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A wave y = A sin(kx - ωt + φ) from its constants: the record `A`, `lambda` = 2π/k, `f` = ω/2π, `T`, `v` = ω/k, `k`, `omega`, `phase` (degrees). NCERT XI Ex 14.1.
A (number) -- the amplitudek (number) -- the angular wave numberomega (number) -- the angular frequencyphi (number) -- the initial phase in degrees (0)The same record from the amplitude, wavelength and frequency.
A (number) -- the amplitudelambda (number) -- the wavelengthf (number) -- the frequencyphi (number) -- the initial phase in degrees (0)The speed of a transverse wave on a string, √(T/μ). NCERT XI Ex 14.2.
T (number) -- the tensionmu (number) -- the mass per unit lengthThe speed of sound in a gas, √(γP/ρ) (Laplace); give gamma 1 for Newton's √(P/ρ). NCERT XI Ex 14.3.
gamma (number) -- Cp/Cv (1 for Newton's formula)P (number) -- the pressurerho (number) -- the densityThe speed of sound from the molar mass and temperature, √(γRT/M).
gamma (number) -- Cp/CvM (number) -- the molar mass in kg/molT (number) -- the temperature in kelvinThe speed at another temperature, v ∝ √T (kelvin).
v0 (number) -- the speed at T0T0 (number) -- the first temperature in kelvinT1 (number) -- the secondThe speed of a longitudinal wave in a solid or liquid, √(Y/ρ) or √(B/ρ).
modulus (number) -- Young's or the bulk modulusrho (number) -- the densityThe phase difference in degrees for a path difference, 360 Δx/λ.
dx (number) -- the path differencelambda (number) -- the wavelengthThe path difference for a phase difference in degrees, λ φ/360.
phi (number) -- the phase difference in degreeslambda (number) -- the wavelengthThe pressure amplitude of a sound wave, ρ v ω s0.
rho (number) -- the densityv (number) -- the speed of soundomega (number) -- the angular frequencys0 (number) -- the displacement amplitudeThe intensity of a sound wave, ½ ρ v ω² s0².
rho (number) -- the densityv (number) -- the speed of soundomega (number) -- the angular frequencys0 (number) -- the displacement amplitudeThe sound level, 10 log10(I/I0) with I0 = 1e-12 W/m².
I (number) -- the intensityThe intensity of a sound level in decibels.
level (number) -- the level in dBTwo waves of one frequency with a phase difference φ: the record `A` = √(A1² + A2² + 2A1A2 cos φ) and `phase`, the resultant's lead on the first in degrees.
A1 (number) -- the first amplitudeA2 (number) -- the secondphi (number) -- the phase difference in degreesTwo waves of intensities I1 and I2: the record `max` = (√I1 + √I2)², `min` = (√I1 - √I2)², `ratio`.
I1 (number) -- the first intensityI2 (number) -- the secondThe amplitude ratio A1/A2 from the ratio of the brightest to the darkest intensity, (√r + 1)/(√r - 1).
ratio (number) -- I_max/I_minThe intensity of two coherent waves of I0 each with a phase difference φ in degrees, 4 I0 cos²(φ/2). NCERT XII Ex 10.3.
I0 (number) -- each wave's intensityphi (number) -- the phase difference in degreesThe frequency of the n-th mode: n v/2L for a string fixed at both ends or an open pipe, (2n - 1) v/4L for a pipe closed at one end. NCERT XI Ex 14.4.
kind (text) -- "string", "closed" or "open"L (number) -- the lengthv (number) -- the wave speedn (number) -- the mode (1 the fundamental)The wavelength of the n-th mode: 2L/n, or 4L/(2n - 1) for a closed pipe.
kind (text) -- "string", "closed" or "open"L (number) -- the lengthn (number) -- the modeThe mode n whose frequency is f, or "none" when f is not a natural frequency (a closed pipe has no even harmonics).
kind (text) -- "string", "closed" or "open"L (number) -- the lengthv (number) -- the wave speedf (number) -- the frequencyThe positions of the displacement nodes of the n-th mode from the left end (a closed pipe's closed end), as a list.
kind (text) -- "string", "closed" or "open"L (number) -- the lengthn (number) -- the modeThe positions of the displacement antinodes of the n-th mode, as a list.
kind (text) -- "string", "closed" or "open"L (number) -- the lengthn (number) -- the modeThe n-th harmonic of a stretched string, (n/2L) √(T/μ) (the sonometer).
L (number) -- the lengthT (number) -- the tensionmu (number) -- the mass per unit lengthn (number) -- the harmonic (1)The first two resonance lengths with a fork of frequency f: the record `v` = 2f(l2 - l1), `lambda`, `e` = (l2 - 3 l1)/2 the end correction.
f (number) -- the fork's frequencyl1 (number) -- the first resonance lengthl2 (number) -- the secondThe n-th resonance length of a tube, (2n - 1) v/4f - e.
f (number) -- the fork's frequencyv (number) -- the speed of sounde (number) -- the end correctionn (number) -- which resonanceTwo notes together: the record `f` = |f1 - f2| beats per second, `T` the beat period, `mean` the tone heard.
f1 (number) -- the first frequencyf2 (number) -- the secondThe two frequencies an unknown fork may have, beating n times a second with a known one: [known - n, known + n].
known (number) -- the known frequencybeats (number) -- the beat frequencyThe Doppler effect for sound: the record `f` = f (v + w + v_o)/(v + w - v_s), `lambda` in front of the source, `shift`. Speeds are positive towards the other party (the source towards the observer, the observer towards the source) and negative away; the wind blows from the source to the observer when positive. NCERT XI Ex 14.6.
f (number) -- the source's frequencyv (number) -- the speed of soundvs (number) -- the source's speed towards the observervo (number) -- the observer's speed towards the source (0)wind (number) -- the wind from the source to the observer (0)A source moving at v_s towards a wall and hearing its echo: the record `atWall` = f v/(v - v_s), `heard` = f (v + v_s)/(v - v_s), `beats` with its own note.
f (number) -- the source's frequencyv (number) -- the speed of soundvs (number) -- the source's speed towards the wallThe wavelengths in front of and behind a moving source: the record `ahead` = λ(1 - v_s/v), `behind` = λ(1 + v_s/v), `lambda`.
f (number) -- the frequencyv (number) -- the speed of soundvs (number) -- the source's speedA travelling wave y = A sin(kx - ωt) on the diagram: the snapshot at t over 2.5 wavelengths with λ and A marked and the arrow v, the snapshot at t + dt dashed, a particle at x with its velocity arrow (v_p = -v ∂y/∂x). The picture has its own scale; the real values are in the labels and caption; the summary gives f, T, k, ω.
d (number) -- the diagramA (number) -- the amplitudelambda (number) -- the wavelengthv (number) -- the wave speedt (number) -- the time of the snapshot (0)dt (number) -- the second snapshot's delay (T/8; 0 for none)x (number) -- the marked particle's position (a quarter wave after a crest)The two graphs of a wave side by side: y against x at t = 0 with λ marked, and y against t at the particle x0 with T marked.
d (number) -- the diagramA (number) -- the amplitudelambda (number) -- the wavelengthf (number) -- the frequencyx0 (number) -- the particle whose y-t graph is drawn (0)A longitudinal wave: a row of particles bunched and spread over two wavelengths with C and R marked, the displacement graph s(x) and the pressure graph Δp(x) = -B ∂s/∂x beneath (a quarter wave ahead).
d (number) -- the diagramlambda (number) -- the wavelength (for the label)A (number) -- the swing kA as a share of λ/2π (0.8; below 1 so the particles never cross)t (number) -- the time as a share of the period (0)A pulse on a string meeting a rigid end (reflected inverted, a phase change of π) or a free end (reflected upright): three rows, arriving, at the end with its image, going back.
d (number) -- the diagramend (text) -- "rigid" or "free"width (number) -- the pulse's width in picture units (0.6)The n-th mode of a string fixed at both ends, a pipe closed at one end or an open pipe: the envelope, N and A marked at the formula's positions, the holder drawn; f_n in the caption when v is given. The word "pressure" draws the pressure picture of a pipe (nodes and antinodes swapped).
d (number) -- the diagramkind (text) -- "string", "closed" or "open"L (number) -- the lengthn (number) -- the mode (1 the fundamental)v (number) -- the wave speed (none)picture (text) -- "displacement" (default) or "pressure"The first modes stacked, each named (fundamental, third harmonic, ...) with its wavelength and, when v is given, its frequency: a closed pipe shows the odd harmonics only.
d (number) -- the diagramkind (text) -- "string", "closed" or "open"L (number) -- the lengthcount (number) -- how many modes (3, up to 8)v (number) -- the wave speed (none)The resonance tube at its first two resonances side by side: the water levels, the fork, the quarter-wave and three-quarter-wave loops, l1, l2 and e marked; v = 2f(l2 - l1) and e = (l2 - 3l1)/2 in the caption.
d (number) -- the diagramf (number) -- the fork's frequencyl1 (number) -- the first resonance lengthl2 (number) -- the secondBeats: the two waves and their sum with the dashed envelope 2A cos(π(f1 - f2)t), the beat period marked; best with frequencies a few hertz apart (NCERT Fig. 14.15 uses 11 and 9).
d (number) -- the diagramf1 (number) -- the first frequencyf2 (number) -- the secondduration (number) -- the time drawn (two beat periods)Two waves of one frequency with a phase difference and their sum over two wavelengths, the resultant amplitude marked: A = √(A1² + A2² + 2A1A2 cos φ), ε in the caption.
d (number) -- the diagramA1 (number) -- the first amplitudeA2 (number) -- the secondphi (number) -- the phase difference in degreesPhasors tip to tail from the origin and the resultant: the list holds amplitude, phase (degrees), amplitude, phase, ... for each wave; A and ε in the caption.
d (number) -- the diagramphasors (list) -- [A1, phi1, A2, phi2, ...]The Doppler effect: the wavefronts emitted at each period by the moving source, bunched ahead, with λ' ahead and behind marked, the observer with its velocity; f' = f (v + v_o)/(v - v_s) in the caption. Speeds positive towards the other party.
d (number) -- the diagramf (number) -- the source's frequencyv (number) -- the speed of soundvs (number) -- the source's speed towards the observervo (number) -- the observer's speed towards the source (0)A source moving towards a wall and hearing its echo: the two shifts, f1 at the wall and f2 heard back, and the beats with its own note.
d (number) -- the diagramf (number) -- the source's frequencyv (number) -- the speed of soundvs (number) -- the source's speed towards the wall