Problem 39 - Algebra and Trigonometry with Analytic Geometry

Question

Vibration of a viotin string Mathematical analysis of a vibrating violin string of length \(I\) 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

\( f(x) = \frac{1}{2} [\sin(\frac{\pi n}{l} (x + kt)) + \sin(\frac{\pi n}{l} (x - kt))] \).

Step 1: Identify the functions involved

The function given is \( f(x) = \sin \left( \frac{\pi n}{l} x \right) \cos \left( \frac{k \pi n}{l} t \right) \). It involves a product of sine and cosine functions, where \(x\) and \(t\) are variables, and \(n\), \(l\), and \(k\) are constants.

Step 2: Use the product-to-sum formula

To express the product of sine and cosine as a sum of sine functions, use the trigonometric identity: \( \sin A \cos B = \frac{1}{2}[\sin(A + B) + \sin(A - B)] \). In this case: \( A = \frac{\pi n}{l} x \) and \( B = \frac{k \pi n}{l} t \).

Step 3: Substitute and simplify

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

Step 4: Final expression

Thus, the function \( f(x) \) can be expressed as a sum of two sine functions: \[ f(x) = \frac{1}{2} \left[ \sin \left( \frac{\pi n}{l} (x + kt) \right) + \sin \left( \frac{\pi n}{l} (x - kt) \right) \right] \].

Key Concepts

Product-to-Sum Formula

The product-to-sum formulas are essential tools in trigonometry, helping to simplify the product of sine and cosine functions into sums. This transformation can make complex equations more manageable. Specifically, the formula \( \sin A \cos B = \frac{1}{2}[\sin(A + B) + \sin(A - B)] \) is used to convert products of trigonometric functions into sums of sine functions.
This formula is quite handy in various fields such as signal processing, physics, and engineering. By converting the product of \( \sin \) and \( \cos \) into a sum, it becomes easier to integrate or differentiate, or to interpret its behavior over time. This is precisely what was done in the step-by-step solution where the function \( f(x) = \sin \left( \frac{\pi n}{l} x \right) \cos \left( \frac{k \pi n}{l} t \right) \) was rewritten as:

• \( \frac{1}{2} \left[ \sin \left( \frac{\pi n}{l} (x + kt) \right) + \sin \left( \frac{\pi n}{l} (x - kt) \right) \right] \)

To effectively apply these formulas, it's crucial to identify the exact expressions for \( A \) and \( B \), which represent specific angles in your function. This knowledge allows for the precise transformation of trigonometric expressions.

Sine Function

The sine function is one of the cornerstone trigonometric functions, often abbreviated as \( \sin \). It describes a smooth, periodic oscillation, which is key in modeling wave-like phenomena such as sound waves or vibrating strings. Its foundational role in the analysis of vibrations and harmonics makes it invaluable.
Sine functions have the form \( \sin(x) \), oscillating between -1 and 1. The function is periodic with a period of \( 2\pi \). In the context of the exercise, \( \sin \left( \frac{\pi n}{l} x \right) \) is used, representing a spatial sine wave, where \( \frac{\pi n}{l} \) defines the frequency related to the length \( l \) of the string, and \( n \) determines the mode of vibration.

• Amplitude: Determines the height of the wave's peaks and troughs.
• Frequency: Indicated by \( \frac{\pi n}{l} \) governing how many oscillations occur over a unit distance.

The sine function's role in this mathematical model helps explain how different modes of vibration form complex sound patterns, which in musical terms, contribute to the timbre and resonance of notes played on the violin.

Cosine Function

The cosine function, abbreviated as \( \cos \), is another fundamental trigonometric function. Often associated with the sine function, it also exhibits periodic oscillation. The cosine function \( \cos(x) \) ranges from -1 to 1, similar to the sine function, but is phase-shifted by \( \frac{\pi}{2} \) radians. It is used to describe various wave phenomena, such as those encountered in circuits or sound.
In the provided exercise, \( \cos \left( \frac{k \pi n}{l} t \right) \) represents a time-dependent cosine wave, where \( t \) stands for time. The term \( \frac{k \pi n}{l} \) controls the frequency, affecting how fast the wave oscillates with time.

• Phase Shift: The cosine function is shifted by \( \frac{\pi}{2} \) compared to the sine.
• Periodic Behavior: Like sine, cosine completes one full cycle within \( 2\pi \) radians.

The interplay of sine and cosine functions in this context models the temporal evolution of a vibrating string. As these trigonometric functions combine, they create complex interference patterns that are vital in understanding the harmonics and resonance characteristics of musical instruments.