"Here we are concerned with one of the most ancient branches of mathematics, the theory of the vibrating string , which has its roots in the ideas of the Greek mathematician Pythagoras." NORBERT WIENER, I Am a Mathematician (1956).
"A through understanding of vibrating strings, its dynamic, its natural vibrations, its response at different frequencies - we are introduced to results and concepts that have their counterparts throughout the realm of physics, including electromagnetic theory, quantum mechanics, and all the rest." A. P. French.
In other words looking at the various aspects of a vibrating string is a starting point for more complicated phenomena.
Take a string with its ends fixed. Our interest is not in the musical effects but in the basic mechanical fact that the string has a number of well defined states of natural vibration. These "stationary vibrations", called this because each point on the string vibrates transversely in simple harmonic motion with constant amplitude, have the same frequency of vibration for all parts of the string.
These stationary vibrations are called the normal modes of the string. Except for the lowest mode, there are points where the displacement of the string remains at zero. Such places are called nodes; while positions of maximum amplitude are called anti nodes.
One way to describe the shape of the string at any given time in a particular mode n, is by noticing that the total length of the string must exactly accommodate an integral number of half-sine curves. From this we can define the wavelength, lambda, associated with mode, n.
This means that all the possible frequencies of a given stretched string are just integral multiples of the lowest frequency. The lowest frequency, w, is called the basic mode or the fundamental. The frequency of this fundamental is what characterizes the pitch, and therefore defines the tension required to obtain a certain note from a string of given mass and length. To begin we will suppose that the string has a uniform linear density (mass per unit length) equal to u and that it is stretched with a tension T. Also, since the end points are fixed then the displacements at the ends must be zero. The number of cycles per unit of time, v, measured in Hz, is equal to w/2pi.
For a more in-depth explanation of equations used to model wave motion in a string. Use your browsers back button to return to this page.
Example: The E string of a violin is to be tuned to a frequency of 640 Hz. Its length and mass are 33 cm and 0.125 g. What tension is required? answer
A timed animation constructed from the exact Fourier series for an ideal plucked string, vibrating in a plane, with uniform tension and initial triangular profile of a guitar unwound g-string. see animation
It is very important to your understanding of waves to appreciate how the motion of a wave along its direction of travel (x) can be the results of particle displacements along the transverse direction (y). As the pulse moves from left to right those points to the left of the peak (wave crest) are decreasing while the points to the right of the peak are increasing in displacement. Therefore, the peak displacement occurs at larger and larger values of x as time goes on. Multimedia Files on String Motion
In this simulation you will be able to:
Go to String Lab