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➗ MathFree
Diagonalize a matrix
For the matrix with rows (2, 1) and (1, 2) you get the eigenvalues 3 and 1, the diagonal matrix D and the eigenvector matrix P, so that A = P·D·P⁻¹.
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Phase difference to path difference calculator
A phase shift of π/2 is a quarter cycle, so with 632.8 nm light the two waves differ by 158.2 nm of path.
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String wave average power calculator
A wave with 0.05 kg/m, 20 m/s, 10 rad/s and a 0.02 m amplitude carries 0.02 W along the string. Enter those four values for your own wave and see the average…
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Wave intensity inverse square law calculator
Move three times farther from a point source and the intensity drops to one ninth: 120 measured at 2 m becomes one ninth of that at 6 m.
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Angular wave number calculator
Go from wavelength to angular wave number in one step: 1 m gives 2π, or 6.28318530718 radians per metre.
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Wave phase value calculator
Find where a travelling wave sits in its cycle at a given place and instant: wave number 2, position 3, angular frequency 4, time 1 and starting phase 0.5 give…
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Diameter of cone
Turn a cone's volume and height into the diameter of its base, which is twice √(3V ÷ (π·h)), without rearranging the formula by hand.
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Closed pipe fundamental frequency calculator
Gives the lowest note of a closed-open air column: a 0.5 m pipe with sound at 343 m/s resonates at 171.5 Hz, with a wavelength of 2 m.
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Group velocity calculator
Turn two measured dispersion points into the speed of a pulse envelope: angular frequencies of 1e15 and 1.2e15 rad/s at wave numbers of 1e7 and 1.1e7 rad/m give…
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Shallow water wave speed calculator
Water 10 m deep carries a long wave at 9.90285312423 m/s under standard gravity. Enter the depth, and a local gravity if you need one, to estimate tides…
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Standing wave antinodes count calculator
The third harmonic of a string fixed at both ends has 3 antinodes, 4 nodes and 3 loops: enter any harmonic number and get the antinode count, which always…
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String linear density from frequency calculator
A string vibrating at 110 Hz over 0.65 m under 80 N has a linear density of 0.003912171744 kg/m. Enter the measured frequency, length and tension to find how…
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String nth harmonic frequency
On a 0.75 m string with a wave speed of 120 m/s, the third harmonic sounds at 240 Hz, three times the 80 Hz fundamental.
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Wave amplitude reflection coefficient calculator
See how strongly a wave bounces back where the impedance changes: from 1.5 to 4.5 the reflection coefficient is 0.5, so a quarter of the power reflects and 0.75…
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Pipe end correction calculator
A tube with a 0.01 m radius gains about 0.006 m of acoustic length at its open end: the calculator multiplies the inner radius by 0.6 so your resonance math…
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Resonance tube speed of sound calculator
A 512 Hz fork resonating at 16 cm and 49.5 cm gives a sound speed of 343.04 m/s. Enter the frequency and the two consecutive lengths; the end correction cancels…
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Speed from Mach number calculator
Mach 2 with a local speed of sound of 340.3 m/s is 680.6 m/s. Enter the Mach number and the speed of sound where the object flies, and get its true speed in…
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Speed of sound in ideal gas calculator
Enter γ = 1.4, 293.15 K and 0.0289652 kg/mol, or the values of your own gas, and get the speed of sound in metres per second and km/h.
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String frequency after tension change
Raise a string's tension from 100 to 144 and a 440 Hz note climbs to 528 Hz: pitch follows the square root of the tension ratio while length and gauge stay the…
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Distance for target sound level
Find the distance at which a sound drops to the level you need: a source measuring 100 dB at 10 m falls to 80 dB at 100 m, because every 20 dB quieter means ten…
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Sound power from power level calculator
A source rated at 120 dB re 1 pW radiates 1 W of acoustic power, which is 1000 mW. Enter the level from a datasheet or report and get watts, picowatts and…
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Specific Acoustic Impedance Calculator
Air at 1.225 kg/m³ carrying sound at 343 m/s has a specific acoustic impedance of 420.175 rayl. Enter the density and sound speed of one medium to compare…
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Average Absorption Coefficient Calculator for Rooms
Gives the average share of sound a room's surfaces absorb: 900 sabins over 3000 square feet is a coefficient of 0.3, or 30%.
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Doppler effect with wind
Hear what a listener really measures outdoors: a 440 Hz source approaching at 20 m/s with a 10 m/s tailwind is heard about 6 % higher, because the wind lifts…
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Mass law transmission loss
Get a first figure for how well a single wall stops sound: a 10 kg/m² panel at 500 Hz gives about 31.603985401312 dB of transmission loss at normal incidence…
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MIDI note to frequency calculator
Type a MIDI note number from 0 to 127 and get its exact pitch in hertz: note 69 is A4 at 440 Hz, and each step is one semitone.
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Schroeder frequency calculator
A 150 m³ room with a 0.6 s reverberation time has a Schroeder frequency of 126.491106 Hz. Below that, individual room modes shape the bass; above it, averaged…
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Semitones between frequencies calculator
From 440 Hz to 880 Hz is exactly 12 semitones, one octave up. Enter any two frequencies to get the signed distance, fractions included, for tuning checks…
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Ultrasound Doppler frequency shift calculator
A target moving at 0.5 m/s toward a 5 MHz probe, at 60° in soft tissue with sound at 1540 m/s, returns an echo 1623.376623377 Hz higher.
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Combined focal length of lenses in contact calculator
Combines two thin lenses into one: 20 cm and 30 cm in contact act like a single 12 cm converging lens, because their powers add.
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Focal length from lens radii
Turn a glass index and two surface radii into a focal length: an index of 1.5 with radii of 0.1 m and -0.1 m gives 0.1 m and 10 diopters, a converging lens.
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Helmholtz resonator volume calculator
Pick the pitch and the neck, and get the cavity to build: 100 Hz with a neck of 0.000314 m² by 0.05 m in air at 343 m/s needs 0.001871492742 m³.
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Object distance thin lens calculator
Give the focal length and the image distance and get the object position: f of 10 with an image at 15 puts the object at 30, with magnification −0.5, so the…
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Object height from magnification calculator
Divide the image height by the magnification to get the object back: an image of −12 mm at magnification −3 comes from an object 4 mm tall, and the image is…
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Pythagorean tuning frequency
From A at 440 Hz, one pure fifth up lands on 660 Hz. Stack fifths up or down, shift by octaves, and read the frequency, ratio and cents from your reference for…
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Sound level exponential decay calculator
A 90 dB sound that fades at 6 dB per second reaches 72 dB after 3 seconds. Enter the starting level, the decay rate and the elapsed time to get the final level…
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Wavelength of a musical note calculator
See how long one cycle of a note really is: concert A at 440 Hz spans 77.954545 cm in air at 343 m/s.
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Compound microscope magnification calculator
Multiplies the two stages of a microscope: a 16 cm tube with a 0.5 cm objective gives 32×, a 2.5 cm eyepiece adds 10×, so the view is enlarged 320 times.
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Lens focal length in a medium calculator
See how much a lens weakens under water: a 0.1 m glass lens of index 1.5 in water of index 1.33 stretches to 0.391176470588235 m, and its power falls from 10 to…
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Mirror focal length from radius
Enter the radius of curvature and get the focal length at once: a mirror with a radius of 40 has a focal length of 20 and is concave.
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Spherical mirror image height calculator
Multiply the object height by the mirror's magnification and you have the image: a 4 cm object at magnification −1.5 gives an image 6 cm tall, inverted.
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Telescope objective focal length calculator
To reach 40x with a 25 mm eyepiece, you need an objective focal length of 1000 mm. Enter your eyepiece and the magnification you want, and compare the result…
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Telescope true field of view calculator
A 50° eyepiece at 40x shows 1.25° of real sky, enough to frame the full Moon with room to spare. Enter the apparent field printed on the eyepiece and your…
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Abbe number calculator from refractive indices
Turns three refractive indices into one dispersion figure: 1.5168, 1.52238 and 1.51432 give an Abbe number of 64.1191067.
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Camera angle of view calculator
Shows how much of a scene a lens takes in: a 50 mm focal length across a 36 mm sensor width covers 39.597752709 degrees.
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Diffraction-limited spot size
Know the smallest spot your optics can focus: the Airy disk measures 2.44 × wavelength × f-number, so light of 5.5 × 10⁻⁷ m at f/2 gives a core 0.000002684 m…
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Prism angle of deviation calculator
With 48° of incidence, 52° of emergence and a 60° apex, a ray leaving the prism has turned 40°. Enter the three angles in degrees and get the signed deviation…
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Refractive index from speed of light
Divide 299792458 m/s by the speed light reaches inside your sample: at two-thirds of that pace, the refractive index is 1.5.
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Total internal reflection critical angle calculator
Light passing from glass of index 1.5 into air of index 1 is fully reflected beyond 41.810315° from the normal.
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Fresnel reflection at normal incidence
See how much light a surface throws back at normal incidence: from air at index 1 into glass at 1.5, reflectance is 0.04, or 4 %, and 96 % passes through, with…
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Grating resolving power calculator
Check how finely a grating can split nearby wavelengths: second order across 1200 illuminated lines gives a resolving power of 2400, with the multiplication…
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Grating wavelength from angle calculator
Identify the light from where it lands: a line spacing of 0.001 mm with a first-order angle of 30° gives a wavelength of 0.0005 mm, in the unit you used for the…
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Single-slit diffraction minima angle
For order 1, a 500 nm wavelength and a 0.1 mm slit, the minimum sits at 0.286480091241°, where the sine of the angle is 0.005.
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Thin prism deviation calculator
A thin prism of index 1.5 with a 4° apex turns a ray by 2°. Enter the refractive index and the apex angle in degrees to get the deviation and the…
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Wavelength from double-slit fringes calculator
Turn a measured interference pattern into a wavelength: fringes 0.0042 m apart, slits 0.00025 m apart and a screen 1.5 m away give 700 nm.
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Anti-reflection coating thickness calculator
Gives the thickness a single-layer AR coating must reach: at 550 nm with a coating index of 1.38, just under 100 nm, for an optical thickness of 137.5 nm.
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Fresnel number calculator
Check which diffraction model fits your setup: a 0.001 m aperture radius, 1 m of distance and 6.328e-7 m light give a Fresnel number of 1.58027812895, labeled…
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Michelson interferometer mirror displacement
Count fringes, get mirror travel: 100 fringes with light of 6.328e-7 m mean the mirror moved 0.00003164 m, half the optical path change, because the light…
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Newton's rings lens radius of curvature
Turn one measured Newton's ring into the lens curvature: a dark ring of radius 0.0025 m at order 5 under 589 nm light means the lens surface has a radius of…
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Photoelectric Threshold Wavelength Calculator
Type a work function such as 2.28 eV and see the longest wavelength that still knocks electrons out of the surface, in nanometres and metres, with the matching…
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