Waves, sound and reflection

Waves, sound and reflection

A wave transfers energy through a disturbance without requiring the medium as a whole to travel with that disturbance. In a transverse wave, vibrations are perpendicular to the direction of propagation. In a longitudinal wave, vibrations are parallel to that direction. Water-surface motion can be more complicated than a simple transverse model, so use the stated model carefully when interpreting examples.

Wavelength is the distance between neighbouring points in the same phase, such as successive crests. Frequency is the number of complete cycles passing a point each second and is measured in hertz. Period is the time for one cycle, with. Wave speed is related to frequency and wavelength by. Always convert centimetres to metres if speed is needed in metres per second.

Sound is a mechanical longitudinal wave and needs a medium. Electromagnetic waves, including visible light and radio waves, can travel through a vacuum. Pitch is mainly related to sound frequency, while loudness depends on factors including intensity and the listener. Amplitude is the maximum displacement from equilibrium; it is not the same measurement as wavelength, which is a distance along the direction of propagation.

Reflection occurs when a wave returns from a boundary. For a light ray at a plane surface, the angle of incidence equals the angle of reflection, with both measured from the normal. Refraction changes a wave’s direction when its speed changes across a boundary, except for normal incidence. The frequency remains unchanged across a stationary boundary, while wavelength changes with speed.

A plane mirror forms a virtual image that is upright, the same size as the object and apparently as far behind the mirror as the object is in front. Lenses and curved mirrors require attention to conventions when using formulae. A ray diagram should identify the principal axis, relevant focus and actual or extended rays. Do not confuse a real meeting of rays with an apparent meeting of their backward extensions.

In wave calculations, write the required relationship before substituting values. If a wave has frequency 5 Hz and wavelength 2 m, its speed is 10 m/s, while its period is 0.2 s. These two results describe different features. Graph labels also matter: a displacement–distance graph can show wavelength, whereas a displacement–time graph can show period; neither axis should be inferred merely from the curve’s shape.

Graph interpretation and phase

A displacement–distance graph is a snapshot at one instant; a displacement–time graph follows one position as time passes. Read wavelength from the first and period from the second. Points a whole wavelength apart are in phase, while points half a wavelength apart are antiphase for a sinusoidal wave. Amplitude is equilibrium-to-crest displacement, not trough-to-crest height. The wave's propagation speed differs from the instantaneous speed of an oscillating medium particle. A graph without labelled axes cannot safely supply a wavelength merely because its curve looks periodic. State which quantity each spacing represents.

Superposition and interference

In a linear medium overlapping disturbances add their signed displacements. Equal crests reinforce and an equal crest and trough can cancel at an instant. Stable interference requires a stable phase relationship: coherent sources share frequency with a constant phase difference. Destructive interference redistributes energy in the overall pattern rather than destroying it. For in-phase sources, path difference nλ gives constructive interference and (n +½)λ destructive interference in the elementary geometry. If source phase changes, those location rules change. Add displacements first rather than assuming that overlapping waves stop one another permanently.

Diffraction and polarisation

Diffraction is spreading at an opening or around an obstacle. It is most noticeable when gap size is comparable with wavelength. Sound bends around everyday openings more conspicuously than visible light because its wavelengths are much longer. Reflection, refraction, interference and diffraction support wave descriptions of many phenomena. Polarisation distinguishes transverse behaviour: a transverse wave can have its vibrations restricted to a chosen direction, while simple longitudinal sound in air cannot be polarised in the same way. A polarising sheet does not change every incoming frequency to one frequency; monochromatic and polarised describe different properties.

Stationary waves and resonance

Opposite-travelling coherent waves of matching frequency and suitable amplitude can form stationary patterns. Nodes have zero amplitude and antinodes maximum amplitude. Adjacent nodes are separated byλ/2. A string fixed at both ends fits half a wavelength in its fundamental, so. Higher ideal modes are integer multiples. An air column closed at one end has a displacement node there and an antinode at the open end, giving ideal fundamental. Open-end corrections affect precise experiments. A resonant response involves energy supplied at a suitable driving frequency, not generation of energy without input.

Sound, echoes and timbre

Sound intensity is power per area, while loudness is perceived response depending on intensity, frequency and hearing. Pitch mainly follows frequency; timbre depends on the mixture of overtones. Echo distance is vt/2 because the pulse travels out and back. Reverberation is persistence from multiple reflections and can help or hinder room acoustics depending on amount and purpose. In a resonance-tube experiment successive resonance lengths differ by approximatelyλ/2, reducing the effect of an unknown end correction. Use the sound speed specified and do not substitute light speed just because both phenomena are waves.

Beats, Doppler effect and string frequency

Nearby coherent frequencies can produce periodic loudness variations called beats, with elementary beat frequency |f₁ −f₂|. Beats can assist tuning by revealing a small frequency difference. The Doppler effect changes observed frequency when source and receiver have relative motion along the propagation direction: approaching sound is generally heard at a higher frequency and receding sound lower under the usual conditions. It is not a change in the source's emitted frequency itself. For an ideal stretched string, speed, so, where T is tension and μ mass per length. Distinguish tension T from period T by context.

The electromagnetic spectrum and reflection

Increasing frequency runs from radio through microwave, infrared, visible, ultraviolet, X-ray and gamma radiation. Vacuum speed is approximately3.0 ×10⁸ m/s for all, so frequency increase means shorter wavelength. Their uses depend on interactions with matter; no band is harmless at unlimited intensity. At a reflecting surface measure angles from the normal. A ray30° to the surface is60° to the normal. A concave mirror can form real or virtual images depending on object position; a convex mirror gives an upright diminished virtual image for a real object. In the spherical paraxial model, focal length is half curvature radius.

Self-check and physical boundaries

An echo arrives0.8 s after a pulse where sound speed is340 m/s. Total path272 m gives wall distance136 m. A0.60 m string with both ends fixed and fundamental100 Hz has wavelength1.20 m and wave speed120 m/s. Its second harmonic has a central node and twice the fundamental frequency. A pair of tones at256 and260 Hz gives4 beats per second. Each result needs a physical interpretation in addition to: boundary conditions determine which wavelengths fit, an echo adds a return trip and a beat compares two frequencies. Sketch the arrangement before choosing a relationship.