in the medium at the same time? The medium does not have to “choose” between them. The principle of superposition says the net displacement is just the plain algebraic sum of what each wave would have produced on its own:

This holds because the wave equation governing small displacements is linear, double a displacement and you double the restoring force that produces it, with no cross-terms getting in the way. Two consequences follow immediately once the waves separate again: each one carries on completely unaffected by the encounter, as if the other had never been there, and while they overlap, the combined pattern can look very different from either wave alone, including constructive reinforcement, cancellation, or anything in between.

This one principle is doing all the work behind the next two sections: a standing wave (§14.5) is just the superposition of two identical waves travelling in opposite directions, and a beat (§14.6) is just the superposition of two waves of slightly different frequency at a single point. Nothing new is added mathematically, only the situation changes.

A wave reaching the end of its medium does not simply vanish, it reflects, and how it reflects depends on what kind of end it meets. At a fixed end (a string tied firmly to a wall), the support cannot move, so the reflected pulse comes back inverted, flipped through a phase of $\pi$. At a free end (the string tied to a frictionless ring that can slide), the end overshoots and the reflected pulse comes back upright, with no phase change at all.

Now apply superposition (§14.4) to a string fixed at both ends, continuously fed with a travelling wave: the original wave and its reflections combine into two identical waves of the same amplitude and frequency moving in opposite directions, and the sum is a standing wave, a pattern that oscillates in place rather than travelling at all:

Superposition does not need opposite directions to produce something striking, two waves of nearly equal frequency travelling the same way and overlapping at one point in space produce beats: a tone whose loudness rhythmically swells and fades. Take two waves of equal amplitude $A$ at a fixed point, oscillating in time only:

Superposing them and applying the sum-to-product identity, $\sin\alpha + \sin\beta = 2\sin\!\left(\frac{\alpha+\beta}{2}\right)\cos\!\left(\frac{\alpha-\beta}{2}\right)$, with $\alpha = 2\pi f_1 t$ and $\beta = 2\pi f_2 t$, turns the sum into a product:

Read this as a fast oscillation riding inside a slow envelope. The $\sin\big(\pi(f_1+f_2)t\big)$ factor oscillates at the average frequency $(f_1+f_2)/2$, that is the pitch you actually hear, close to either original tone since $f_1\approx f_2$. The bracketed $2A\cos\big(\pi(f_1-f_2)t\big)$ factor varies far more slowly (since $f_1-f_2$ is small) and acts as a time-varying amplitude, the envelope, exactly the swelling-and-fading pattern traced by the orange sum curve in the wave-beats simulation, where it plots $(y_1+y_2)/2$ for two waves of unit amplitude.

One subtlety is worth getting right: loudness depends only on the envelope’s size, $|\cos(\pi(f_1-f_2)t)|$, not its sign, and a bare cosine reaches its peak size twice every cycle, once at $+1$ and once at $-1$. So a loud moment is heard twice as often as the envelope’s own mathematical frequency $(f_1-f_2)/2$ would suggest, which exactly doubles it back to the familiar result:

and correspondingly $T_{\text{beat}} = 1/|f_1-f_2|$ is the time between one loud moment and the next.

The Doppler effect is the shift in a wave’s observed frequency caused by relative motion between the source and the observer, even though the source itself never changes what frequency it is emitting. The everyday giveaway is a train sounding a fixed, steady whistle: its pitch sounds distinctly higher while it approaches a platform and drops to distinctly lower the instant it passes and recedes, despite the driver hearing the exact same note the whole time.

A source moving toward a stationary observer effectively “crowds” each successive wavefront closer to the one before it (it has moved forward slightly before emitting the next crest), shortening the wavelength reaching the observer and raising the frequency heard; moving away does the reverse. For a source moving at speed $v_s$ and sound speed $v$ in the medium:

A moving observer instead changes how often wavefronts are intercepted, not their spacing: approaching the source at speed $v_o$ sweeps up extra wavefronts per second, while receding misses some, giving $f' = f_0(v\pm v_o)/v$. One detail trips up almost everyone at first: the actual speed of sound through the medium, $v$, is a property of the medium alone (its elasticity and density) and is completely unaffected by how fast the source or observer happens to be moving.

Two of the chapter’s own classic exercise questions, with original worked solutions.