As you may recall, the speed of a wave depends on the medium through which it is traveling. The speed of a wave on a string therefore depends on the characteristics of the string. These characteristics are the tension in the string, and the mass per unit length (linear density) of the string.
Using T to represent the tension and μ to represent the linear density of the string, the velocity of a wave on a string is given by the equation:
In order for a standing wave to form on a string that is fixed at both ends, you need just the right combination of string length, vibrational frequency, string tension, and string linear density. This is because the wavelength depends on the wave velocity and frequency, and the velocity depends on the string's tension and linear density.
Use the simulation below to see how changing the tension, linear density, and vibrational frequency affects the wave speed. The standing wave will only form if the wavelength of the string is equal to some integer number of half-wavelengths.
To set up one possible fundamental mode vibration, set the Linear Density to its lowest value (0.1 x 10-3 m) and the Tension to its highest value (100 N). Use the velocity equation to verify that the wave would have a speed of 1000 m/s.
Keeping in mind that the fundamental mode occurs when the length of the string (L) is equal to one half a wavelength (λ), you can verify that the wavelength of the fundamental is 8 meters.
Using the velocity equation you can see that velocity divided by the wavelength yeilds the required frequency. With a velocity of 1000 m/s and a wavelength of 8 m, you will need a frequency of 125 Hz to produce the fundamental.