Oscillations and waves

Equilibrium and repetition

A system is in equilibrium when the relevant resultant is zero. In stable equilibrium, a small displacement produces a tendency to return. If the system repeatedly passes from one side of this position to the other, it describes an oscillation. Not every periodic motion is a straight-line oscillation: uniform rotation also repeats, but does not cross an equilibrium position in the same way.

Amplitude A is the maximum displacement from equilibrium, in m when describing position. Period T is the duration of one complete cycle, in s. Frequency f = 1/T measures cycles per second, in Hz. Going from one extreme to the other is half a cycle in a symmetric oscillator, not an entire period.

Phase indicates the stage of the cycle: two oscillators can have the same frequency and amplitude but reach maxima at different instants. Knowing the period alone is not enough to know position at a given instant. The initial condition, such as position and direction of motion, completes the description.

Force proportional to displacement

For an ideal spring, F = −kx. x is deformation relative to equilibrium and k measures stiffness in N/m. The negative sign shows that force points opposite to displacement. With mass m and no other net forces along that axis, the second law gives a = −(k/m)x.

This proportionality between acceleration and position, with opposite directions, characterises simple harmonic motion. One representation is x(t) = A cos(ωt + φ), with ω = √(k/m) and φ the initial phase. T = 2π√(m/k) and f = 1/T. The trigonometric function’s argument must be an angle in radians.

At equilibrium, x = 0 and a = 0, but speed is greatest. At the extremes, speed is zero and acceleration magnitude is greatest. Do not confuse “equilibrium position” with “being at rest”. The spring relationship requires the linear elastic regime; large deformations may invalidate the proportionality and predicted period.

Energy and another model: the pendulum

For an ideal mass–spring oscillator, E = mv²/2 + kx²/2 is constant. At the extremes, all mechanical energy is elastic potential energy: E = kA²/2. At equilibrium it is kinetic: E = m maximum v²/2. Thus maximum v = A√(k/m) = Aω. Doubling A quadruples energy while retaining the same period in the linear model.

A simple pendulum is a point mass attached to a light, inextensible string of length L. For small angles and neglecting resistance, its motion approximates simple harmonic motion with T ≈ 2π√(L/g). We use g = 9,8 m/s² in this lesson. Length runs from the suspension point to the centre of the mass.

The pendulum’s approximate period does not depend on mass, but does depend on L and g. The small-angle condition is essential: the approximation uses sin θ ≈ θ when θ is in radians. At large angular amplitudes, the actual period depends more on amplitude and the simple formula loses accuracy.

A disturbance propagates

A wave carries a disturbance and energy between regions. On a string, each small section can move up and down while the pulse advances horizontally. Local matter motion is not propagation speed. In the small-wave model, the entire string does not travel along with the pulse.

Mechanical waves need a material medium. Electromagnetic waves, such as light, can propagate through a vacuum. In a transverse wave, the disturbance is perpendicular to propagation direction; in a longitudinal wave it is parallel. Sound in air is longitudinal, with pressure compressions and rarefactions.

Wavelength λ is the spatial separation between consecutive points in phase, such as two crests. Period T is the time between two crests passing the same point. During one period, the shape advances λ, so v = λ/T = λf. Amplitude must not be confused with λ: one measures the strength of disturbance, the other repetition in space.

The medium governs propagation

In many linear models, propagation speed depends on the medium’s properties rather than the frequency chosen by the source. For an ideal stretched string, v = √(tension F/μ), where μ is mass per length in kg/m. Greater tension increases speed; greater mass per length decreases it.

If a wave crosses into another medium, the source-imposed frequency stays the same at a stationary interface. Since v changes, λ changes according to v = λf. This distinction prevents the idea that a wave “always carries the same length” into any material.

Energy carried depends on amplitude and system properties. For linear harmonic waves, holding other conditions fixed, energy or intensity is proportional to A². Doubling amplitude can quadruple intensity. This does not mean propagation speed should double: speed and energy per wave are different properties.

Adding disturbances and forming patterns

In the linear regime, when two waves meet, the resultant displacement at a point is the sum of the displacements each would produce separately. This is the superposition principle. Two in-phase crests can reinforce each other; a crest and a trough of equal magnitudes can cancel displacement at that point and instant.

Local cancellation does not destroy total energy. After crossing, ideal pulses continue propagating. Energies and fluxes must be analysed together; total energy is not obtained simply by squaring the sum of amplitudes without considering interference and the region.

Two identical waves travelling in opposite directions can form a standing wave. Nodes remain at zero displacement and antinodes have maximum amplitude. For a string fixed at both ends, L = nλ/2, with positive integer n. Thus the allowed frequencies are fₙ = nv/(2L). The distance between consecutive nodes is λ/2, not λ.

Real oscillations and resonance

Friction and resistance remove mechanical energy from the oscillator, transforming it into internal energy: damping occurs. With weak damping, amplitude decreases over successive cycles. With very strong damping, the system may return to equilibrium without oscillating. The critical condition separates regimes and corresponds to a rapid return without oscillations in this model.

An external periodic force can sustain oscillations by supplying energy. Response amplitude depends on driving frequency, natural frequency and damping. Near a resonant frequency, energy transfer may be more effective and amplitude may increase.

Resonance does not mean infinite amplitude in a real system. Damping limits the response, and large amplitudes may alter behaviour or cause deformation. A swing is an accessible example: push timing matters, because pushing outside the appropriate phase may supply little energy or even remove it. Understanding phase and losses is as important as comparing frequencies.

Worked examples, without skipping steps

Follow the reasoning first. Then cover the solution and try to reconstruct it in your own words.

Now try it yourself

There is no timer. Write what you know, choose a relationship and justify each step. Hints and solutions are available whenever you need them.

Mistakes worth avoiding

At equilibrium the oscillator is always at rest.

It can pass through at maximum speed.

The wave carries the entire medium with it.

The disturbance and energy propagate; particles oscillate locally in the model.

Frequency and speed are the same property.

Frequency is cycles per time; speed is distance per time.

A node means there is no energy anywhere in the system.

It is a position of zero displacement in a standing pattern.

Resonance requires unlimited amplitude.

Losses and physical limits determine the real response.