Problem 53 - Algebra 2

Question

For sound waves, the period and the frequency of a pitch are reciprocals of each other: period \(=\frac{\text { seconds }}{\text { cycle }}\) and frequency \(=\frac{\text { cycles }}{\text { second }} .\) Write an equation for each pitch. Let \(\theta=\) time in seconds, Use \(a=1\). the lowest pitch easily heard by humans: 30 cycles per second

Step-by-Step Solution

Answer

The frequency equation is \(f(\theta) = 30\) and the period equation is \(P(\theta) = \frac{1}{30}\).

Step 1: Understand the Relationship

The period and frequency of any sound wave are reciprocals. This implies that if the frequency is \(f\), then the period \(P\) is \(\frac{1}{f}\).

Step 2: Equation for frequency

Given that \(a = 1\), the frequency \(f\) can be written as a function of \(\theta\) (time in seconds) as follows: \(f(\theta) = a\), where \(a = 30\). So, \(f(\theta) = 30\). This means that for any given \(\theta\) (time in seconds), the frequency of the pitch remains constant at 30 cycles per second.

Step 3: Equation for period

The period \(P\) is the reciprocal of the frequency, that is \(P = \frac{1}{f}\). Substitute \(f(\theta)\) from Step 2 into this equation, we have \(P(\theta) = \frac{1}{f(\theta)} = \frac{1}{30}\). So the period, for any given \(\theta\), remains constant at \(\frac{1}{30}\) seconds per cycle.

Key Concepts

Frequency

Frequency refers to how often a wave repeats itself in one second. For sound waves, we measure frequency in cycles per second, also known as Hertz (Hz). It's like counting how many hoops a sound wave jumps through every second.
Understanding frequency is important because it determines the pitch of a sound. Higher frequencies mean higher pitches, while lower frequencies mean lower pitches. Consider how different musical notes have different pitches due to their varying frequencies.
In the given exercise, we have a frequency of 30 cycles per second. This means the wave repeats its pattern 30 times each second. This consistent repetition is a key factor in how we perceive sound.

Period

The period of a sound wave is the time it takes to complete one full cycle. Think of it as the duration for one repetition of the wave. We measure the period in seconds per cycle.
Period and frequency are tightly connected. If you know one, you can find the other because they are reciprocals. In simple terms, if frequency goes up, period goes down, and vice-versa. This is because the wave is cramming more cycles into the same amount of time.
For the exercise, where the frequency is 30 cycles per second, the period can be found using the equation: period = \(\frac{1}{\text{frequency}}\). So, the period equals \(\frac{1}{30}\) seconds per cycle. This indicates that each cycle of the wave takes approximately 0.033 seconds to complete.

Reciprocal

Reciprocal is a mathematical term used to describe a particular inverse relationship between two numbers. For any non-zero number, its reciprocal is \(\frac{1}{\text{that number}}\).
In the context of sound waves, the relationship between frequency and period is described using reciprocals. This means if you multiply the frequency by its period, you get 1, as \(\text{frequency} \times \text{period} = \frac{1}{\text{frequency}} \times \text{frequency} = 1\).
This concept is essential in wave mechanics and helps us understand how changing one aspect of a wave affects the other. In practical terms, if the frequency of a sound wave changes, you can instantly calculate the new period by finding its reciprocal and vice-versa.