Estimating Frequency and Identifying Musical Notes Using Wave Equations

Math Problem Statement

The frequencies for the seven “white key” notes produced on the tempered scale of a piano (starting with middle C) are shown in Table 1.4. A computer analyzes the pressure displacement versus time for the wave produced by a tuning fork and gives its equation as y = 1.23 sin (2073.55x - 0.49) + 0.44. (a) Estimate the frequency of the note produced by the tuning fork. (b) Identify the note produced by the tuning fork.

Solution

Part (a) Estimate the frequency of the note produced by the tuning fork

The general equation for a simple harmonic wave is:

$y = A \sin (kx - \omega t + \phi) + C$

Where:

$A$ is the amplitude,
$k$ is the wavenumber,
$\omega$ is the angular frequency,
$t$ is the time,
$\phi$ is the phase shift,
$C$ is the vertical shift.

In the equation given by the computer, ( y = 1

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Math Problem Analysis

Mathematical Concepts

Trigonometry
Wave Analysis
Harmonic Motion
Frequency Calculation

Formulas

General wave equation: y = A sin(kx - ωt + ϕ) + C
Frequency formula: f = ω / (2π)

Theorems

Simple Harmonic Motion Principles

Suitable Grade Level

Grades 10-12

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