Problem 39 - Precalculus: Functions and Graphs

Question

Mathematical analysis of a vibrating violin string of length \(l\) involves functions such that $$f(x)=\sin \left(\frac{\pi n}{l} x\right) \cos \left(\frac{k \pi n}{l} t\right)$$ where \(n\) is an integer, \(k\) is a constant, and \(t\) is time. Express \(f\) as a sum of two sine functions.

Step-by-Step Solution

Answer

The function is \(f(x) = \frac{1}{2} \left( \sin \left( \frac{\pi n}{l} x + \frac{k \pi n}{l} t \right) + \sin \left( \frac{\pi n}{l} x - \frac{k \pi n}{l} t \right) \right).\)

Step 1: Use Trigonometric Identity

To express the function as a sum of sines, apply the trigonometric identity for the product of sine and cosine: \[ \sin(A) \cos(B) = \frac{1}{2} \left( \sin(A + B) + \sin(A - B) \right). \] Here, \(A = \frac{\pi n}{l} x\) and \(B = \frac{k \pi n}{l} t\).

Step 2: Substitute into the Identity

Substitute \(A\) and \(B\) into the trigonometric identity. The expression becomes: \[ \frac{1}{2} \left( \sin \left( \frac{\pi n}{l} x + \frac{k \pi n}{l} t \right) + \sin \left( \frac{\pi n}{l} x - \frac{k \pi n}{l} t \right) \right). \]

Step 3: Rewrite the Function

Now, replace the original function expression with its equivalent using the identity: \[ f(x) = \frac{1}{2} \left( \sin \left( \frac{\pi n}{l} x + \frac{k \pi n}{l} t \right) + \sin \left( \frac{\pi n}{l} x - \frac{k \pi n}{l} t \right) \right). \]This is the form of \(f(x)\) expressed as a sum of two sine functions.

Key Concepts

The Sine Function

Understanding the sine function is critical in analyzing vibrations and waves. The sine function, denoted as \( \sin(\theta) \), is a fundamental trigonometric function, characterized by its periodic wave-like pattern. It is typically used to describe oscillating or wave-like phenomena
like sound waves or alternating currents.

Key characteristics of the sine function:

• Periodicity: The sine function repeats its values in regular intervals, or periods. For \( \sin(x) \), the period is \(2\pi\).
• Range: The sine function has a range of [-1, 1], meaning it oscillates between -1 and 1 as it repeats through its cycles.
• Wave form: The graph of the sine function is a smooth, continuous wave that crosses the x-axis at regular intervals.

The vibration of a violin string can be modeled using the sine function because it naturally represents the harmonic oscillations that occur as the string vibrates.

The Cosine Function

Much like the sine function, the cosine function is a crucial trigonometric function in wave analysis. The cosine function is denoted by \( \cos(\theta) \) and, just like sine, it represents periodic oscillations.

The main features of the cosine function include:

• Periodicity: The cosine function repeats every \(2\pi\), making its period the same as the sine function.
• Range: Cosine also oscillates between -1 and 1, similar to the sine function.
• Wave form: The graph of the cosine function resembles a sine wave, but it is shifted to the left by \(\pi/2\) radians.

In the context of the vibrating strings, the cosine function can represent certain aspects of the wave's motion, such as its starting phase or how it damping behaves over time. Working in conjunction with the sine function, it allows a full representation of the wave's behavior.

Vibrating Strings

Vibrating strings are a common phenomenon in musical instruments such as guitars, pianos, and violins. These vibrations produce sound waves that we hear. The analysis of a vibrating string involves understanding how the string moves and the frequencies it generates.

Here's how vibrating strings work:

• Wave Production: When a string vibrates, it produces a standing wave, characterized by a pattern of nodes and antinodes along the string length.
• Frequency Dependency: The frequency at which the string vibrates is influenced by its length, tension, and mass per unit length. Shorter, tighter, or lighter strings vibrate at higher frequencies, producing higher pitches.
• Harmonics: A vibrating string naturally divides itself into integer segments, called harmonics, as it vibrates. Each harmonic corresponds to a certain frequency. The sine and cosine functions model these harmonics due to their inherent wave-like characteristics.

In mathematical terms, the motion of a vibrating string can be modeled using a combination of sine and cosine functions, allowing for a well-rounded representation of its dynamic behavior.