Describing a wave
A wave carries energy from place to place through a repeating disturbance. Its amplitude is the size of the disturbance, its wavelength (λ) is the distance between neighbouring crests, and itsfrequency (f) is how many crests pass each second. The three are linked by the wave speed:v = f·λ. The phase tells you where in its cycle a wave is at a given place and time.
Superposition
The principle of superposition says that when waves overlap, the total displacement at every point is the sum of the individual displacements. In the string view, the bottom graph is always exactly wave 1 plus wave 2. After they pass, each wave carries on unchanged.
Constructive and destructive interference
- Constructive interference: waves meet in phase (crest on crest). The amplitudes add, making a larger wave.
- Destructive interference: waves meet 180° out of phase (crest on trough). They partially or completely cancel.
- Any phase in between gives partial interference.
Destructive interference does not destroy energy. In the ripple tank you can see that energy missing from the dark regions is concentrated in the bright ones.
Beats
When two waves have slightly different frequencies, they drift in and out of step. The sum grows and shrinks regularly — a pattern called beats. The beat frequency is simply the difference between the two frequencies. With sound, you hear a “wah-wah” throb in loudness.
Two-source interference
Two sources vibrating together send out circular waves that overlap. At each point, the waves have travelled different distances. Where the path difference is a whole number of wavelengths they arrive in step and reinforce (bright fringes); where it is a half-number they cancel (dark lines called nodal lines). Thomas Young used this effect with light in about 1801 to show that light behaves as a wave.
Formulas
v = f · λ beat frequency f_beat = |f₁ − f₂|Resultant amplitude of two waves with the same frequency and phase difference φ:
A = √(A₁² + A₂² + 2·A₁·A₂·cos φ)For two in-phase sources, bright fringes appear where the path difference is:
Δr = n·λ (n = 0, 1, 2 …) dark fringes at Δr = (n + ½)·λKey terms
- Phase difference
- How far one wave is shifted relative to another, measured in degrees (360° = one full cycle).
- Coherent sources
- Sources with the same frequency and a constant phase relationship, needed for a stable interference pattern.
- Node and antinode
- Points of zero and maximum vibration in an interference pattern.
- Path difference
- The difference in distance travelled by two waves to reach the same point.
Real-world applications
- Noise-cancelling headphones record incoming sound and play an inverted copy, producing destructive interference.
- Tuning instruments: musicians listen for beats between two notes and adjust until the beats disappear.
- Soap bubbles and oil films show colours because light reflecting from the top and bottom surfaces interferes.
- Gravitational-wave detectors such as LIGO use laser interference to measure changes in length far smaller than a proton.
- Wi-Fi and radio engineers design antenna arrays that use interference to beam signals in chosen directions.
Common misconceptions
- “Waves bounce off each other.” They pass through each other unchanged.
- “Destructive interference destroys energy.” Energy is redistributed to places of constructive interference.
- “Any two waves make a stable pattern.” A steady pattern needs coherent sources with the same frequency.
Key takeaways
- Overlapping waves add point by point (superposition).
- In phase → constructive; 180° out of phase → destructive.
- Slightly different frequencies produce beats at f₁ − f₂.
- Two coherent sources produce fringes determined by the path difference.
Model note: waves are idealized — they do not weaken with distance, and the ripple tank uses a single frequency for both sources.