How Do Engineers Design AntiNoise to Effectively Reduce Ambient Noise?

[Decentralized]

Summary:: AmbientNoise + AntiNoise combined calculation

I am having trouble with this question:
Noise cancelling headphones use both passive (insulated earphones) and active (electronic “anti-noise”) methods to nullify ambient noise. One task of a sound engineer is to design low-energy anti-noise signals that help cancel ambient noise. Consider anti-noise that is to be combined (to cancel) ambient-noise.

AmbientNoise = 100 sin(ω t) Amplitude 100 and frequency ω .
AntiNoise = A sin(ω t + φ) Amplitude A is a positive number. −π < φ ≤ π

1. Choose AntiNoise so the sum AmbientNoise + AntiNoise has a combined amplitude of 20 (much quieter than AmbientNoise). Guess/choose the phase φ that minimizes A (minimum A decreases hearing fatigue and energy consumption).

AmbientNoise = 100 sin(ω t)
AntiNoise = A sin(ω t + φ)
A = ____ φ = _____rad
CombinedSound = ____sin( ____ )

2. It is difficult for AntiNoise to be perfectly out of phase with AmbientNoise (i.e., difficult for φ to be exactly π). Consider AntiNoise = 100 sin(ω t + π + δ). Determine the maximum δ between 0 and π to create a combined noise/anti-noise sound of amplitude 20, i.e.,
CombinedSound = 100 sin(ω t) + 100 sin(ω t + π + δ) = 20 sin(ω t + SomePhase)
Show δ is governed by the following equation – and solve for δ.

sqr(2 − 2 cos(δ))= 0.2
δ ≈ 0.2 rad ≈ 11.5◦

For the first question, if I want to cancell ambient noise with anti noise down to 20, I am assuming it is going to be -80 sin(ω t)?
But A cannot be a negative number. I am not sure how to approach this problem. We've only learned Asin(x) + Bsin(x) = C sin(x+φ) where C = sqr( A^2 + B^2).

Appreciate your input here.

 

[berkeman]

For the first question, if I want to cancell ambient noise with anti noise down to 20, I am assuming it is going to be -80 sin(ω t)?
But A cannot be a negative number. I am not sure how to approach this problem.

Welcome to PF.

What is the value of $sin(\omega + \pi)$ compared to $sin(\omega)$ ?

 

[Decentralized]

Welcome to PF.

What is the value of $sin(\omega + \pi)$ compared to $sin(\omega)$ ?

Oh, It's going to be $sin(\omega + \pi)$ = $-sin(\omega)$
Should I make it $sin(\omega t + \pi \omega)$ = $-sin(\omega t)$ so that φ = ω π, A = 80?

But if that's the case, φ depends on ω, and in the second question it gives out:
CombinedSound = 100 sin(ω t) + 100 sin(ω t + π + δ) = 20 sin(ω t + SomePhase),
in which A = 100, φ = π+ δ. It is kind of contradict with what I just got. Am I on the right direction so far?

Appreciate your reply!

 

[berkeman]

Correct.

No, just $sin(\omega t + \pi) = -sin(\omega t)$