Standing Waves, Double Slit and Diffraction Grating Calculator: String and Pipe Harmonics

Standing waves and light interference with the formulas worked.

Calculated from standard textbook formulas, with every step shown. The formulas are fixed, so the working does not need checking: what matters is the numbers you type in. Use these calculators only if you are competent to choose the inputs (loads, material strength, dimensions, supports) and check your units and assumptions. For learning and checking your own working: design and sign-off are for a qualified engineer. Drafting Central does not store the numbers you type.

How this calculator works

Choose a string, a pipe, a double slit, a grating or a single slit.

Enter the lengths, the tension and the temperature, or the wavelength and the slit details.

The calculator gives the harmonics, the fringe positions or the angles, with the working.

Formulas used

  • f_n = \frac{nv}{2L},\quad f_n = \frac{nv}{4L}\ (\text{closed})
  • \Delta y = \frac{\lambda D}{d}
  • d\sin\theta = m\lambda
  • a\sin\theta = m\lambda

Worked example

With the default numbers in the calculator above:

  1. Speed of the wave on the string: 374.17 m/s
  2. Frequencies of the harmonics (the fundamental and the overtones): 287.82 Hz for harmonic 1; 575.64 Hz for harmonic 2; 863.46 Hz for harmonic 3; 1151.3 Hz for harmonic 4; 1439.1 Hz for harmonic 5
  3. Wavelength of the fundamental (twice the length of the string): 1.3 m

Change the numbers in the calculator to see the full working, every step, for your own case.

A second worked example

A different case, with these numbers: What do you want? Harmonics of a pipe open at both ends; Length of the string or the pipe 0.5 m.

  1. Speed of sound in the air at that temperature: 343.42 m/s
  2. Frequencies of the harmonics of a pipe open at both ends (all the harmonics): 343.42 Hz for harmonic 1; 686.84 Hz for harmonic 2; 1030.3 Hz for harmonic 3; 1373.7 Hz for harmonic 4; 1717.1 Hz for harmonic 5

Common questions

What are the harmonics of a closed pipe?

Only the odd ones: 1, 3, 5 and so on, and the fundamental is half the frequency of an open pipe of the same length.

How does a double slit make fringes?

Light from the two slits travels different distances, so the waves add or cancel depending on the position on the screen.

What does a diffraction grating do?

It splits light into sharp maxima at angles given by d sin theta = m lambda, so it separates wavelengths.

Related calculators

Got a question about this? Ask an engineering question and get a plain-language answer with references.

Further reading: Any first-year physics text (waves and interference).

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