Question

Determine whether the given function is periodic. If so, find its fundamental period. $$ \sin \pi x / L $$

Short answer

Answer: The fundamental period of the given function is \(P=2L\).

Step-by-step solution

Define periodic function

A function \(f(x)\) is periodic if there exists a non-zero constant P such that \(f(x+P)=f(x)\) for all x in its domain. The smallest such constant P is called the fundamental period of the function.

Determine if the given function is periodic

The sine function is known for having periodic nature. For a general sine function \(\sin(kx)\), it is periodic because \(\sin(kx + 2\pi n) = \sin(kx)\) for all integers n. Now, we have the function \(\sin(\frac{\pi x}{L})\). Comparing it with the general sine function, we can see k here is \(\frac{\pi}{L}\). Since k is not zero, the given function is periodic.

Find the fundamental period

To find the fundamental period, we want the smallest non-negative constant P such that $$ \sin\left(\frac{\pi (x+P)}{L}\right) = \sin\left(\frac{\pi x}{L}\right) $$ for all x. By comparing this with the general sine function, we want to find P such that $$ \frac{\pi (x+P)}{L} = \frac{\pi x}{L} + 2\pi n $$ for some integer n. This simplifies to $$ \frac{\pi P}{L} = 2\pi n $$ Now divide both sides by \(\pi\) and solve for P $$ \frac{P}{L} = 2n $$ $$ P = 2nL. $$ Here, the smallest positive integer value for n is 1. Therefore, the fundamental period P for the given function is \(P=2L\).

Key concepts

Fundamental Period

A periodic function repeats its values in regular intervals or periods. To determine if a function is periodic, look for a non-zero constant \( P \) such that \( f(x + P) = f(x) \) for all \( x \) in its domain. This constant \( P \) is the fundamental period. It represents the smallest interval after which the function starts repeating itself. For example, if a function has a fundamental period of 4, then its values at \( x \), \( x + 4 \), \( x + 8 \), and so on, will be identical.
Understanding the fundamental period helps to analyze wave patterns, oscillations, and other cyclic behaviors in mathematics and science. It's crucial in identifying the rhythm of these repetitive processes. Finding the fundamental period involves algebraically manipulating the function to find the smallest repeating interval, as demonstrated in periodic functions like sine or cosine.

Sine Function

The sine function is a classic example of a periodic function. It repeats its values in a smooth, wave-like pattern every \( 2\pi \) radians. The general form of a sine function is \( \sin(kx) \), where \( k \) influences the frequency or how quickly the function oscillates. The function \( \sin\left( \frac{\pi x}{L} \right) \) is a variation of the sine function that still retains its periodic nature.
The sine wave is fundamental in mathematics and physics due to its continuous and smooth oscillation, making it perfect for modeling waves such as sound, light, and water. When we observe functions like \( \sin(\theta) \), the values oscillate between -1 and 1, creating a predictable and repeating pattern. This nature of the sine function is utilized extensively in trigonometry and calculus. The periodicity of sine leads to intriguing applications in signal processing and different areas of engineering.

Mathematical Constants

Mathematical constants are numbers with a fixed value, often appearing in various mathematical contexts. Constants like \( \pi \) play a crucial role in trigonometry and periodic functions. In the function \( \sin\left( \frac{\pi x}{L} \right) \), \( \pi \) serves as a scaling factor in the argument of the sine function. This enriches the function's ability to map periodic phenomena to real-world scenarios.
The constant \( \pi \) is approximately 3.14159 and represents the ratio of the circumference of a circle to its diameter. It's ubiquitous in equations involving circles and waves. In the context of periodic functions, \( \pi \) helps determine oscillation periods and frequencies.

• \( e \), Euler's number, is another mathematical constant with immense significance, particularly in the contexts of growth models and calculus.
• Constants provide the foundation for numerous mathematical principles and are pivotal in helping simplify and solve complex mathematical problems.
• Recognizing these constants' roles facilitates a deeper understanding of the equations and models they appear in.

These constants are not just numbers; they are keys to unlocking deeper insights in mathematics and its applications.