Open the Fourier Series Demo and select the "Sinusoid" function. The reason the Fourier series representation (shown in green) and analytical representation (shown in red) overlay exactly when the the number of Fourier terms is
≥
2 is because:

Question

Open the Fourier Series Demo and select the "Sinusoid" function. The reason the Fourier series representation (shown in green) and analytical representation (shown in red) overlay exactly when the the number of Fourier terms is 
≥
2 is because:

BlackTom AI · Accepted Answer

Let’s unpack what the question is saying about the Fourier Series Demo and the overlay of the green (Fourier) and red (analytical) representations.
The provided statement asserts that the analytical function is a mode 2 sine wave, and therefore the Fourier representation matches exactly once the number of Fourier terms n is at least 2.
First, recall that a Fourier series represents a periodic function as a sum of sine and cosine terms with integer harmonics. If the target function is a pure sine wave at a specific harmonic (here, mode 2), then including the corresponding harmonic in the series can capture that component exactly.
Second, the phrase 'mode 2 sine wave' implies the signal has its fundamental content at the second harmonic (n = 2) rather than at n = 1. In practice, a Fourier series truncated to include terms up to n = 1 would not include the second harmonic, so it would fail to reproduce the mode-2 content precisely. By including terms with n ≥ 2, the series can represent the second harmonic component exactly, assuming the basis functions align with the signal's frequency.
Third, if the function truly is a single mode-2 sine wave (and no additional harmonics or components exist), then once you allow the n = 2 term (and possibly higher harmonics), the Fourier sum can reproduce the target exactly, resulting in the overlay with the analytical curve.
Now, consider potential misunderstandings: if one were thinking that a pure sine wave is always captured exactly with n = 1, that would be incorrect here because the statement specifies a mode-2 sine wave, which requires inclusion of the second harmonic to capture its frequency content properly. Conversely, asserting that infinite terms are needed would be inconsistent with the given modal description; a single nonzero second-harmonic component is exactly representable by including that harmonic in the finite sum.
In summary, the reasoning behind the statement is that the function is characterized by the second harmonic (mode 2), and the Fourier series must include up to at least n = 2 to reproduce that content exactly, leading to an exact overlay when n ≥ 2.

AMME2000 BMET2960 BMET9960 (ND) Week 4 Pre-work

Open the Fourier Series Demo and select the "Sinusoid" function. The reason the Fourier series representation (shown in green) and analytical representation (shown in red) overlay exactly when the the number of Fourier terms is ≥ 2 is because:

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