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The lowest frequency, called the fundamental frequency    , is thus for the longest wavelength, which is seen to be λ 1 = 2 L size 12{λ rSub { size 8{1} } =2`"L"} {} . Therefore, the fundamental frequency is f 1 = v w / λ 1 = v w / 2 L size 12{f rSub { size 8{1} } =v rSub { size 8{w} } /λ rSub { size 8{1} } =v rSub { size 8{w} } /2`"L"} {} . In this case, the overtones    or harmonics are multiples of the fundamental frequency. As seen in [link] , the first harmonic can easily be calculated since λ 2 = L size 12{λ rSub { size 8{2} } =L} {} . Thus, f 2 = v w / λ 2 = v w / 2 L = 2 f 1 size 12{f rSub { size 8{2} } =v rSub { size 8{w} } /λ rSub { size 8{2} } =v rSub { size 8{w} } /2`"L"=2f rSub { size 8{1} } } {} . Similarly, f 3 = 3 f 1 size 12{f rSub { size 8{3} } =3f rSub { size 8{1} } } {} , and so on. All of these frequencies can be changed by adjusting the tension in the string. The greater the tension, the greater v w size 12{v rSub { size 8{w} } } {} is and the higher the frequencies. This observation is familiar to anyone who has ever observed a string instrument being tuned. We will see in later chapters that standing waves are crucial to many resonance phenomena, such as in sounding boxes on string instruments.

The graph shows a wave with wavelength lambda one equal to L, which has two loops. There three nodes and two antinodes in the figure. The length of one loop is L.
The figure shows a string oscillating at its fundamental frequency.
first overtone is shown as the wave length if lambda two is L and there are three nodes and two antinodes in the figure. For first overtone the frequency f two is equal to two times f one.
First and second harmonic frequencies are shown.

Beats

Striking two adjacent keys on a piano produces a warbling combination usually considered to be unpleasant. The superposition of two waves of similar but not identical frequencies is the culprit. Another example is often noticeable in jet aircraft, particularly the two-engine variety, while taxiing. The combined sound of the engines goes up and down in loudness. This varying loudness happens because the sound waves have similar but not identical frequencies. The discordant warbling of the piano and the fluctuating loudness of the jet engine noise are both due to alternately constructive and destructive interference as the two waves go in and out of phase. [link] illustrates this graphically.

The graph shows the superimposition of two similar but non-identical waves. Beats are produced by alternating destructive and constructive waves with equal amplitude but different frequencies. The resultant wave is the one with rising and falling amplitude over different intervals of time.
Beats are produced by the superposition of two waves of slightly different frequencies but identical amplitudes. The waves alternate in time between constructive interference and destructive interference, giving the resulting wave a time-varying amplitude.

The wave resulting from the superposition of two similar-frequency waves has a frequency that is the average of the two. This wave fluctuates in amplitude, or beats , with a frequency called the beat frequency    . We can determine the beat frequency by adding two waves together mathematically. Note that a wave can be represented at one point in space as

x = X cos t T = X cos ft , size 12{x=X" cos"` left ( { {2π t} over {T} } right )=X" cos " left (2π ital "ft" right )","} {}

where f = 1 / T size 12{f= {1} slash {T} } {} is the frequency of the wave. Adding two waves that have different frequencies but identical amplitudes produces a resultant

x = x 1 + x 2 . size 12{x=x rSub { size 8{1} } +x rSub { size 8{2} } "."} {}

More specifically,

x = X cos f 1 t + X cos f 2 t . size 12{x=X"cos" left (2π`f rSub { size 8{1} } t right )+X"cos" left (2π`f rSub { size 8{2} } t right )"."} {}

Using a trigonometric identity, it can be shown that

x = 2 X cos π f B t cos f ave t , size 12{x=2X"cos" left (π`f rSub { size 8{B} } t right )"cos" left (2π`f rSub { size 8{"ave"} } t right )","} {}

where

f B = f 1 f 2 size 12{f rSub { size 8{B} } = lline f rSub { size 8{1} } - f rSub { size 8{2} } rline } {}

is the beat frequency, and f ave size 12{f rSub { size 8{"ave"} } } {} is the average of f 1 size 12{f rSub { size 8{1} } } {} and f 2 size 12{f rSub { size 8{2} } } {} . These results mean that the resultant wave has twice the amplitude and the average frequency of the two superimposed waves, but it also fluctuates in overall amplitude at the beat frequency f B size 12{f rSub { size 8{"B"} } } {} . The first cosine term in the expression effectively causes the amplitude to go up and down. The second cosine term is the wave with frequency f ave size 12{f rSub { size 8{"ave"} } } {} . This result is valid for all types of waves. However, if it is a sound wave, providing the two frequencies are similar, then what we hear is an average frequency that gets louder and softer (or warbles) at the beat frequency.

Practice Key Terms 8

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Source:  OpenStax, Physics 101. OpenStax CNX. Jan 07, 2013 Download for free at http://legacy.cnx.org/content/col11479/1.1
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