Question

Will they have exactly the same fundamental (lowest) frequency? Explain

Answer 1

The string on a musical instrument is (almost) fixed at both ends, so any vibration of the string must have nodes at each end. Now that limits the possible vibrations. For instance the string with length L could have a standing wave with wavelength twice as long as the string (wavelength ? = 2L) as shown in the first sketch in the next series. This gives a node at either end and an antinode in the middle.

This is one of the modes of vibration of the string ("mode of vibration" just means style or way of vibrating). What other modes are allowed on a string fixed at both ends? Several standing waves are shown in the next sketch.

A sketch of the first four modes of vibration of an idealised* stretched string with a fixed length. The vertical axis has been exaggerated.

Let's work out the relationships among the frequencies of these modes. For a wave, the frequency is the ratio of the speed of the wave to the length of the wave: f = v/?. Compared to the string length L, you can see that these waves have lengths 2L, L, 2L/3, L/2. We could write this as 2L/n, where n is the number of the harmonic.

The fundamental or first mode has frequency f₁ = v/?₁ = v/2L,