Question

Noise-cancelling headphones are designed to give you maximum listening pleasure by cancelling ambient noise and actively creating their own sound waves. These waves mimic the incoming noise in every way, except that they are out of sync with the intruding noise by \(180^{\circ}\). Suppose that the amplitude and period for the sine waves created by the outside noise are 4 and \(\frac{\pi}{2},\) respectively. Determine the equation of the sound waves the headphones produce to effectively cancel the ambient noise.

Short answer

The equation is \( y = -4 \sin(4x) \).

Step-by-step solution

Determine the equation for the ambient noise

The general form of the equation for a sine wave is \[ y = A \sin(Bx + C) + D \]. Given that the amplitude (A) is 4 and the period is \( \frac{\pi}{2} \), we can use the formula for the period of a sine function, \( \frac{2\pi}{|B|} \). Solving for B, we get \( B = 4 \). Therefore, the equation for the ambient noise can be written as \[ y = 4 \sin(4x) \].

Adjust for the phase shift

To cancel out the noise, the headphones need to produce a sound wave that is out of phase by \(180^{\circ} \), or \( \pi \) radians. The phase shift can be included in the sine function by adding \( \pi \) inside the sine function. Therefore, the equation becomes \[ y = 4 \sin(4x + \pi) \].

Simplify using sine properties

Using the trigonometric identity \( \sin(\theta + \pi) = -\sin(\theta) \), we can simplify the equation. Thus, \[ 4 \sin(4x + \pi) = 4(-\sin(4x)) = -4 \sin(4x) \].

Final equation for the headphones

The sound wave produced by the headphones to cancel the ambient noise is \[ y = -4 \sin(4x) \].

Key concepts

Sine Wave Equation

A sine wave is a mathematical curve that describes a smooth, periodic oscillation. It can be used to model various periodic phenomena, such as sound waves. The general form of the sine wave equation is \(y = A \sin(Bx + C) + D\). In this equation:

• \(A\) represents the amplitude, which is the peak value of the wave.
• \(B\) affects the period of the wave. The period is the length of one complete cycle.
• \(C\) represents the phase shift, which moves the wave left or right.
• \(D\) is the vertical shift, moving the wave up or down.

For example, in the noise-cancelling headphones problem, the ambient noise was modeled as \(y = 4 \sin(4x)\), where:

• The amplitude \(A\) is 4.
• The period is determined by \(B = 4\)
• There's no phase shift \(C = 0\).
• There's no vertical shift \(D = 0\).

Phase Shift

Phase shift refers to the horizontal movement of a wave left or right along the x-axis. For a sine wave \(y = A \sin(Bx + C) + D\), the phase shift is determined by the value of \(C\). A positive \(C\) shifts the wave left, and a negative \(C\) shifts it right.
In the noise-cancelling headphones, the goal was to create a wave exactly out of phase with the incoming noise. Being out of phase by 180° means adding \(\pi\) radians to the sine function: \(y = 4 \sin(4x + \pi)\).
This phase shift ensures that the peaks of one wave align with the troughs of the other, effectively canceling the noise.
We can further simplify using the identity \ \sin(\theta + \pi) = -\sin(\theta) \, turning \(4 \sin(4x + \pi)\) into \(-4 \sin(4x)\).

Trigonometric Identity

Trigonometric identities are equations involving trigonometric functions that are true for all values of the included variable. One key identity used in the noise-cancelling problem is \( \sin(\theta + \pi) = -\sin(\theta) \). This identity tells us that adding 180° (or \(\pi\) radians) to any angle \(\theta\) in a sine function results in the negative of the original sine of that angle.
In practice, for the problem at hand: \(4 \sin(4x + \pi) = -4 \sin(4x)\).
This simplifies the computation and helps in understanding how the headphones' sound wave counteracts the ambient noise by being perfectly out of phase.

Amplitude and Period

Amplitude and period are two important characteristics of sine waves. The amplitude \(A\) is the maximum distance from the wave's central axis to its peak. In our problem, the amplitude of the ambient noise's sine wave is 4, implying the variation between the maximum and minimum points of the wave.
The period of a wave is the length of one complete cycle, calculated as \( \frac{2\pi}{|B|} \). In our example, with \(B = 4\), the period is \( \frac{2\pi}{4} = \frac{\pi}{2} \).
Understanding these characteristics helps in accurately modeling and then cancelling the noise with an inverse wave produced by the headphones.