11.3. Resonance with Sound Waves http://www.ck12.org
Strings Fixed at One End and Opened at One End
A string fixed at one only end displays a different standing wave pattern. In this case, the unbounded end of a
string of lengthLis an antinode. The fundamental mode (the first harmonic) for the lengthLof string contains only
one-fourth of a wavelength as shown inFigure11.20. Therefore L, the length of the unstretched string, is equal to
one-quarter the wavelength, which is^14 λ 1 =L→λ 1 = 4 L.
FIGURE 11.20
The second harmonic contains three-quarters of a wavelength^34 λ 2 =L→λ 2 =^43 Las shown inFigure11.21.
FIGURE 11.21
The third harmonic contains five-fourths of a wavelength^54 λ 2 =L→λ 2 =^45 Las shown inFigure11.22.
If the pattern continues, then the fourth harmonic will have a wavelength of^74 λ 4 =L→λ 4 =^47 L. Looking at the
expressions for the length of the string in terms of the wavelength, a simple pattern emerges^14 λ 1 ,^34 λ 2 ,^54 λ 3 ,^74 λ 4.. ..
We can express the condition for resonance asL=n 4 λnorλn=^4 nL, whereLis the length of string andn= 1 , 3 , 5.. ..
As long as the tension in the string remains fixed, the velocity of the wave along the string remains constant. Does it
seem reasonable that a sagging string will not support the same wave velocity as a taut string? Sincev=λfproduct
λfis constant as long as the wave velocity remains constant. Therefore, for a string vibrating in many different
modes, we havev=λ 1 f 1 =λ 2 f 2 =λ 3 f 3.. ..