Fourier Analysis

(339) |

where is the angular frequency of the fundamental (i.e., ) harmonic, and the and are the amplitudes and phases of the various harmonics. The preceding expression can also be written

where and . The function is periodic in time with period . In other words, for all . This follows because of the mathematical identities and , where is an integer. [Moreover, there is no for which for all .] Can any periodic waveform be represented as a linear superposition of sine and cosine waveforms, whose periods are integer subdivisions of that of the waveform, such as that specified in Equation (340)? To put it another way, given an arbitrary periodic waveform , can we uniquely determine the constants and appearing in Equation (340)? It turns out that we can. Incidentally, the decomposition of a periodic waveform into a linear superposition of sinusoidal waveforms is commonly known as

The problem under investigation is as follows. Given a periodic waveform , where for all , we need to determine the constants and in the expansion

where

(342) |

It can be demonstrated that [cf., Equation (277)]

where and are positive integers. Thus, multiplying Equation (341) by , and then integrating over from 0 to , we obtain

where use has been made of Equation (343)-(345), as well as Equation (278). Likewise, multiplying Equation (341) by , and then integrating over from 0 to , we obtain

Hence, we have uniquely determined the constants and in the expansion (341). These constants are generally known as

In principle, there is no restriction on the waveform in the previous analysis, other than the requirement that it be periodic in time. In other words, we ought to be able to Fourier analyze any periodic waveform. Let us see how this works. Consider the periodic sawtooth waveform

(348) |

with for all . (See Figure 35.) This waveform rises linearly from an initial value at to a final value at , discontinuously jumps back to its initial value, and then repeats ad infinitum. According to Equations (346) and (347), the Fourier harmonics of the waveform are

(349) | ||

(350) |

where . Integration by parts (Riley 1974) yields

(351) | ||

(352) |

Hence, the Fourier reconstruction of the waveform is written

(353) |

Given that the Fourier coefficients fall off like , as increases, it seems plausible that the preceding series can be truncated after a finite number of terms without unduly affecting the reconstructed waveform. Figure 35 shows the result of truncating the series after 4, 8, 16, and 32 terms (these cases correspond the top-left, top-right, bottom-left, and bottom-right panels, respectively). It can be seen that the reconstruction becomes increasingly accurate as the number of terms retained in the series increases. The annoying oscillations in the reconstructed waveform at , , and are known as

We can slightly generalize the Fourier series (341) by including an term. In other words,

(354) |

which allows the waveform to have a non-zero average. There is no term involving , because when . It can be demonstrated that

(355) | ||

(356) |

where , and is a positive integer. Making use of the preceding expressions, as well as Equations (343)-(345), we can show that

and also that Equations (346) and (347) still hold for .

As an example, consider the periodic ``tent'' waveform

where for all . (See Figure 36.) This waveform rises linearly from zero at , reaches a peak value at , falls linearly, becomes zero again at , and repeats ad infinitum. Moreover, the waveform has a non-zero average. It can be demonstrated, from Equations (346), (347), (357), and (358), that

(359) |

and

(360) |

for , with for . In fact, only the odd- Fourier harmonics are non-zero. Figure 36 shows a Fourier reconstruction of the ``tent'' waveform using the first 1, 2, 4, and 8 terms (in addition to the term) in the Fourier series (these cases correspond to the top-left, top-right, bottom-left, and bottom-right panels, respectively). The reconstruction becomes increasingly accurate as the number of terms in the series increases. Moreover, in this example, there is no sign of Gibbs' phenomena, because the tent waveform is completely continuous.

In our first example--that is, the sawtooth waveform--all of the Fourier coefficients are zero, whereas in our second example--that is, the tent waveform--all of the coefficients are zero. This occurs because the sawtooth waveform is odd in --that is, for all --whereas the tent waveform is even--that is, for all . It is a general rule that waveforms that are even in only have cosines in their Fourier series, whereas waveforms that are odd only have sines (Riley1974). Waveforms that are neither even nor odd in have both cosines and sines in their Fourier series.

Fourier series arise quite naturally in the theory of standing waves, because the normal modes of oscillation of any uniform continuous system possessing linear equations of motion (e.g., a uniform string, an elastic rod, an ideal gas) take the form of spatial cosine and sine waves whose wavelengths are rational fractions of one another. Thus, the instantaneous spatial waveform of such a system can always be represented as a linear superposition of cosine and sine waves: that is, a Fourier series in space, rather than in time. In fact, the process of determining the amplitudes and phases of the normal modes of oscillation from the initial conditions is essentially equivalent to Fourier analyzing the initial conditions in space. (See Sections 5.3 and 6.2.)