Problem 21 - Advanced Engineering Mathematics

Question

A real-valued function \(f\) is said to be periodic with period \(T\) if \(f(x+T)=f(x)\). For example, \(4 \pi\) is a period of \(\sin x\) since \(\sin (x+4 \pi)=\sin x\). The smallest value of \(T\) for which \(f(x+T)=f(x)\) holds is called the fundamental period of \(f\). For example, the fundamental period of \(f(x)=\sin x\) is \(T=2 \pi\). What is the fundamental period of each of the following functions? (a) \(f(x)=\cos 2 \pi x\) (b) \(f(x)=\sin \frac{4}{L} x\) (c) \(f(x)=\sin x+\sin 2 x\) (d) \(f(x)=\sin 2 x+\cos 4 x\) (e) \(f(x)=\sin 3 x+\cos 2 x\) (f) \(f(x)=A_{0}+\sum_{n=1}^{\infty}\left(A_{n} \cos \frac{n \pi}{p} x+B_{n} \sin \frac{n \pi}{p} x\right), A_{n}\) and \(B_{n}\) depend only on \(n\)

Step-by-Step Solution

Answer

(a) 1, (b) \(\frac{L\pi}{2}\), (c) \(2\pi\), (d) \(\pi\), (e) \(2\pi\), (f) \(2p\)

Step 1: Understanding Periodicity of Cosine Function

The function given is \( f(x) = \cos 2\pi x \). The cosine function, \(\cos(x)\), has a fundamental period of \(2\pi\). Therefore, \(\cos(2\pi x)\) has a period \(T\) such that \(T = \frac{2\pi}{2\pi} = 1\). This means the smallest positive period for this cosine function is \(1\).

Step 2: Finding the Period of Adjusted Sine Function

The function given is \(f(x)=\sin\left(\frac{4}{L} x\right)\). The sine function \(\sin(x)\) has a fundamental period of \(2\pi\). Adjusting it to \(\sin\left(\frac{4}{L} x\right)\) gives a period \(T\) such that \(T = \frac{2\pi}{\frac{4}{L}} = \frac{L\pi}{2}\).

Step 3: Combining Sine Functions

The function is \(f(x)=\sin x + \sin 2x\). The sine functions \(\sin(x)\) and \(\sin(2x)\) have periods of \(2\pi\) and \(\pi\), respectively. The fundamental period of \(f(x)\) will be the least common multiple of these periods, which is \(2\pi\).

Step 4: Period of Combined Sine and Cosine Terms

The function is \(f(x)=\sin 2x + \cos 4x\). \(\sin(2x)\) has a period of \(\pi\), and \(\cos(4x)\) has a period of \(\frac{\pi}{2}\). The least common multiple of these periods is \(\pi\), which is the fundamental period of the function.

Step 5: Period of Sine and Cosine Addition

The function is \(f(x) = \sin 3x + \cos 2x\). \(\sin(3x)\) has a period of \(\frac{2\pi}{3}\), and \(\cos(2x)\) has a period of \(\pi\). The least common multiple of these periods is \(2\pi\), which is the fundamental period of the function.

Step 6: Periodicity in Trigonometric Series

The function is a Fourier series \(f(x) = A_0 + \sum_{n=1}^{\infty} \left(A_n \cos \frac{n\pi}{p}x + B_n \sin \frac{n\pi}{p}x\right)\). Each term in the series like \(\cos\left(\frac{n\pi}{p} x\right)\) and \(\sin\left(\frac{n\pi}{p} x\right)\) has a fundamental period of \(2p\). Therefore, the fundamental period of the entire function is \(2p\).

Key Concepts

Fundamental Period

To comprehend periodic functions, understanding the fundamental period is key. A function is periodic if it repeats its values at regular intervals, known as periods. The fundamental period is the smallest positive interval, denoted by \( T \), for which the function satisfies \( f(x + T) = f(x) \) for all \( x \). This concept is crucial for determining how functions like sine and cosine repeat over their domains.
For example, the sine function \( f(x) = \sin(x) \) has a fundamental period of \( 2\pi \). This means that every \( 2\pi \) units along the x-axis, the sine function looks exactly the same. Similarly, for the function \( f(x) = \cos(2\pi x) \), the fundamental period is \( 1 \), because every increase of 1 along the x-axis returns the same value.

Cosine Function

The cosine function, \( \cos(x) \), is an essential trigonometric function with a fundamental period of \( 2\pi \). This means that the graph of the cosine function repeats itself every \( 2\pi \) units along the x-axis.
The cosine function can be stretched or compressed horizontally by a factor. For example, in the function \( f(x) = \cos(2\pi x) \), the regular period \( 2\pi \) of \( \cos(x) \) is divided by the coefficient of \( x \), which results in a period of \( \frac{2\pi}{2\pi} = 1 \). Therefore, the fundamental period of \( \cos(2\pi x) \) is 1, which is much shorter than the original cosine function.

Sine Function

The sine function, denoted \( \sin(x) \), is another core trigonometric function with a fundamental period of \( 2\pi \). This fundamental period indicates that the pattern of the sine wave repeats every \( 2\pi \) along the x-axis.
Like the cosine function, the sine function can also be stretched or compressed. Consider the example function \( f(x) = \sin\left(\frac{4}{L}x\right) \). Here, the fundamental period is adjusted by the factor of \( \frac{4}{L} \). Calculate the new period by dividing \( 2\pi \) by this factor, giving a fundamental period of \( \frac{L\pi}{2} \). Such manipulations are common practice to fit trigonometric functions to specific periodic data sets.

Fourier Series

Fourier series are incredibly versatile tools in mathematics that allow us to express complex periodic functions as sums of simple sine and cosine functions. This decomposition is central to many fields, including signal processing and heat transfer.
A typical Fourier series is expressed as:

• \( f(x) = A_0 + \sum_{n=1}^{\infty} \left(A_n \cos\left(\frac{n\pi}{p} x\right) + B_n \sin\left(\frac{n\pi}{p} x\right)\right) \)

The coefficients \( A_n \) and \( B_n \) depend on the function's properties and determine the amplitude of each sine and cosine term.
For each term in this series, the fundamental period is determined largely by the denominator \( p \) in the argument of the sine and cosine functions. The smallest period over which the entire series repeats is \( 2p \), demonstrating how Fourier series can consolidate seemingly chaotic patterns into understandable repetitions.