Wave Motion & Waves on a String

Say “hello” to a friend across the room. No air from your mouth reaches their ear, yet your voice does. Energy has travelled through the air while the air itself stayed put. That is a wave.

Energy without the stuff#

A moving particle carries its energy with it. A wave carries energy in a different way. When you speak you squeeze and stretch the air next to your lips; that air pushes and pulls the next layer, and so on, until the air by your friend’s ear is disturbed. The disturbance travels; the air molecules near you stay near you.

Picture a restless queue at a closed ticket window. The person at the back leans forward and shoves the next person, then stands straight again. That person stumbles into the one in front, and recovers. The jolt runs all the way to the front, carrying energy with it, while everyone ends up roughly where they started. Raindrops on a pond send out circular ripples in the same way.

A pulse on a string#

Tie one end of a long string to a wall, pull the other end tight and flick your hand quickly up and down once. A bump forms near your hand and runs along the string toward the wall. For a uniform, elastic string and a small bump, it travels at a constant speed without changing shape. No part of the string moves toward the wall: each bit simply moves up and back down as the bump passes, receiving energy from the bit behind it and handing it on.

A disturbance confined to a small region at any moment is a wave pulse. Keep shaking for a while (say ten times) and a wave train of ten loops runs down the string.

The equation of a travelling wave#

Put the hand end at x = 0, the string along +x, and let the hand’s displacement be $y(0, t) = f(t)$. The disturbance travels at speed v, so whatever the end does at time t reaches the point x at time t + x/v. Turned around: the point x, at time t, is doing what the end did at time t − x/v.

The same wave can be written using the shape of the string at t = 0, g(x):

So $y = A\sin\frac{t - x/v}{T}$ and $y = Ae^{-(t - x/v)/T}$ are travelling waves, but $y = A\sin\frac{x^2 - vt}{L}$ is not.

Keep moving the end of the string up and down in simple harmonic motion and the whole string comes alive with a sine wave, the shape behind almost every wave in physics.

From a shaking end to a sine wave#

If the end x = 0 performs SHM, $f(t) = A\sin\omega t$ (15.5), with period T = 2π/ω and frequency ν = 1/T, the wave that travels out is

Every particle of the string performs SHM with the same amplitude and frequency, but each copies the motion of a particle to its left after a delay x/v. When one particle reaches its top, a particle just ahead is still rising and one just behind is already coming down.

The velocity of the particle at x (x held fixed, so a partial derivative) is

Wavelength, frequency and speed#

A particle a distance vT further along copies the first after a delay of exactly one period, which makes no difference: they move in perfect step (same phase). This separation is the wavelength.

Other ways to write it#

These all assume t = 0 when the end crosses its mean position going up. A different choice of t = 0 adds a phase constant: $y = A\sin[\omega(t - x/v) + \phi]$ (15.13). With φ = π/2 (end at its top at t = 0) this becomes $A\cos\omega(t - x/v)$; with φ = π, $y = A\sin(kx - \omega t)$ (15.15).

Tighten a guitar string and its note rises; the thick bass strings sound low. Both come from how fast waves travel on the string, which depends on just two things.

What should the speed depend on?#

When part of the string is disturbed, its elasticity makes it pull on the neighbouring part (measured by the tension F), and the neighbour responds according to its inertia (measured by the mass per unit length μ). Stronger pull, faster response; heavier string, slower response.

Deriving v = √(F/μ)#

Ride alongside the wave at its speed v. In your frame the wave shape stands still and the string slides backward through it at speed v, like a snake through a fixed curve. Look at a short element of length Δl at the top of a crest. Any small curve is nearly a circular arc; let its radius be R. The element is moving round this arc at speed v.

The tension F pulls on each end of the element along the string, each tilted down by a small angle θ from the horizontal, so the net downward force is $2F\sin\theta \approx 2F\theta = 2F\dfrac{\Delta l/2}{R} = \dfrac{F\,\Delta l}{R}$. The element’s mass is μΔl, and it needs a centripetal acceleration v²/R:

Power carried by a sine wave#

At position x the string to the left pulls on the string to the right with the tension F, along the string. Its vertical component is $F_y = -F\sin\theta \approx -F\,\partial y/\partial x$, and it pushes the string to the right up and down at speed $\partial y/\partial t$. The power delivered across x is therefore

The cos² term averages to ½ over a cycle. Using F = μv² and ω = 2πν:

Two billiard balls collide and bounce. Two waves meet, overlap, and then carry on as if they had never met. That ghostly ability is what makes interference possible.

Pulses that pass through each other#

Two people holding opposite ends of a string each flick a pulse toward the other. The pulses meet, overlap and separate, and each emerges with exactly its original shape, travelling exactly as before. While they overlap, the displacement at each point is simply the sum of what each pulse alone would produce: 0.2 cm up from one and 0.4 cm up from the other gives 0.6 cm up.

If one pulse is an upside-down copy of the other, there is an instant when they cancel everywhere and the string is perfectly straight. But the particles are still moving, and a moment later the two pulses re-emerge. If one person shakes up and down and the other side to side, the displacements are at right angles and add as vectors.

This holds for small disturbances (linear waves). Stretch a string too far and displacements no longer simply add; such nonlinear waves are beyond this course. When waves overlap we say they interfere, and superposition makes interference remarkably simple.

Two sine waves going the same way#

Two sources send waves of the same frequency (so the same k) along +x, one started a little later or placed elsewhere, so they differ in phase by δ:

Exactly as when we combined two SHMs, the sum is another sine wave $y = A\sin(kx - \omega t + \varepsilon)$ with

Send a pulse down a rope tied to a wall and it comes back. But it comes back upside down. Untie the rope and slip the end on a smooth pole, and it comes back the right way up. Why?

Reflection at a fixed end#

Along the string, each element is pulled by the string on either side and moves to follow the pulse. At a clamped end the last element is pulled by the string on one side and by the clamp on the other. Since the end can’t move, the clamp’s force exactly opposes the string’s. That opposing force launches a pulse back along the string with the same shape but inverted.

At the end itself the incident pulse tries to lift the string and the reflected one pulls it down by the same amount, so the end never moves. You can find the string’s shape during reflection by adding to the real pulse an imaginary upside-down pulse approaching from beyond the wall.

Reflection at a free end#

Now tie the end to a light ring that slides without friction on a vertical rod. When the pulse arrives, the string pulls the ring up and nothing pulls it back (the rod exerts no vertical force), so the end overshoots, rising higher than the pulse itself. Equivalently, a pulse of the same shape, not inverted, is reflected; at the end the two displacements add.

Partly reflected, partly transmitted#

Most ends are neither perfectly fixed nor perfectly free. Join a light string to a heavier one and send a pulse along the light string: part of it is reflected (inverted) and part continues into the heavy string. Send it the other way, from heavy to light, and the reflected part is not inverted.

The rule in terms of speed

Waves are slower on the heavier string (v = √(F/μ)), so the rule can be stated using wave speed:

A fixed end is the extreme case of an infinitely heavy string (speed zero); a free end, of a massless one.

Pluck a guitar string and it blurs into a shape that shimmers in place rather than running along. That stay-at-home pattern is two travelling waves added together.

Two waves in opposite directions#

Take two sine waves of equal amplitude and frequency travelling in opposite directions on a long string (phase difference zero for now):

By superposition, and the identity sin a + sin b = 2 sin ½(a + b) cos ½(a − b):

Nodes and antinodes#

The amplitude |2A cos kx| varies from point to point:

• Nodes: where cos kx = 0, i.e. $kx = (n + \tfrac12)\pi$, so $x = (n + \tfrac12)\lambda/2$. These points never move, though nothing holds them.
• Antinodes: where |cos kx| = 1; the amplitude is greatest, 2A.

Adjacent nodes (and adjacent antinodes) are λ/2 apart. When sin ωt = 1 every particle is at its maximum displacement (those with cos kx > 0 up, the others down); when sin ωt = 0 every particle is passing through its mean position. Because nodes never move, no energy crosses them.

Standing versus travelling waves#

A guitar string can’t vibrate at just any frequency. Its length, tension and weight pick out a ladder of special frequencies, and those are the notes it plays.

How resonance builds up#

Tie one end of a string of length L to a wall and the other to a tuning fork vibrating gently along the string. The fork sends sine waves toward the wall; they reflect (inverted) at the fixed end and travel back; the heavy, barely moving fork acts like another fixed end and reflects them again (inverted again). After two reflections a wave has the same shape as the fork’s new output but has travelled an extra 2L.

If 2L is a whole number of wavelengths (L = nλ/2), the old wave and the new one are in step: they add to amplitude 2A, then 3A, and so on. The string soaks up energy from the fork until losses (air drag, stiffness) balance the input. The steady state is a standing wave with large antinodes: the string is in resonance with the fork. If the condition fails, successive reflections arrive out of step, the build-up doesn’t happen, and the string barely moves.

A string fixed at both ends#

Analytically, waves travelling each way add to a standing wave. Both ends must be nodes: y = 0 at x = 0 fixes the phase, giving $y = 2A\sin kx\cos\omega t$ (15.24), and y = 0 at x = L needs $\sin kL = 0$, so $kL = n\pi$:

These are the string’s natural (normal, resonant) frequencies. Frequencies above the fundamental are overtones; integer multiples of a frequency are its harmonics. For a string fixed at both ends, every harmonic is an allowed overtone: the first overtone is the 2nd harmonic, the second overtone the 3rd harmonic, and so on. That neat harmonic ladder is why strings make such musical sounds, from the violin to the sitar.

Normal modes

Vibrating at its nth natural frequency, the string is in its nth normal mode:

• Fundamental (n = 1): nodes only at the ends, one antinode in the middle.
• First overtone (n = 2): a node at the middle, antinodes at L/4 and 3L/4.
• Second overtone (n = 3): nodes at L/3 and 2L/3, antinodes at L/6, L/2, 5L/6.
• In general the nth overtone has n nodes between the ends and n + 1 antinodes.

A string fixed at one end only#

Fix x = 0 but let x = L move freely sideways (in practice, tie it to a very light thread). Then x = L must be an antinode: $\sin kL = \pm1$, so $kL = (n + \tfrac12)\pi$ and

Violinists slide their fingers to shorten a string; tuners twist pegs to change its tension; bass strings are made thick. Each trick is one of three simple laws.

Three laws#

From $\nu = \dfrac1{2L}\sqrt{\dfrac F\mu}$ for a string fixed at both ends:

The sonometer#

A sonometer is a hollow wooden sound box with two fixed bridges at its ends. An auxiliary wire is welded tight between them. An experimental wire is fixed at one end, runs over the other bridge and carries a hanger for weights, which set its tension. Pairs of movable bridges under each wire mark out the vibrating length. Wires can be swapped to change μ.

Strike a tuning fork on a rubber pad and press its stem on the box: the box passes the fork’s vibration to the wire. If the fork’s frequency equals a natural frequency of the wire, large standing waves build up: the fork is in unison with the wire.

Spotting resonance

• Paper rider: a small folded paper on the middle of the wire. In the fundamental mode the middle is an antinode, and the rider shakes violently and usually flies off. (It also sits on an antinode in the 3rd harmonic, but shakes less.)
• Beats: sound the fork and pluck the wire together. If the frequencies are close but not equal, the sound swells and fades periodically (beats, next chapter). Adjust the length until the beats disappear.

Testing the law of length

Load the experimental wire (say 3–4 kg), then for each of several tuning forks slide the bridges until the wire resonates in its fundamental. The product νL should come out the same each time.

Testing the laws of tension and mass

Here the experimental wire’s length is fixed, so its frequency changes in a continuous way as you vary the tension (or swap wires). Tuning forks only come in a few fixed frequencies, so instead use the auxiliary wire as an adjustable source: pluck it and slide its bridges until it resonates with the experimental wire, and note its length l′. Since its frequency ∝ 1/l′:

• Law of tension: ν ∝ √F means $l'\sqrt F$ stays constant as you change the load.
• Law of mass: ν ∝ 1/√μ means $l'/\sqrt\mu$ stays constant as you change wires (μ found by weighing a known length).

A wave travels one way; the medium wiggles another. Comparing those two directions sorts waves into kinds, and for some waves the wiggle direction can even be filtered.

Transverse and longitudinal#

Not every wave is one or the other. Disturb the surface of a still lake and the water particles move in circles or ellipses as the wave passes, with components both along and across the direction of travel.

Polarization#

Thread a stretched string through a narrow slit in a card held perpendicular to the string (the string along x, the card in the y–z plane). If the string vibrates along y and the slit is also along y, the string slides freely in the slit and the wave passes. Turn the card through 90° so the slit is along z: the card now blocks the y motion, and the wave can’t get through. At an intermediate angle only part of the wave passes, and what passes vibrates parallel to the slit. (Two chairs with the string through the gap between their backs make a homemade version.)

Waves are classified by how their particles move in the plane perpendicular to travel:

• Linearly (plane) polarized: always along one fixed direction, e.g. y = A sin ω(t − x/v), or z = A sin ω(t − x/v).
• Circularly polarized: each particle moves in a small circle.
• Elliptically polarized: each particle moves in an ellipse.
• Unpolarized: the displacement direction changes at random.