TOPIC 4 WAVES


string at different locations along its length. These patterns showed nodes (zero displacement) at the two fixed ends of the string. But when looking at the case of air columns, a closed end in a column of air can be compared to the fixed end of a vibrating string. In other words, at the closed end of an air column, air is not free to undergo displacement and therefore is forced into becoming the node. However, the opposite is true at the open end where air is free to undergo its full longitudinal motion. As such, the standing wave patterns will show antinodes at the open ends of air columns.


All higher frequencies are integer–multiples of the lowest (fundamental) frequency for the system (f, 2f, 3f, 4f …). The sequence of progression of harmonics from one resonant frequency to the next will define the overtone frequencies for the whole system. Therefore, as long as you can work out the fundamental frequency for the pipe, you can calculate how many times you multiply f for the next harmonic.


pipe 1 pipe 2


pipe 3 pipe 4


pipe 5 • Pipe 1: length of pipe l = 1


FIG. 17.18 Antinodes are present at both ends of open pipes


fundamental H Open pipes and harmonics


Open pipes are pipes that have both ends open. In these, antinodes are at both ends. The harmonic patterns shown in Fig. 17.18 can be set up in these types of pipes.


To work out the possible harmonics in a pipe, calculate the fundamental frequency f in terms of c and l. Use the wave formula c = f λ to calculate f at the beginning. Once you know the fundamental frequency, you can calculate what multiples of it are possible.


The following calculations apply to the open pipes shown in Fig. 17.18:


__ 2 λ = 1


__ 2 λ1 ∴ λ1 = 2l, so that f1 = c


• Pipe 2: length of pipe l = λ2 ∴ λ2 = l, so that f2 = c __


• Pipe 3: length of pipe l = 3


__ 2 λ3 ∴ λ3 = 2


• Pipe 4: length of pipe l = 2λ4 ∴ λ4 = 1 __


• Pipe 5: length of pipe l = 5


FIG. 17.19 A trumpet is an example of an open pipe instrument


Closed pipes


Closed pipes are pipes with one end closed. In this case, a node develops at the closed end and an antinode always exists at the open end.


In the same way as for open pipes, by looking at the possible waveforms (Fig. 17.20) and calculating the multiples of the fundamental frequency, we can see the possible harmonics.


200 FUSION l = 2


2 l, so that f4 = 2c ___


__ 2 λ5 ∴ λ5 = 2


__ 5 l, so that f5 = 5c


( c ___


__ 3 l, so that f3 = 3c


__ 2l = f


2l ) = 2f


___ 2l = 3


l = 4


___ 2l = 5


( c ___


( c ___


( c ___


2l ) = 3f 2l ) = 4f 2l ) = 5f


This shows that all harmonics are possible in an open pipe (f, 2f, 3f, 4f, 5f …).


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