Standing Waves
Oscillations that propagate through space — energy travels, matter does not.
Transverse waves: particles oscillate perpendicular (⊥) to the direction of wave propagation. Examples: light, waves on a string, surface water waves. Longitudinal waves: particles oscillate parallel (∥) to the direction of propagation. Example: sound. In longitudinal waves, regions of compression (high pressure) and rarefaction (low pressure) travel through the medium.
Wavelength λ: distance between two consecutive identical points in phase (e.g. crest to crest) — unit: m. Frequency f: number of complete waves passing a point per second — unit: Hz. Period T = 1/f: time for one complete wave to pass — unit: s. Amplitude A: maximum displacement from equilibrium. Wave speed v: how fast the wave pattern moves through the medium.
The wave equation v = fλ is always true for any wave in any medium. f is determined by the source, λ is determined by the medium (speed). If the wave moves into a different medium (speed changes), f stays the same but λ changes. In exams: if given wave speed and frequency, find λ = v/f. Or if given λ and v, find f = v/λ.
When two or more waves are simultaneously present at the same point, the resultant displacement equals the algebraic sum of the individual displacements at that point. This is purely mathematical addition of amplitudes — applies to all types of waves. Constructive interference: waves in phase → amplitudes add. Destructive interference: waves out of phase → amplitudes cancel.
A standing wave forms when two identical waves travel in opposite directions in the same medium and superimpose. The wave does NOT travel — it oscillates in place. Formed when a wave reflects off a fixed end and interferes with the incoming wave. The pattern has permanent nodes and antinodes. Energy is stored in the standing wave — it does not propagate.
Nodes (knopen): points of zero displacement — particles never move. At fixed/closed ends, there is always a NODE (the wave reflects with phase inversion). Antinodes (buiken): points of maximum displacement — particles oscillate with maximum amplitude. At free/open ends, there is always an ANTINODE (reflection without phase inversion). Distance between adjacent node and antinode = λ/4.
Both ends are fixed → both ends must be nodes. Harmonics: n = 1, 2, 3... (all harmonics present). Wavelength: λ_n = 2L/n. Frequency: f_n = nv/2L. The fundamental (n=1) has λ = 2L. Second harmonic (n=2) has λ = L. Same formulas apply to a pipe closed at BOTH ends (antinode-antinode would be open pipe — see next subtopic).
Both ends are open → both ends are antinodes. All harmonics present: n = 1, 2, 3... Wavelength: λ_n = 2L/n. Frequency: f_n = nv/2L. Identical formulas to the string! The fundamental (n=1) has λ = 2L, one antinode in the middle. This is why flutes and organs with open pipes behave like strings.
One end closed (node), one end open (antinode). Only ODD harmonics present: n = 1, 3, 5... Wavelength: λ_n = 4L/n (n odd). Frequency: f_n = nv/4L (n odd). The fundamental (n=1) has λ = 4L — the pipe fits one quarter of a wavelength. No even harmonics! This is why a clarinet (effectively closed at one end) sounds different from a flute.
When two waves of slightly different frequencies f₁ and f₂ are superimposed, they alternately interfere constructively and destructively. The result is a wave whose amplitude pulses at the beat frequency: f_beat = |f₁ − f₂|. You hear a throbbing sound |f₁ − f₂| times per second. As f₁ → f₂, beats slow and disappear. Used by musicians to tune instruments: when beats disappear, the frequencies are equal. Note: beats occur when f₁ ≈ f₂ — if the difference is too large, you hear two separate notes.
Intensity I = power / area = P/A (unit: W/m²). For a point source radiating equally in all directions: I = P/(4πr²) — intensity follows an inverse-square law (I ∝ 1/r²). Intensity ∝ amplitude²: I ∝ A². Doubling amplitude quadruples intensity. Energy transmitted per second by a wave per unit area. Exam tip: when comparing intensities at different distances or amplitudes, use ratios: I₂/I₁ = A₂²/A₁² = r₁²/r₂².
y–x Graph (Snapshot of Wave)
Shows displacement of all particles at one instant in time. Wavelength λ is read from crest to crest (or trough to trough). Amplitude A is the maximum displacement from equilibrium.
y–t Graph (Oscillation at One Point)
Shows how one particular particle's displacement changes over time. Period T is read from peak to peak. This looks like the x–t graph of SHM — because each particle IS oscillating in SHM.
Standing Wave Diagrams (n = 1, 2, 3)
Draw the mode pattern showing nodes (N) and antinodes (A). For string/open pipe: n=1 has 2 nodes and 1 antinode; n=2 has 3 nodes and 2 antinodes; n=3 has 4 nodes and 3 antinodes. For closed-one-end pipe: n=1 has 1 node and 1 antinode; n=3 has 2 nodes and 2 antinodes.
Amplitude–Frequency Graph (Resonance)
Shows sharp peaks at each natural frequency (harmonic). For a string: peaks at f₁, 2f₁, 3f₁... For a closed pipe: peaks at f₁, 3f₁, 5f₁... Peak amplitude is limited by damping. The first peak is the fundamental frequency.
Every point in a standing wave oscillates in SHM. The wave's f and T are the oscillating particles' f and T. Resonance in standing waves is the same concept as resonance in oscillations.
💡Exam tip: f comes from the source. λ comes from wave geometry. v = fλ connects them.
Click any formula to see symbol definitions.
All formulas for this topic are in BINAS BINAS 35B2. In the exam you don't need to memorise the equations — but you must know which table to open and what every symbol means.