Problem 54 - A Graphical Approach to Precalculus with Limits

Question

Hearing Different Tones When a musical instrument creates a tone of \(110 \mathrm{Hz}\), it also creates tones at \(220,330\) \(440,550,660, \dots\) Hz. A small speaker cannot reproduce the \(110-\mathrm{Hz}\) vibration, but it can reproduce the higher frequencies, called the upper harmonics. The low tones can still be heard, because the speaker produces difference tones of the upper harmonics. The difference between consecutive frequencies is \(110 \mathrm{Hz}\), and this difference tone will be heard by a listener. We can model this phenomenon with a graphing calculator. (a) In the window \([0,0.03]\) by \([-1,1]\), graph the upper harmonics represented by the pressure $$ \begin{aligned} P=& \frac{1}{2} \sin [2 \pi(220) t]+\frac{1}{3} \sin [2 \pi(330) t] \\ &+\frac{1}{4} \sin [2 \pi(440) t] \end{aligned} $$ (b) Estimate all \(t\) -coordinates where \(P\) is maximum. (c) What does a person hear in addition to the frequencies of \(220,330,\) and \(440 \mathrm{Hz} ?\) (d) Graph the pressure produced by a speaker that can vibrate at \(110 \mathrm{Hz}\) and above in the window \([0,0.03]\) by \([-2,2]\) (Image can't copy)

Step-by-Step Solution

Answer

The listener hears 220, 330, 440 Hz, and the difference tone of 110 Hz.

Step 1: Understand the System of Harmonic Frequencies

When a musical instrument creates a fundamental frequency of \(110\, \text{Hz}\), it also creates harmonics at integer multiples of this frequency: \(220, 330, 440, \ldots\, \text{Hz}\). These are the upper harmonics, and a speaker that cannot produce the fundamental frequency can still emit these frequencies.

Step 2: Write the Equation for Pressure

The pressure produced by the upper harmonics at time \(t\) is given by the equation: \[P = \frac{1}{2} \sin [2 \pi(220) t] + \frac{1}{3} \sin [2 \pi(330) t] + \frac{1}{4} \sin [2 \pi(440) t]\]

Step 3: Graph the Function in a Specific Window

On a graphing calculator, set the window to \([0, 0.03]\) for the \(x\)-axis and \([-1, 1]\) for the \(y\)-axis. Plot the equation for pressure, \(P\). You should observe oscillations, representing the combined effect of the harmonics.

Step 4: Estimate Maximum Points for P

Examine the plotted graph to identify where the function \(P\) attains maximum values. These occur at the points where the sinusoids align constructively. You may use a calculator’s feature to find these maximum points.

Step 5: Calculate or Estimate Contributing Frequencies

A person hears both the harmonics \(220, 330, \text{and } 440\, \text{Hz}\) and the difference tone. The difference tone is the frequency difference between the harmonics, which is \(110\, \text{Hz}\). This is perceived even if not directly generated by the speaker.

Step 6: Graph Pressure with Fundamental Frequency Included

Now, graph the pressure \[P = \sin [2 \pi(110) t] + \frac{1}{2} \sin [2 \pi(220) t] + \frac{1}{3} \sin [2 \pi(330) t] + \frac{1}{4} \sin [2 \pi(440) t]\] in the window \([0, 0.03]\) by \([-2, 2]\). This graph includes the effect of the \(110\, \text{Hz}\) frequency.

Key Concepts

Graphing Calculator

When solving problems involving harmonic frequencies, a graphing calculator can be an invaluable tool. For instance, in the context of modeling sound waves, a graphing calculator allows you to visualize complex sinusoidal functions over time.

By setting the graphing calculator's window correctly, you can view the overlap of different harmonics. In our example, setting the window to

• X-axis: [0, 0.03], representing time in seconds,
• Y-axis: [-1, 1] for the pressure due to the sinusoidal function.

produces a graphical representation of pressure. The peaks and troughs shown on the graph represent points of constructive and destructive interference between harmonics. This visual aid helps in identifying where these interferences occur so they can be analyzed further.

Additionally, modern graphing calculators have features like maximum finding tools, which can be used to estimate where peaks occur. This is crucial for determining maximum pressure points from the sinusoidal output.

Fundamental Frequency

The fundamental frequency is the base frequency from which harmonics are derived. In musical terms, it is the lowest note or pitch that an instrument produces, around which other harmonics are built. For example, if the fundamental frequency produced by an instrument is 110 Hz, the associated harmonics will be multiples of this frequency, such as 220 Hz, 330 Hz, and 440 Hz.

These harmonics play an essential role in enriching the sound, giving musical notes their unique timbre and character. Although a small speaker might not be able to reproduce the 110 Hz fundamental frequency, it can still emit the higher harmonics successfully. The listener, thanks to auditory processes, perceives the overall sound as if the lower tones are present, due to hearing the difference tone between harmonics.

Understanding the fundamental frequency's significance helps in acoustics and audio engineering, where designing speakers and instruments is tailored to exploit both fundamental and harmonic frequencies effectively.

Sinusoidal Functions

Sinusoidal functions are mathematical functions that describe oscillations and waves. They are crucial when dealing with sound waves as they model how the sound pressure varies over time. In terms of the exercise, the sinusoidal equation given by:

\[P = \frac{1}{2} \sin [2 \pi(220) t] + \frac{1}{3} \sin [2 \pi(330) t] + \frac{1}{4} \sin [2 \pi(440) t]\]
represents the pressure variations due to harmonic frequencies.

In this equation:

• The sine functions are standard ways to represent waves mathematically.
• The coefficients (\(\frac{1}{2}\), \(\frac{1}{3}\), \(\frac{1}{4}\)) determine the amplitude of each harmonic wave, reflecting how much each frequency contributes to the overall sound wave.
• The arguments (\(2\pi nt\)) represent the angular frequency of each harmonic, with \(n\) being the harmonic number, which defines the number of complete cycles per second.

By combining multiple sinusoids, you get a complex wave that models the sound comprehensively, capturing the rich texture produced by musical instruments. Understanding sinusoidal functions is essential in signal processing, acoustics, and many areas of physics and engineering.