http://www.ck12.org Chapter 11. Vibrations and Sound
FIGURE 11.17
If you compare the wave shape of the first harmonic to the wave shape ofFigure11.17, it will be apparent that the
first harmonic contains one-half of a wavelength,λ. ThereforeL, the length of the unstretched string, is equal to
one-half the wavelength, which is^12 λ 1 =L→λ 1 = 2 L.
The second harmonic contains an entire wavelength^22 λ 2 =L→λ 2 =Las shown inFigure11.18.
FIGURE 11.18
And the third harmonic contains one and one-half wavelengths^32 λ 3 =L→λ 3 =^23 L.
FIGURE 11.19
If the pattern continues then the fourth harmonic will have a wavelength of^42 λ 4 =L→λ 4 =^12 L. Looking at the ex-
pressions for the length of the string in terms of the wavelength, a simple pattern emerges:L=^12 λ 1 ,^22 λ 2 ,^32 λ 3 ,^42 λ 4.. ..
We can express the condition for standing waves (and of resonance, as well) asL=n 2 λnorλn=^2 nL, whereLis the
length of string andn= 1 , 2 , 3.. ..
Check Your Understanding
- How many nodes and anti-nodes are shown inFigure11.19?
Answer:There are three nodes and two anti-nodes.
- If the length of the unstretched string is 20 cm, what is the wavelength for the 10th harmonic?
Answer:λn=^2 nL→λ 10 = 102 L=^15 (20 cm) =4 cm