Physics · Oscillations and waves · Grade 10

Waves on a rope and a spring

When you shake a rope, what travels along it — matter or energy?

The situation

In a sports hall the coach tied one end of a long rope to a steel post and jerked the other end sharply: a hump ran along the rope, reached the post and came back upside down. When she shook the rope at a steady rhythm, it seemed to split into “loops” frozen in place. On a spring hanging beside it the coils do not move across the spring but crowd together and spread apart along its own axis. The rope itself goes nowhere — so what is it that actually runs along it?

Topic

Mechanical waves. Slinky spring and rope: transverse and longitudinal waves, reflection

Key ideas

Transverse and longitudinal waves
Standing wave
Huygens' principle

Formulas

v = \sqrt{\frac{T}{μ}}, λ = v T_{0} = \frac{v}{f}

The speed of a transverse wave on a rope ( $T$ — tension, $μ$ — mass of 1 m) and the wavelength. A longitudinal wave on a spring: $v = L \sqrt{k / m}$ .

y (x, t) = A \sin (ω t - k x), \frac{\partial^{2} y}{\partial t^{2}} = v^{2} \frac{\partial^{2} y}{\partial x^{2}}

The equation of a plane travelling wave ( $k = 2 π / λ$ , $ω = 2 π f$ ) and the wave equation.

w = \frac{1}{2} μ ω^{2} A^{2}, P = w v = \frac{1}{2} μ ω^{2} A^{2} v

The linear energy density of a wave and the energy flow (Umov–Poynting vector $\vec{S} = w \vec{v}$ ).

f_{n} = n \frac{v}{2 L} (fixed), f_{n} = (2 n - 1) \frac{v}{4 L} (free)

Frequencies of a standing wave: the distance between adjacent nodes is $λ / 2$ .

v_{f} = \frac{ω}{k}, v_{g} = \frac{d ω}{d k} \approx \frac{Δ ω}{Δ k}

Phase and group velocities; in deep water $ω^{2} = g k$ , so $v_{g} = v_{f} / 2$ . Diffraction minimum: $\sin θ = λ / d$ .

Theory

What travels through a medium?

In an elastic medium each particle pulls or pushes its neighbours, so an oscillation in one place is passed on to neighbouring particles. The particles oscillate in place, but energy and the state of oscillation (the phase) travel far away. Switch on “Track particles”: the trace of the markers on the rope is a vertical line (a transverse wave), and on the spring it is a line along the axis (a longitudinal wave — compressions and rarefactions). A transverse wave travels only where there is resistance to shear (a solid, a string, the surface of water); a longitudinal wave travels in gases, liquids and solids too.

Speed, wavelength, the wave equation

The speed is set by the medium: if the rope is tightened or made lighter, $v = \sqrt{T / μ}$ increases. The frequency is set by the source. In one period the wave moves a distance $λ = v T_{0}$ , so if $f$ stays the same and $T$ increases, $λ$ gets longer — check with the ruler. The travelling wave $y = A \sin (ω t - k x)$ is a solution of the wave equation $\partial^{2} y / \partial t^{2} = v^{2} \partial^{2} y / \partial x^{2}$ ; electromagnetic waves obey the same equation, with $v = c = 1 / \sqrt{ε_{0} μ_{0}}$ . A surface of points with the same phase is a wavefront: in the ripple tank it is a straight line for a plane wave and a circle for a point source. In space the amplitude of a spherical wave falls as $1 / r$ , and for a circular wave on the surface of water as $1 / \sqrt{r}$ — the energy is spread over an expanding front.

Wave energy and intensity

Each metre of the rope has oscillation energy $w = \frac{1}{2} μ ω^{2} A^{2}$ . This energy flows with the wave at speed $v$ , and the flow $P = w v$ is the form of the Umov–Poynting vector $\vec{S} = w \vec{v}$ on the rope (the yellow arrow). If the amplitude is doubled, the energy flow becomes 4 times greater, and it also grows 4 times if the frequency is doubled. In space the intensity $I = P / S$ is the energy passing through a unit area in 1 s, in W/m².

Reflection, superposition and the standing wave

A fixed end cannot move: so that the sum of the incident and reflected waves is always zero, the pulse returns inverted. A free end (a light ring on the post) rises twice as high and the pulse returns without inversion. When two waves meet, their displacements simply add (superposition). Continuous incident and reflected waves add to form a standing wave: the nodes do not move at all, the antinodes oscillate most, and the distance between adjacent nodes is $λ / 2$ . The end of the vibrator is also almost a node, so resonance occurs at $f_{n} = n v / (2 L)$ . In the model the rope has a small damping ( $α = 0.03$ 1/m) — otherwise the amplitude at resonance would grow without limit.

Huygens' principle and diffraction

Huygens' principle: every point of a wavefront is a source of secondary waves, and their envelope (common tangent) is the front at the next moment. In the tank, switch on “Huygens wavelets”: at the wall the envelope of the secondary waves gives the reflected front, and the angle of incidence equals the angle of reflection. The water in the model calculates a wave passing through a slit in exactly this way — the sum of 8–64 secondary sources in the slit. If the slit is much wider than the wavelength, the wave passes almost straight through; if $d ≲ λ$ , it spreads out as a semicircle (diffraction). The direction of the first minimum: $d \sin θ_{1} = λ$ . Diffraction of light is in the “Diffraction” experiment.

Beats, phase and group velocity

When two waves of close frequency are added, the amplitude of the resultant wave changes slowly — these are beats; at one point the beat frequency is $| f_{1} - f_{2} |$ . The wave “group” (the envelope) moves at the group velocity $v_{g} = Δ ω / Δ k$ , and the crests at the phase velocity $v_{f} = ω / k$ . Energy is carried at the group velocity. For water waves $ω^{2} = g k th (k h)$ : in shallow water ( $k h ≪ 1$ ) there is no dispersion and $v_{f} = v_{g} = \sqrt{g h}$ ; in deep water $v_{g} = v_{f} / 2$ — the crests appear at the back of the group and vanish at the front. In the channel the wave height is magnified 3 times (less in very shallow water).

Did you know?

In an earthquake the longitudinal P-waves travel faster than the transverse S-waves, so a seismograph records the P-wave first. S-waves do not pass through liquid — this is how it was found that the Earth's outer core is liquid.
The wavelength of a tsunami is much greater than the depth of the ocean, so it travels like a “shallow-water” wave at $v = \sqrt{g h}$ : at a depth of 4 km about 200 m/s, that is about 700 km/h.

Inside the experiment

The experiment starts from a real-life situation: you predict the result, change the controls, run the experiment and draw a conclusion. Inside are missions, theory, a quiz and a guided tour in which the camera moves and explains the key points.

More experiments in this section

Swing — Grade 10
Spring pendulum — Grade 10
Sound: speed, pitch, ultrasound — Grade 10
The dutar string and sound — Grade 11
Waves and the Doppler effect — Grade 11
Bridge resonance — Grade 11
Coupled oscillations — Grade 11
Nonlinear oscillations — Year 2
Fourier: the spectrum of sound — Year 3