Problem 71 - Precalculus: Functions and Graphs

Question

Pressure on the eardrum If a tuning fork is struck and then held a certain distance from the eardrum, the pressure \(p_{1}(t)\) on the outside of the eardrum at time \(t\) may be represented by \(p_{1}(t)=A\) sin \(\omega t,\) where \(A\) and \(\omega\) are positive constants. If a second identical tuning fork is struck with a possibly different force and held a different distance from the eardrum (see the figure), its effect may be represented by the equation \(p_{2}(t)=B \sin (\omega t+\tau),\) where \(B\) is a positive constant and \(0 \leq \tau \leq 2 \pi .\) The total pressure \(p(t)\) on the eardrum is given by $$ p(t)=A \sin \omega t+B \sin (\omega t+\tau) $$ (a) Show that \(p(t)=a \cos \omega t+b \sin \omega t,\) where $$ a=B \sin \tau \quad \text { and } \quad b=A+B \cos \tau $$ (b) Show that the amplitude \(C\) of \(p\) is given by $$ C^{2}=A^{2}+B^{2}+2 A B \cos \tau $$ (IMAGE CAN'T COPY)

Step-by-Step Solution

Answer

(a) Use angle addition: expand and rearrange to get given form. (b) Expand and simplify using trigonometric identities to find \(C^2\).

Step 1: Understanding the Problem and the Given Equations

We are given two pressure functions: \( p_1(t) = A \sin \omega t \) and \( p_2(t) = B \sin (\omega t + \tau) \). The total pressure is given by \( p(t) = p_1(t) + p_2(t) = A \sin \omega t + B \sin (\omega t + \tau) \). The goal is to express \( p(t) \) in the form \( p(t) = a \cos \omega t + b \sin \omega t \) and find the amplitude \( C \).

Step 2: Applying the Angle Addition Formula

Use the angle addition formula for sine: \( \sin(\omega t + \tau) = \sin \omega t \cos \tau + \cos \omega t \sin \tau \).By substituting this in the expression for \( p(t) \), we have:\[ p(t) = A \sin \omega t + B (\sin \omega t \cos \tau + \cos \omega t \sin \tau). \]

Step 3: Rearranging the Expression

Substitute from the previous step:\[ p(t) = A \sin \omega t + B \sin \omega t \cos \tau + B \cos \omega t \sin \tau \]Combine like terms:\[ p(t) = (B \sin \tau) \cos \omega t + (A + B \cos \tau) \sin \omega t \].Given: \( a = B \sin \tau \) and \( b = A + B \cos \tau \).

Step 4: Deriving the Amplitude Formula

The amplitude \( C \) is given by the formula for the amplitude of a sine wave: \(C = \sqrt{a^2 + b^2}.\)Substitute for \( a \) and \( b \):\(C^2 = (B \sin \tau)^2 + (A + B \cos \tau)^2.\)Expand the squared term:\( b^2 = (A + B \cos \tau)^2 = A^2 + 2AB \cos \tau + (B \cos \tau)^2 \).Combine terms:\[ C^2 = B^2 \sin^2 \tau + A^2 + 2AB \cos \tau + B^2 \cos^2 \tau \].

Step 5: Simplifying the Amplitude Expression

Recognize that \( \sin^2 \tau + \cos^2 \tau = 1 \), which simplifies the expression:\[ C^2 = B^2 (\sin^2 \tau + \cos^2 \tau) + A^2 + 2AB \cos \tau \].Thus, \(C^2 = B^2 + A^2 + 2AB \cos \tau. \)

Key Concepts

Amplitude in Trigonometric Functions

Amplitude is a crucial concept when dealing with trigonometric functions like sine and cosine waves. Think of amplitude as the height of a wave. It represents the maximum displacement from the horizontal axis. In simpler terms, it's the "loudness" of the wave.

In the context of our problem, the amplitude of the resulting pressure function on the eardrum is given by the formula:

• \( C^2 = A^2 + B^2 + 2AB \cos \tau \)

This equation helps us find the total amplitude \(C\), which determines how the combined effect of both tuning forks behaves. You square the total amplitude to derive it from the given functions.

The key components at play are:

• \(A\) and \(B\) are the original amplitudes from the two tuning forks.
• \(\tau\) is the phase shift between the two sources.
• \(C\) signifies the total amplitude of the resultant wave.

Recognizing amplitude helps visualize how the peak pressure on the eardrum shifts when multiple waves interact.

Angle Addition Formula

The angle addition formula is a tool used to break down the sine or cosine of a sum of angles into a manageable form. This is particularly useful in our problem when working with the pressure functions.

For sine, the formula is:

• \( \sin(\omega t + \tau) = \sin \omega t \cos \tau + \cos \omega t \sin \tau \)

This decomposition enables us to separate components that involve \(\cos\) and \(\sin\) separately, which simplifies combining terms further down the line.

In our problem:

• The function \( p_2(t) = B \sin(\omega t + \tau) \) is expanded using the angle addition formula.
• This allows us to combine like terms in \( p(t) = A \sin \omega t + B \sin(\omega t + \tau) \) by separately considering the contributions from \( \cos \omega t \) and \( \sin \omega t \).

By understanding this formula, students can approach complex trigonometric expressions more systematically.

Understanding Sine and Cosine Waves

Sine and cosine waves are the fundamental building blocks of trigonometry. Appearing as smooth, periodic curves, they model a wide range of natural phenomena. Here are some basics to keep in mind when analyzing these waves:

• **Sine Wave:** Begins at the origin and oscillates up and down symmetrically.
• **Cosine Wave:** Starts at its maximum value and follows a similar oscillatory pattern.
• **Period:** The distance along the horizontal axis before the wave repeats itself. For sine and cosine, the period is typically \(2\pi\).
• **Phase Shift:** Indicates how far the wave shifts horizontally from a standard position.

In the problem:

• The pressure functions \(p_1(t)\) and \(p_2(t)\) represent sine waves, altered by factors \(A\), \(B\), and \(\tau\).
• The combination of these waves leads to a new resultant wave, determined by considering both amplitude and phase shift.

Understanding how these waves behave and interact helps demystify their real-world applications and mathematical beauty.