Problem 55 - Algebra 2

Question

Sound 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 highest pitch heard by bats: 120,000 cycles per second.

Step-by-Step Solution

Answer

The equation for the frequency of the bat's pitch is \(f = 120000\). The equation for the period of the sound wave is \(T = \frac{1}{\theta}\).

Step 1: Identify given values

Take note of the values provided: \(a = 1\) and the frequency \(f = 120,000 \, \text{cycles per second}\). Also, let \(\theta = \) time in seconds.

Step 2: Write the equation for frequency

Using the conventional expression for frequency \(f = \frac{\text{cycles}}{\text{second}}\), and given that \(a = 1\) and the identified frequency of the bat's pitch, the equation of the frequency of the bat's pitch is \(f = a \theta\), where \(f = 120000\). Remember that \(\theta =\) time in seconds.

Step 3: Write the equation for period

The period \(T\) of a wave is the reciprocal of the frequency. In mathematics, the reciprocal of a number x is simply \(1/x\). Therefore, the equation for period can be written as \(T =\frac{1}{f} = \frac{1}{a\theta}\). For \(a = 1\) and \(\theta =\) time in seconds, the equation simplifies to \(T = \frac{1}{\theta}\).

Key Concepts

Sound Waves

Sound waves are fascinating phenomena that surround us constantly. They are mechanical waves, meaning they require a medium like air, water, or solids to travel through. Sound waves are generated when an object vibrates, creating compressions and rarefactions in the medium. For instance, a vibrating guitar string displaces air molecules, creating pressure waves that travel to our ears, allowing us to perceive sound as different pitches and volumes. The speed and intensity of these waves determine the type of sound we hear.

• Tonal Quality: Related to the arrangement of sound waves and overtones, affecting the "color" of the sound.
• Volume: Determined by the amplitude of the waves; larger amplitude results in a louder sound.
• Pitch: Defined by the frequency of the sound waves; higher frequency leads to a higher pitch sound.

Understanding sound waves is essential for various fields, such as acoustics, music production, and even ultrasound imaging in medicine. Each area utilizes sound in different ways to achieve specific outcomes.

Period and Frequency

In the realm of waves, two fundamental terms that often come up are period and frequency. These properties are crucial for characterizing waves, especially sound waves. The period of a wave refers to the time it takes for one complete cycle or oscillation. It is usually measured in seconds per cycle. On the other hand, frequency is related to how often a wave repeats in a given time frame. It is usually expressed in cycles per second or Hertz (Hz).

• Period (T): In simple terms, it is the time for one full wave cycle. For sound waves, shorter periods mean higher frequencies.
• Frequency (f): This is the number of cycles of the wave passing a point in one second. Higher frequencies result in higher pitches.

Both period and frequency are directly related to how we perceive sound; for example, music notes are identified by their specific frequencies. The understanding of these properties is also crucial in applications such as telecommunications and audio engineering.

Reciprocal Relationship

A fascinating aspect of the relationship between period and frequency is that they are reciprocals of each other. This means that the frequency of a wave can be found by taking the reciprocal of the period and vice versa. Mathematically, this reciprocal relationship is expressed as:\[ f = \frac{1}{T} \]\[ T = \frac{1}{f} \]This reciprocal relationship allows us to easily convert between period and frequency. For example, if you know the frequency of a wave, you can quickly find the period by dividing 1 by the frequency. This relationship is crucial in problem-solving related to wave phenomena, making it easier to switch between these two interconnected properties.

• Example: A sound wave with a frequency of 1,000 Hz has a period of \( \frac{1}{1000} \) seconds or 0.001 seconds.

This concept is applied across various disciplines, from tuning musical instruments to designing electronic circuits.

Equation Formulation

Formulating equations for sound waves involves understanding the relationship between variables such as frequency, period, and time. Using the properties of waves, particular equations can be devised to describe sound behaviors accurately. In the exercise at hand, we see the usage of variables such as \(\theta\) for time and specifics like a frequency value.

Frequency Equation

For sound waves at a defined pitch, frequency is expressed using the equation:\[ f = a \theta \]This equation highlights that frequency impacts the number of cycles per unit time. For instance, with the given highest pitch heard by bats, 120,000 cycles per second, this equation models that sound's frequency.

Period Equation

The period can be represented mathematically as the reciprocal of frequency:\[ T = \frac{1}{f} = \frac{1}{a\theta} \]This equation underscores how periods stretch or shrink inversely with frequency.Such equations are foundational in fields dealing with wave analysis, allowing for calculations related to sound wave propagation, music, and much more. Understanding how to construct and utilize these equations is vital for practical applications in science and engineering.