Problem 101 - Precalculus

Question

If a sound wave is represented by \(y=0.008 \sin (750 \pi t) \mathrm{cm},\) what are its amplitude and frequency? See Exercise 99.

Step-by-Step Solution

Answer

Amplitude is 0.008 cm; frequency is 375 Hz.

Step 1: Identify the Amplitude

The amplitude of a sine wave in the equation \( y = A \sin(Bt) \) is the coefficient \( A \). For the given sound wave equation \( y = 0.008 \sin(750 \pi t) \), the amplitude is \( 0.008 \) cm.

Step 2: Determine the Frequency

In the general form \( y = A \sin(Bt) \), the frequency of the wave can be calculated from \( B \). The formula for frequency \( f \) is \( f = \frac{B}{2\pi} \). Here, \( B = 750\pi \), so the frequency is \( f = \frac{750\pi}{2\pi} = 375 \). Therefore, the frequency is 375 Hz.

Key Concepts

Amplitude

Amplitude in sound waves refers to the height of the wave and is a measure of how much energy or loudness the wave possesses. In the context of a sine wave represented by an equation like \(y = A \sin(Bt)\), the amplitude is denoted as \(A\). This value reflects how far the wave's peak deviates from the horizontal axis.

• In the equation \(y = 0.008 \sin(750 \pi t)\), the amplitude is \(0.008\) cm.
• This small value indicates the wave isn't very loud, as there's less energy.

Amplitude is essential in understanding sound volume. Higher amplitude means a louder sound, while lower amplitude represents a quieter one. It's important to remember that amplitude influences, but does not directly dictate, the pitch of a sound. Instead, it reflects the intensity.

Frequency

Frequency describes how many times a wave oscillates or completes a cycle per second. Measured in hertz (Hz), it's vital for understanding the pitch of the sound. In sine wave equations like \( y = A \sin(Bt) \), frequency is extracted using the coefficient \(B\) and the formula \( f = \frac{B}{2\pi} \).

• For the wave equation \( y = 0.008 \sin(750 \pi t) \), \(B\) is \(750\pi\).
• Plug \(B\) into the formula to find the frequency: \( f = \frac{750\pi}{2\pi} = 375 \) Hz.

A frequency of 375 Hz means the sound wave vibrates 375 times per second. The higher the frequency, the higher the perceived pitch. Musicians often use frequency to tune instruments and ensure they produce the correct pitch. Remember, frequency gives us the pitch, whereas amplitude affects the loudness.

Sine Wave

A sine wave is a mathematic curve that describes smooth periodic oscillations. As one of the purest forms of a wave, it's characterized by its consistent and predictable shape. In sound, sine waves are fundamental because they illustrate a singular pure tone. The equation form of a sine wave is \( y = A \sin(Bt) \).

• The \(A\) value represents amplitude, the height from the center line to the peak.
• The \(B\) value connects to the wave's frequency, determining how quickly it oscillates.

Sine waves are pivotal in sound engineering and acoustics since they form the building blocks of complex waves. By combining multiple sine waves of various frequencies and amplitudes, we create the diverse sounds heard in music and everyday life. Understanding sine waves allows us to break down and synthesize sound in technology and musical applications.