Consider the transverse oscillations of a uniform string of length and mass per unit length that is stretched between two immovable walls. It is again convenient to define a Cartesian coordinate system in which measures distance along the string from the left wall, and measures the transverse displacement of the string. Thus, the instantaneous state of the system at time is determined by the function for . This function consists of an infinite number of different values, corresponding to the infinite number of different values in the range 0 to . Moreover, all of these values are free to vary independently of one another. It follows that we are indeed dealing with a dynamical system possessing an infinite number of degrees of freedom.
Let us try to reuse some of the analysis of the previous section. We can reinterpret as , as , and as , assuming that and . Moreover, becomes ; namely, a second derivative of with respect to at constant . Finally, , where is the tension in the string, can be rewritten as , because . Incidentally, we are again assuming that the transverse displacement of the string remains sufficiently small that the tension is approximately constant in . Thus, the equation of motion of the beaded string, (4.8), transforms into
However, Taylor expanding in at constant (see Appendix B), we obtain(4.27) 
(4.28) 
(4.29) 
(4.31) 
By analogy with Equation (4.12), let us search for a solution of the wave equation of the form
where , , , and are constants. We can interpret such a solution as a standing wave of wavenumber , wavelength , angular frequency , peak amplitude , and phase angle . Substitution of the preceding expression into Equation (4.30) yields the dispersion relation [cf., Equation (4.16)](4.33) 
The standing wave solution (4.32) is subject to the physical constraint that the two ends of the string, which are both attached to immovable walls, remain stationary. This leads directly to the spatial boundary conditions
It can be seen that the solution (4.32) automatically satisfies the first boundary condition. However, the second boundary condition is only satisfied when , which immediately implies that(4.36) 
It follows, from the previous analysis, that the th normal mode of the string is associated with the pattern of motion
where Here, and are constants that are determined by the initial conditions. (See Section 4.4.) How many unique normal modes are there? The choice yields for all and , so this is not a real normal mode. Moreover,(4.39)  
(4.40) 

Figure 4.6 illustrates the spatial variation of the first eight normal modes of a uniform string with fixed ends. It can be seen that the modes are all smoothlyvarying sine waves. The lowmodenumber (i.e., longwavelength) modes are actually quite similar in form to the corresponding normal modes of a uniformlybeaded string. (See Figure 4.3.) However, the highmodenumber modes are substantially different. We conclude that the normal modes of a beaded string are similar to those of a uniform string, with the same length and mass per unit length, provided that the wavelength of the mode is much larger than the spacing between the beads.

Figure 4.7 illustrates the temporal variation of the normal mode of a uniform string. All points on the string attain their maximal transverse displacements, and pass through zero displacement, simultaneously. Moreover, the mode possesses five nodes (i.e., points where the string remains stationary). Two of these are located at the ends of the string, and three in the middle. In fact, according to Equation (4.37), the nodes correspond to points where . Hence, the nodes are located at
(4.41) 
Finally, Figure 4.8 shows the first eight normal frequencies of a uniform string with fixed ends, plotted as a function of the mode number. It can be seen that the angular frequency of oscillation increases linearly with the mode number. Recall that the lowmodenumber (i.e., longwavelength) normal modes of a beaded string also exhibit a linear relationship between normal frequency and mode number of the form [see Equation (4.25)]
(4.42) 
(4.43) 