16.6 Standing waves and resonance  (Page 3/17)

The lab setup shows a string attached to a string vibrator, which oscillates the string with an adjustable frequency f . The other end of the string passes over a frictionless pulley and is tied to a hanging mass. The magnitude of the tension in the string is equal to the weight of the hanging mass. The string has a constant linear density (mass per length) $\mu$ and the speed at which a wave travels down the string equals $v=\sqrt{\frac{F_T}{\mu }}=\sqrt{\frac{mg}{\mu }}$ [link] . The symmetrical boundary conditions (a node at each end) dictate the possible frequencies that can excite standing waves. Starting from a frequency of zero and slowly increasing the frequency, the first mode $n=1$ appears as shown in [link] . The first mode, also called the fundamental mode or the first harmonic, shows half of a wavelength has formed, so the wavelength is equal to twice the length between the nodes ${\lambda }_1=2L$ . The fundamental frequency    , or first harmonic frequency, that drives this mode is

where the speed of the wave is $v=\sqrt{\frac{F_T}{\mu }}.$ Keeping the tension constant and increasing the frequency leads to the second harmonic or the $n=2$ mode. This mode is a full wavelength ${\lambda }_2=L$ and the frequency is twice the fundamental frequency:

The next two modes, or the third and fourth harmonics, have wavelengths of ${\lambda }_3=\frac{2}{3}L$ and ${\lambda }_4=\frac{2}{4}L,$ driven by frequencies of $f_3=\frac{3v}{2L}=3f_1$ and $f_4=\frac{4v}{2L}=4f_1.$ All frequencies above the frequency $f_1$ are known as the overtone     s . The equations for the wavelength and the frequency can be summarized as:

The standing wave patterns that are possible for a string, the first four of which are shown in [link] , are known as the normal mode     s , with frequencies known as the normal frequencies. In summary, the first frequency to produce a normal mode is called the fundamental frequency (or first harmonic). Any frequencies above the fundamental frequency are overtones. The second frequency of the $n=2$ normal mode of the string is the first overtone (or second harmonic). The frequency of the $n=3$ normal mode is the second overtone (or third harmonic) and so on.

The solutions shown as [link] and [link] are for a string with the boundary condition of a node on each end. When the boundary condition on either side is the same, the system is said to have symmetric boundary conditions. [link] and [link] are good for any symmetric boundary conditions, that is, nodes at both ends or antinodes at both ends.